This file has been bundled using the "shar" program. Instructions for unbundling it follow. Once unbundled it will produce the files: defs.tex driver.tex fig1.tex fig2.tex head.tex intro.tex refs.tex sec3.tex sec4.tex sec5.tex title.tex The completed preprint can then be recovered by typing "tex driver". The preprint was prepared with plain TeX, version 3.1. The figures in this manuscript were prepared with the "pictex" program. If that is not available on your machine, remove the lines \input fig1.tex \input fig2.tex from the file driver.tex, and you can produce the preprint (less the figures) as above. The figures can be obtained by sending e-mail to "wayne@math.psu.edu". ######################################################## #! /bin/sh # This is a shell archive. Remove anything before this line, then unpack # it by saving it into a file and typing "sh file". To overwrite existing # files, type "sh file -c". 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X\fi\fi X \if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE, X\else\period \fi\fi X \if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi X \if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if XT\isyrJPE \ \else\period \fi\fi\fi X \if T\istoappearJPE (To appear)\period \fi X \if T\ispreprintJPE Preprint\period \fi X \if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi X \if T\isvolJPE \unhbox\volboxJPE\unskip, \fi X \if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi X \if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi X \if T\issecondpaperJPE \hfill\break\unhbox\secondpaperboxJPE\unskip\period \fi X \if T\issecondjourJPE \unhbox\secondjourboxJPE\unskip\ \fi X \if T\issecondvolJPE \unhbox\secondvolboxJPE\unskip, \fi X \if T\issecondpagesJPE \unhbox\secondpagesboxJPE\unskip\ \fi X \if T\issecondyrJPE (\unhbox\secondyrboxJPE\unskip)\period \fi X X X} X X%%%%%%%%%%%%%%%%%%%%%%%%%% X X END_OF_FILE if test 9659 -ne `wc -c <'head.tex'`; then echo shar: \"'head.tex'\" unpacked with wrong size! fi # end of 'head.tex' fi if test -f 'intro.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'intro.tex'\" else echo shar: Extracting \"'intro.tex'\" \(9364 characters\) sed "s/^X//" >'intro.tex' <<'END_OF_FILE' X X X X X X\SECTION Introduction: X XIn this paper we prove the existence of periodic Xsolutions of nonlinear wave equations of the form X$$ X \partial_t^2 u = \partial_x^2 u - g(x,u)~, X\EQ(origeqn) X$$ Xfor nonlinearities $g(x,u)$ which satisfy certain Xconditions of nonresonance and genuine nonlinearity. XWe assume that one has either periodic or Dirichlet Xboundary conditions at the ends of the interval X$[0,\pi]$. This problem is called the free vibration Xproblem for the nonlinear string and has been extensively studied. XThe review [B], contains over 60 references, and in the Xeight years since it was written the number has increased Xmuch further. In this introduction we will try to describe Xin very general terms the methods we use to study this Xproblem and how these methods relate to, and differ Xfrom, previous approaches to this problem. X XThe first real breakthrough on this problem was due to XRabinowitz [R]. He rephrased the problem as a variational Xproblem and was able to prove that under appropriate Xassumptions on the non-linearity $g(x,u)$ one had Xperiodic solutions whenever the time period was Xa rational multiple of $\pi$. Many authors have Xused Rabinowitz's variational methods to obtain related Xresults. None, however, Xwere able to circumvent the restriction on the period. XIn addition, these variational techniques have not yet Xshed light on the existence of quasi-periodic solutions Xof \equ(origeqn). On the other hand, the variational Xtechniques are global, and they place few Xrestrictions on the strength of the non-linear term. X XMore recently, a quite different approach which uses the XKolmogorov, Arnold, Moser (KAM) theory has been developed Xby Kuksin [K] and Wayne [W]. This approach uses the fact Xthat \equ(origeqn) is a hamiltonian system and modifies Xthe classical KAM ideas to work in this infinite dimensional Xcontext. This has two advantages--first, it allows one to Xconstruct solutions whose periods are irrational multiples of X$\pi$ and second, it easily extends to give quasi-periodic Xas well as periodic solutions. A disadvantage is Xthat since the KAM theory has an essentially Xperturbative character, it is restricted to equations with Xweak non-linearity, or equivalently, to solutions of Xsmall norm. X XIn this paper we propose yet a third approach. Our method Xis reminiscent of the Lyapunov-Schmidt method of classical Xbifurcation theory in that we split the problem into Xtwo pieces, one of which is finite dimensional and corresponds Xto the null space of the linearized operator, and the other Xpiece infinite dimensional. In contrast to the XLyapunov-Schmidt method in our problem the linearization of the Xinfinite dimensional piece does not have bounded inverse. XIndeed, it's failure to have bounded inverse is related to Xthe restriction in the work of Rabinowitz and others to Xsolutions whose period is a rational multiple of $\pi$. X XOur method is perturbative and begins by expanding Xthe nonlinear term in \equ(origeqn) as $g(x,u) = Xg_1(x) u + g_2(x) u^2 + \dots$. If one ignores terms of X$\OO(u^2)$ or higher, \equ(origeqn) becomes a linear equation Xwhose solutions can be explicitly computed in terms of the eigenvectors, X$\{ \psi_j(x) \}$, and eigenvalues, $\{ \omega_j^2 \}$, of the XSturm-Liouville operator $L = (-{{d^2}\over{dx^2}} + g_1(x) )$. XWe then make the {\it Ansatz} that a periodic solution of \equ(origeqn) Xwith angular frequency $\Omega$ exists and write it as X$$ Xu(x,t) = \sum_{j,k} \hat{u}(j,k) \psi_j(x) e^{i k \Omega t}~~. X\EQ(ex) X$$ XSubstituting \equ(ex) into \equ(origeqn) gives an infinite system Xof nonlinear algebraic equations which the coefficients X$\{ \hat{u} (j,k) \}$ must solve. We construct solutions of this Xsystem of equations using Newton's method taking as our initial Xapproximation to the solution linear combinations Xof Kronecker $\delta$-functions at the lattice sites X$(j,k) = (1, \pm 1)$. This initial guess corresponds to a periodic Xsolution of the linearized equation with angular frequency X$\omega_1$. We are able to prove that our interative scheme Xconverges to a solution of the equations for $\{ \hat{u} (j,k) \}$, Xwhich, Xwhen substituted into \equ(ex) gives a periodic solution of \equ(origeqn). XThis construction yields a family of periodic orbits with frequencies Xin a Cantor set of positive measure. One interesting point Xis that the linear operator which must be inverted in order Xto apply Newton's method is closely related to the lattice XSchr\"odinger operators studied by Fr\"ohlich and Spencer [FS] Xin their work on Xlocalization theory. From a technical point Xof view, this connection between localization theory and Xdynamical systems strikes us as one of the more interesting Xaspects of the present approach. Similar ideas, with Xapplications to partial differential-difference equations have Xbeen previously developed in the work of Albanese, Fr\"ohlich Xand Spencer [AFS]. X XOur method, like the KAM method is perturbative, and Xis restricted to the study of Xsolutions of small norm. However, it differs from the XKAM method in several ways. XFirst of all, Xthe present theory is not a transformation theory--we do Xnot proceed by making a sequence of canonical transformations, Xnor do we transform the system to some normal form. XSecondly, the present Xmethod makes no direct use of the hamiltonian nature of the Xproblem. Thus, we hope it will be applicable Xto non-hamiltonian problems. X XThe existence theorem does not apply to all choices of Xnonlinearity $g$, it requires that certain conditions of Xlinear nonresonance and genuine nonlinearity are satisfied. XThese are analogs of well known difficulties in the theory Xof dynamical systems. These conditions depend only upon the X{\it 3-jet} of $g$, and are finite in number. The set of Xnonlinearities which satisfy them is generic, indeed it Xis open and dense, and is described more precisely in section 6. XThese conditions can in principle be checked in explicit Xcases. For the classical examples of the nonlinear XKlein Gordon and the sine Gordon equations, the Xnonlinearity depends upon one parameter, and we Xshow that, for an open set of full measure of this Xparameter, the conditions are satisfied and the Xexistence theorem applies. X X XThinking of the analogy with the Lyapunov center theorem Xfor finite dimensional hamiltonian systems in the neighborhood Xof an elliptic equilibrium point, one expects to obtain Xa family of periodic orbits bifurcating from the orbit of the Xlinearized equations whenever no integer multiple of the frequency Xof the solution being perturbed coincides with the frequency Xof any other normal mode. XThere is a significant Xdifference when the problem has infinitely Xmany degrees of freedom. In the finite dimensional Xcase there are smooth families of periodic solutions bifurcating Xfrom the solution of the linear problem. In the case of the Xwave equation we will construct a smooth curve bifurcating from the Xsolution of the linear equation. However we cannot prove Xthat all points on this curve give rise to solutions of the Xwave equation--only that there is a (Cantor) set of frequencies Xof positive measure such that for any point on the curve whose Xfrequency lies in this Cantor set one has a solution. XThis difference arises from the fact that if there are Xonly finitely many frequencies $\omega_1, \omega_2, X\omega_3, \dots, \omega_N$, and if $m\omega_1 \ne X\omega_j$, for $j=2,3,\dots,N$, then $m\Omega X\ne \omega_j$, for all $\Omega$ in some interval surrounding X$\omega_1$. Thus, one typically obtains solutions for the Xnonlinear problem for an Xinterval of frequencies. For infinitely many frequencies, Xhowever, even if $m \omega_1 \ne \omega_j$, for X$j = 2, 3, \dots$, there will in general be a dense Xset of $\Omega$'s for which $m \Omega = \omega_j$, for some X$j$. It is the process of excising these resonant Xfrequencies which gives rise to the Cantor set on which the Xconstruction of solutions is successful. X XAs remarked above, the KAM approach to this problem also Xyields the existence of quasi-periodic solutions for equations Xlike \equ(origeqn). We believe that an extension of the Xpresent method will also yield quasi-periodic solutions, and plan Xfurther work on this problem. We remark that Xsuch results would be of interest even for finite dimensional Xsystems since they would give a proof of the existence Xof invariant tori for hamiltonian systems near an Xelliptic equilibrium point whose dimension is less than Xor equal to Xthe number of degrees of freedom of the systems, different Xfrom that of [E], or [P]. X XWe conclude with an outline of the remainder of the paper. XIn the next section we state our principle results, Xtransform the wave equation to a problem on a two-dimensional Xlattice, and apply our results to discuss Xtwo well known examples, the Klein-Gordon and sine-Gordon Xequations. In Section 3 we state the induction hypotheses Xwhich allow us to derive the results in Section 2. Section X4 contains the verification of these Xinduction hypotheses, while Section 5 explains Xhow to control the inverse of the linearized operator which Xarises in Newton's method. This analysis is connected Xwith the theory Xof localization in Schr\"odinger operators. Finally, in XSection 6 we derive some estimates we need which link the wave Xequation to the lattice problem, and discuss the issues of Xgenericity of the nonlinearity. X X END_OF_FILE if test 9364 -ne `wc -c <'intro.tex'`; then echo shar: \"'intro.tex'\" unpacked with wrong size! fi # end of 'intro.tex' fi if test -f 'refs.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'refs.tex'\" else echo shar: Extracting \"'refs.tex'\" \(3134 characters\) sed "s/^X//" >'refs.tex' <<'END_OF_FILE' X X X X\SECTIONNONR References X X X X\ref X \no AFS X \by Albanese, C. and Fr\"ohlich, J. and X Albanese, C. Fr\"ohlich, J. and Spencer, T. X \paper Periodic solutions of some infinite-dimensional X hamiltonian systems associated with non-linear X partial difference equations: Parts I and II X \jour Commun. Math. Phys. X \vol 116 X \pages 475-502 X \secondvol 119 X \secondpages 677-699 X \secondyr 1988 X\endref X X\ref X \no B X \by Brezis, H. X \paper Periodic solutions of nonlinear vibrating X strings and duality principles X \jour Bull. AMS X \vol 8 X \pages 409-426 X \yr 1983 X\endref X X\ref X \no C X \by Chierchia, L. X \paper A direct method for constructing solutions of X the Hamiltonian-Jacobi equation X \preprint X \yr November 1989 X\endref X X\ref X \no CW X \paper Nonlinear Waves and the KAM Theorem: Nonlinear X Degeneracies X \jour To appear in the {\bf Proceedings of the Conference X on Nonlinear Waves}, Villefranche France X \yr January 1991 X\endref X X X X\ref X \no E X \by Eliasson, H. X \paper Perturbations of stable invariant tori X \jour Ann. Sc. Super. Pisa, Cl. Sci. X \vol IV Ser. 15 X \pages 115-147 X \yr 1988 X\endref X X\ref X \no FS X \by Fr\"ohlich, J. and Spencer, T. X \paper Absence of diffusion in the Anderson tight binding X model for large disorder or low energy X \jour Commun. Math. Phys. X \vol 88 X \pages 151-184 X \yr 1983 X\endref X X\ref X \no Ka X \by Kato, T. X \book Perturbation Theory for Linear Operators; 2nd ed. X \publisher Springer Verlag; Berlin X \yr 1976 X\endref X X\ref X \no KT X \by Keller, J. and Ting, L. X \paper Periodic vibrations of systems governed by X non-linear partial differential equations X \jour Commun. Pure Appl. Math. X \vol 19 X \pages 371-420 X \yr 1966 X\endref X X\ref X \no K X \by Kuksin, S. X \paper Perturbation of quasiperiodic solutions of X infinite-dimensional linear systems with an X imaginary spectrum X \jour Funct. Anal. Appl. X \vol 21 X \pages 192-205 X \yr 1987 X\secondpaper Perturbation theroy for quasiperiodic solutions X of infinite-dimensional hamiltonian systems; Parts I-III X \secondjour Preprint of Max-Plank-Institut, Bonn X \secondyr 1990 X\endref X X X\ref X \no P X \by P\"oschel, J. X \paper On elliptic lower dimensional tori in X hamiltonian systems X \jour Math. Z. X \vol 202 X \pages 559-608 X \yr 1989 X\endref X X\ref X \no P2 X \by P\"oschel, J. X \paper On Fr\"ohlich Spencer estimates of Green's X function X \jour Manuscripta Math. X \vol 70 X \pages 27-37 X \yr 1990 X\endref X X\ref X \no PT X \by P\"oschel, J. and Trubowitz, E. X \book Inverse Spectral Theory X \publisher Academic Press; Boston, MA X \yr 1987 X\endref X X\ref X \no R X \by Rabinowitz, P. X \paper Free vibrations for a semilinear wave equation X \jour Commun. Pure Appl. Math. X \vol 30 X \pages 31-68 X \yr 1977 X\endref X X\ref X \no Re X \by Rellich, F. X \book Perturation Theory for Eigenvalue Problems X \publisher Gordon and Breach; New York X \yr 1969 X\endref X X\ref X \no W X \by Wayne, C. E. X \paper Periodic and quasi-periodic solutions X of nonlinear wave equations via KAM theory X \jour Commun. Math. Phys. X \vol 127 X \pages 479-528 X \yr 1990 X\endref X X END_OF_FILE if test 3134 -ne `wc -c <'refs.tex'`; then echo shar: \"'refs.tex'\" unpacked with wrong size! fi # end of 'refs.tex' fi if test -f 'sec2.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sec2.tex'\" else echo shar: Extracting \"'sec2.tex'\" \(29844 characters\) sed "s/^X//" >'sec2.tex' <<'END_OF_FILE' X X X X\SECTION Results X XWe present results for nonlinear wave equations, which Xare obtained by a reduction of the problem to one for nonlinear Xequations on lattices. The reduction to the lattice equations Xand the corresponding existence results are also given in this section. XSolutions are obtained by an induction procedure based on the XNash Moser method which is described in Sections 3 and 4. X X\SUBSECTION Nonlinear wave equations X XThe equation we study in this paper is the nonlinear Xwave equation on a bounded interval X$0 \leq x \leq \pi$ in one space dimension. X$$ X \partial^2_t u = \partial^2_x u - g(x,u) X\EQ(NLW) X$$ XWe will assume that the nonlinear term is Xanalytic in both variables in the region X$\{ (x,u); |{\rm Im} \ x| < \overline \gs \}$, Xperiodic in $x$ with period $\pi$, and with Taylor Xexpansion in $u$, X$$ X g(x,u) = g_1(x)u + g_2(x)u^2 + g_3(x)u^3 + \cdots X\EQ(NL term) X$$ XWe are seeking solutions which are periodic in time, with period X$2\pi / \Omega$, Xwhich satisfy certain self adjoint boundary Xconditions on a spatial interval. XIf we linearize about the solution $u \equiv 0$ we obtain X$$ X \partial^2_t v = \partial^2_x v -g_1(x) v. X\EQ(LW) X$$ XTwo examples of boundary conditions that we will address are XDirichlet conditions, $u(0,t) = 0 = u(\pi,t)$, and periodic Xconditions $u(x+\pi,t) = u(x,t)$. XSolutions of the linear equation \equ(LW) are given by Xseparation of variables and an eigenfunction expansion. XLet $\{ \psi_j(x) \}_{j=1}^\infty $ be normalized eigenfunctions Xfor the the linear differential operator X$$ X L(g_1) \psi = (-{d^2 \over dx^2} + g_1(x)) \psi X\EQ(LOp) X$$ Xwith the proper boundary conditions, with eigenvalues X$\{ \omega^2_j \}_{j=1}^\infty$. X(For periodic boundary conditions, it is more convenient Xto begin labeling the eigenvalues and eigenfunctions Xwith $j=0$--this causes no essential difference.) XSolutions of \equ(LW) are given by X$$ X v(x,t) = \sum_{j=1}^\infty r_j X \cos(\omega_j t + \xi_j) \psi_j(x) X\EQ(LSoln) X$$ Xwhich are parametrized by the amplitudes $r_j$ and the phases X$\xi_j$. XFor real $\omega_j$ each function X$\cos(\omega_j t + \xi) \psi_j(x)$ Xis time periodic, with frequency $\omega_j$. XA more general solution to the linear equation \equ(LW) is time Xperiodic only if for all nonzero amplitudes X$r_j$ there exist a full set of rational Xrelations between the associated frequencies $\omega_j$; Xthat is, there exists $\omega$ and integers $k_j$ such that Xfor all $j$ with $r_j \not= 0$, $\omega_j = k_j \omega$ . XUnless a full set of resonance conditions are satisfied the Xgeneral solution is quasiperiodic or almost periodic. XWe seek periodic solutions to the full nonlinear problem X\equ(NLW) near the linear solutions X$r \cos(\omega_j t + \xi) \psi_j(x)$, with frequency near the real Xlinear frequency $\omega_j$. In this process any coincidence or Xnear coincidence of linear frequencies causes resonance and other Xphenomena related to small divisors in the full nonlinear Xproblem. XHowever the following results demonstrate that for most Xnonlinearities $g(x,u)$ an iterative construction can overcome Xthese difficulties, to prove the existence of periodic solutions Xto \equ(NLW) of small amplitude. X X X\CLAIM Theorem (NLWperiodic) Consider equation \equ(NLW) with Xperiodic boundary conditions on the interval $0 \leq x \leq \pi$. XFor an open dense set of nonlinear terms $g(x,u)$ there exist Xtime periodic real analytic solutions. More precisely there exist Xparameters $(\Omega, r, \xi)$ and solutions $u(x,t;r)$ such that X$$\eqalign{ X |u(x,t;r) - &r \cos(\Omega t + \xi) X \psi_j(x)| < Cr^2 \cr X &|\Omega - \omega_j| < Cr^2 \cr } X\EQ(2.3) X$$ Xfor a set of $r$ of positive measure. X XIn section 6 the topology of the class of nonlinear Xequations will be discussed, and a more precise statement Xregarding the open dense set of nonresonant, Xgenuinely nonlinear terms will be described. XA similar statement holds for solutions of Xthe nonlinear wave equation satisfying XDirichlet boundary conditions. X XWe note that not only will the set of amplitudes $r$ Xof the solutions we construct have positive measure, Xbut the set of frequencies of the periodic solutions Xwill also have positive measure. Thus, we are assured Xof having solutions of irrational period. X X X\CLAIM Theorem (NLWDir) Consider Dirichlet boundary Xconditions for equation \equ(NLW). XAdditionally ask that $g(x,u) = -g(-x,-u)$. XAmong this class of nonlinearities there is Xan open dense set Xsuch that there exist time periodic solutions Xof \equ(NLW). That is, there exist parameters $(\Omega,r,\xi)$ Xand solutions $u(x,t;r)$ satisfying X$$\eqalign{ X |u(x,t;r) - &r \cos(\Omega t + \xi) \psi_j(x)| < Cr^2 \cr X &|\Omega - \omega_j| < Cr^2 \cr } X\EQ(2.4) X$$ Xfor a set of $r$ of positive measure. X X\noindent X{\bf Remarks:} The conditions on the nonlinear Xterm $g(x,u)$ in X\clm(NLWperiodic) and \clm(NLWDir) are quite explicit. XThey depend only Xupon the coefficients $g_1(x), \, g_2(x), \, g_3(x)$, Xin other words only upon the $3-jet$ of $g$. XRoughly, there is a condition on $g_1$ in order Xto avoid certain primary resonances in the Xlinear equation, and a condition of genuine nonlinearity Xplaced upon $g_1, \ g_2$ and $g_3$. XBoth are open conditions, excluding sets Xwhich are essentially of codimension $1$. The Xprecise nature of the good set Xwill be described in more detail below. XUnfortunately the Xcase $g_1(x) = 0$ is too resonant for the present methods to Xhandle, and is in the excluded set. XOn the other hand for an open set of constants X$m^2$ of full Lebesgue measure the case X$g_1(x) = m^2$ is included Xin the conditions of the theorems, thus Xthe nonlinear Klein Gordon-equation Xand the sine-Gordon equations, and Xnonlinear perturbations of them, Xare covered by our results. XIt is possible to prove similar existence Xtheorems for other boundary Xconditions as well; these should be self adjoint as well as Xsatisfying other conditions on the Xeigenfunction expansion and Xthe description of the nonlinearity. XThe precise conditions that are required Xare discussed in Section 6. X X X XThe inverse of the linearized operator in \equ(LW) plays Xa role in the existence results. XWhen applied to time-periodic Xfunctions with frequency $\BGO$ the point spectrum of the Xoperator is X$\{ \omega_j^2 - \BGO^2 k^2; 1 \leq j < \infty, \ X -\infty < k < \infty \}$. XFor most choices of $\BGO$ and coefficient $g_1(x)$ Xthis is a dense set in ${\bf R}$. In particular, spectrum Xwill accumulate at zero, a phenomenon which is Xoften called the small divisor problem. XIt is in this case that the results \clm(NLWperiodic) and X\clm(NLWDir) are most interesting. XBecause of the small divisors, Xthe method of solution is of the Nash Moser type, Xalternating a Newton iteration with approximate Xinversion of the linearized operator. XBoth theorems above follow from a more general Xresult in the form Xof a Nash-Moser type theorem for nonlinear equations Xposed on the lattice $\zsquared$. XTwo dimensional lattices are not special, Xand the theorem is easily generalized; Xin our situation two lattice directions suffice to Xindex the temporal and spatial eigenfunctions Xused to describe the above Xproblems in nonlinear waves. X XThe lattice problem arises by expanding solutions X$u(x,t)$ of \equ(NLW) in eigenfunction-Fourier Xseries. The coefficients $U(i,j)$ in this expansion must Xsatisfy nonlinear equations on the lattice of the form X$$ X W(U) + V(\Omega)U =0~~. X\EQ(NLlattice) X$$ XDenoting lattice sites $ x = (j,k) \in \zsquared $ Xand $\Omega \in \real$ a frequency parameter, the form of X$V(\Omega)$ is a diagonal linear operator on sequences $U(x)$, X$$ X V(\Omega)(x,y) = (\omega_j^2 - \Omega^2 k^2) \delta(x,y). X\EQ(VOp) X$$ Xwhere $\delta(x,y)$ is the Kronnecker delta. XThe sequence of frequencies X$\{ \omega_j \}_{j=1}^\infty$ satisfies X$$ X \qquad |\omega_j - j| < C_g, X$$ Xwhich is the case for the eigenvalues Xof the linear operator \equ(LOp). XLet $\HH_\gs = \{ U(x) \in \l2(\zsquared) ; \sum_{x \in \zsquared} Xe^{2\gs|x|}|U(x)|^2 < \infty \}$, Xa Hilbert space of sequences. XFor $\gs < \overline \gs$ the nonlinear term X$W(U)$ is a real analytic mapping from X$\HH_\gs \rightarrow \HH_{\gs-\ga}$, Xfor any $0 < \ga \le \gs$. XWe ask that $W(0)=0, \, D_U W(0)=0$, Xand furthermore that X$W(U)$ satisfy certain natural conditions of genuine Xnonlinearity, best explained below. XFor the nonlinear wave equation the Xconstant $\overline \gs$ is determined Xby the analyticity properties of the term $g(x,u)$. X XAgain we are led to linearize the nonlinear problem X\equ(NLlattice) about the solution $U(x)=0$, obtaining X$$ X V(\Omega) \varphi = 0. X$$ XSolutions of this are simply $\varphi(x) = \gd_y(x)$, Xwith $y = (j,k)$, and $\Omega = (\omega_j / k)$. XEach nonzero eigenspace of $V(\Omega)$ Xis at least two dimensional, Xfrom the form of $V(\Omega) = (\omega_j^2-\Omega^2 k^2)$, Xwhich is spanned by the vectors in X$\HH_{\gs}$ supported on the lattice sites X$y = (j,k)$ and $\overline y = (j,-k)$. X XThe nonlinear existence theorem focuses on Xsolutions near the linearized solution space, Xwith frequency near the value X$\Omega = \omega_j$ of the linearized problem. XIn fact with little loss Xof generality we will assume that $\omega_1$ Xis real and focus on a neighborhood of X$ \Omega = \omega_1$, Xwith solutions supported near $y=(1,1)$ and X$\overline y = (1,-1)$. X X X X X%%%%%%%%%%%%% XCentral to the construction is the Xinversion of the linearized Xoperator $H(U) = V(\Omega) + DW(U)$ of X, about an approximate solution $U$. XThis involves an analysis of the small Xeigenvalues of $H(U)$. XThese are connected with the geometry Xof the lattice points $x$ at Xwhich is close to zero. X%%%%%%%%%%%%%%%%%%%%% X XA major part of this paper is the analysis of the Xlinearized operator $H(U) = V(\Omega) + DW(U)$ Xof \equ(NLlattice). XWe assume that $DW(u)$ is selfadjoint, which will Xbe the case for the lattice problems which come from the Xnonlinear wave equation. By an analogy with quantum mechanics Xwe call $H(U)$ a {\bf Hamiltonian operator}, and the Xmatrix of the inverse operator $G(U)(z) = (H(U) - z\11)^{-1}$ Xthe {\bf Green's function}. Central to the construction Xis the approximate inversion of $H(U)$ about an approximate Xsolution $U$. This involves an analysis of the small Xeigenvalues of $H(U)$, and the geometry of the lattice Xsites $x$ at which $V(\Omega)(x,x)$ is close to zero. XWe define a {\bf singular site} Xto be a lattice point $x=(j,k)$ at which X$|V(\Omega)(x,x)| = |\omega_j^2 - \Omega^2 k^2| < d_s$, Xwhere $d_s$ is a small parameter which is specified Xin the next section. XAny connected set of singular sites will be Xcalled a singular region. XWe will show that by restricting the Xfrequency $\BGO$ appropriately, singular Xregions for the Dirichlet problem Xconsist only of isolated sites, Xwhile for the periodic problem singular regions will Xconsist of no more than pairs of adjacent sites. X XIn order to make the first step in an existence Xtheorem for \equ(NLW) with periodic or XDirichlet boundary conditions we ask Xfor certain conditions of Xnonresonance among the linear frequencies X$\{ \omega_j \}_{j=1}^\infty$. XThis does not have to be a condition among Xinfinitely many of them, but at least a large Xenough number of the initial frequencies. X X\CLAIM Definition(L-nonresonance) XA sequence $\{ \omega_j \}_{j=1}^\infty$ is X$(L_0,d_0)$-nonresonant with $\omega_1$ Xif there exists some $\tau > 5$ such that Xfor all $|j|+|k|~\leq~L_0$ Xthe following conditions hold: X$$ X |\omega_1^2 k^2 - \omega_j^2| > d_0~~ X{\rm if}~~(j,k)~\ne~(1,\pm 1). X\EQ(2.65) X$$ Xand X$$ X |k\omega_1 - j| > X {d_0 \over (|j| + |k|)^\tau},~~{\rm for}~~ X(j,k)~\ne~(0,0). X\EQ(2.6) X$$ X X X\CLAIM Proposition(dense-L-nonres) An open dense set of Xfrequency sequences $ \{ \omega_j \}_{j=1}^\infty $ Xare $(L_0,d_0)$-nonresonant with $\omega_1$ for Xsome $L_0,d_0$ with $d_0 = o(L_0^{-1/2})$. X X X XThis condition is on the equation linearized about X$U = 0$, thus depends only upon the coefficient X$g_1(x)$. We defer to Section 6 the discussion of the Xprecise topology in which the above set of Xfrequency sequences X$ \{ \omega_j \}_{j=1}^\infty $ is dense, and the Xproof of this proposition; however the set of coefficients X$g_1$ satisfying \equ(2.6) is open and dense Xin $L^2(0,\pi)$, Xand has open intersection with the $\pi$ periodic, Xanalytic potentials. X X\SUBSECTION Symmetries of the equation X XThe wave equation \equ(NLW) has certain elementary Xproperties of Xsymmetry, relevant Xto this paper, that are reflected in the nonlinear Xlattice systems \equ(NLlattice). XThe sequences $u \in \HH_\gs$ among Xwhich we construct solutions Xare complex, however they will correspond to Xreal solutions of Xthe wave equation. XDenote the involution on the lattice X$$ X x = (j,k) \rightarrow \overline x = (j,-k) X\EQ(invol) X$$ Xand the complex conjugate of $U$ by $\overline U$, then the Xreality condition on sequences is that X$\overline{ U(x)} = U(\overline x)$. XWe will require that the lattice equation X\equ(NLlattice) is covariant Xwith respect to this symmetry, X$$\eqalign{ X \overline {V(\Omega) U(x)} &= X V(\Omega) \overline {U(x)} = X V(\Omega) U(\overline x ) \cr X \overline {W(U(x))} &= X W(\overline {U(x)}) = W(U(\overline x)). \cr } X\EQ(sym1) X$$ X XThe wave equation respects an additional Xtranslational symmetry; X$t \rightarrow t+T, \, T \in \real$. XThat is, time translation leaves the Xequation and the boundary Xconditions invariant. XWe will consider lattice systems which also Xpossess a continuous Xsymmetry of this form. XThe translation operator on sequences in X$\l2(\zsquared)$ is the diagonal operator X$$ X T_{\xi} U(x) = e^{ik\xi} U(x), X$$ Xwhere $x = (j,k)$. X$T_{\xi}$ is a unitary operator on $\l2(\ZZ)$ and on all the X$\HH_\gs$ spaces, and it preserves the reality Xcondition. From the nature of the diagonal operator notice that X$T_{\xi} V(\Omega) U = V(\Omega) T_{\xi} U$. XWe will further require that the nonlinearity $W$ satisfy X$$ X T_{\xi} W(U) = W(T_{\xi} U). X\EQ(sym2) X$$ XThe nonlinear terms in lattice problems arising from the Xnonlinear wave equation satisfy \equ(sym2), Xbecause the system is autonomous. XOur construction will be of families of Xsolutions invariant with Xrespect to this translation. XThe interpretation is that this is the Xconstruction of embedded Xinvariant circles of solutions of \equ(NLW) Xin the space $\HH_\gs$. X XThe final requirement on the lattice problem is that Xthe linearized operator $H(U) = V(\Omega) + DW(U)$ Xis selfadjoint. Since $V(\Omega)$ is real and Xdiagonal, this is the condition on the nonlinear Xterm $W(U)$ that X$$ X DW(U)(x,y) = {\overline {DW(U)}}(y,x)~~. X\EQ(selfadjDW) X$$ XAgain, this condition holds for problems stemming Xfrom the nonlinear wave equation. X X XWe expect that similar constructions can be Xobtained for invariant tori of Xhigher dimension, giving rise to Xsolutions of the nonlinear wave Xequation \equ(NLW) which are quasiperiodic in time. XIn a second publication we plan to address this and Xother problems from our point of view. X X X X X\SUBSECTION Results for lattice problems X XThe main existence theorem for periodic solutions for nonlinear Xlattice problems \equ(NLlattice) can now be stated. We fix Xthe exponent $1/2 < \eta < 1$. X X\CLAIM Theorem (NLLat) Consider equations \equ(NLlattice) which Xsatisfy the reality and translation invariance conditions X\equ(sym1), \equ(sym2), \equ(selfadjDW). XSuppose further that the nonlinear Xterm in \equ(NLlattice) satisfies hypotheses {\bf H1}-{\bf H3} Xof Section 6. There is a constant $L_*$ such that Xif $\{ \omega_j \}_{j=1}^\infty$ is X$(L_0,d_0)$-nonresonant with $\omega_1$ for $d_0 > L_0^{-\eta}$, Xfor some $L_0 > L_*$, then Xthere is an open set of nonlinearities X$W$ such that there exist uncountably many solutions X$U(x) \in \HH_{{\overline \gs}/2}$ of \equ(NLlattice). XMore precisely there exist $(r,\theta,\Omega)$ Xsuch that these solutions satisfy X$$\eqalign{ X \Vert U &- rT_{\theta}(\gd_y X + \gd_{\overline y})\Vert_{\overline{\sigma}/2} X \leq Cr^2 \cr X |\Omega &- \omega_1| \leq Cr^2 \cr} X\EQ(2.8) X$$ XThese sequences remain solutions when acted upon by the Xtranslation $T_\xi$; they are embedded circles in Xthe space $\HH_{{\overline \gs}/2}$. X XBoth \clm(NLWperiodic) and \clm(NLWDir) Xfollow from this theorem. XIndeed, consider solutions to the nonlinear problem X\equ(NLW) described in terms of the eigenfunctions, Xor normal modes, of the linearized equations. XOne expands a function $u(x,t)$ which is X$2\pi / \Omega$ periodic in time, Xsatisfying the correct spatial Xboundary conditions (Dirichlet on $[0,\pi]$, or periodic), Xin terms of the eigenfunction expansion. XLet $\xi = \Omega t$, then X$$ X u(x,\xi) = \sum_{(j,k) \in \zsquared} X U(j,k) e^{ik \xi} \psi_j(x). X\EQ(e_funct) X$$ XSquare integrable time periodic solutions X$u$ to the wave equation \equ(NLW) Xcorrespond to sequences $U(j,k) \in \l2(\zsquared)$ Xwhich solve the lattice equation \equ(NLlattice). XThe nonlinear term in the wave equation is X$g(x,u) - g_1(x)u$. XThis corresponds to the nonlinearity Xfor the lattice system X$$ X W(U)(j,k) = \int_0^{2\pi} \int_0^\pi X \psi_j(x) e^{-ik \xi} (g(x,u) - g_1(x)u) dx d\xi. X\EQ(Wdef) X$$ X XWe call equation \equ(NLlattice) the X{\bf mode interaction equation} for the Xnonlinear wave equation \equ(NLW). XIf $U(x) \in \HH_\gs$ then the solution $u(x,\xi) $ Xgiven by \equ(e_funct) is analytic. XIf $U({\overline x}) = \overline {U(x)}$, then X$u(x,\xi)$ is real, and vice versa. XFurthermore, with $g$ analytic in the region X$\{ (x,u); |{\rm Im} \ x| < \overline \gs \}$, Xand with our choices of boundary Xconditions, the lattice nonlinearity X$W \in C^{\omega}(\HH_\gs;\HH_{\gs-\ga})$ Xfor all $0 < \ga \leq \gs < \overline \gs$. XOther boundary conditions will also result in analytic Xnonlinearities on the spaces $\HH_\gs$, Xthis is discussed in detail in section 6. XFor these, theorems similar to \clm(NLWDir) Xand \clm(NLWperiodic) also hold. X XConversely, starting from a solution X$U(x) \in \HH_\gs$ of the Xlattice equation \equ(NLlattice), Xconsider the function X$u(x,\xi) = \sum_{(j,k) \in \zsquared} XU(j,k) \psi_j(x) e^{ik \xi} $ Xand its translates by $T_\theta$. XSince $U \in \HH_\gs$ Xthey form an analytic family, in fact an embedded circle. XSetting $\xi = \Omega t$, Xa real analytic solution of the nonlinear wave Xequation \equ(NLW) is obtained. X X XWe can now give a more precise Xdescription of the existence result. XThe proof of \clm(NLLat) Xis by a Nash Moser iteration scheme, Xproving a result which is in spirit Xvery close to the theorem of XKolmogorov, Arnold and Moser. XIn fact the conclusions are reminiscent of these results; Xwe construct families of solutions of X\equ(NLlattice) invariant Xunder translation $T_\xi$, Xparametrized by $r \in C$ a Cantor set. XThat is, the periodic solutions Xthat we find occur not in smooth Xcurves but in totally disconnected families, XCantor sets foliated by invariant circles. XThere is a set of positive measure of amplitudes X$r$ for which Xthere are solutions. XThis feature of totally disconnected Xfamilies of solutions is Xfamiliar in the study of Xinvariant tori of quasiperiodic orbits Xfor Hamiltonian systems near Xelliptic stationary points. XThe fundamental reason behind this phenomenon Xis that Hamiltonian systems Xpossessing infinitely many degrees of freedom Xhave the possibility for the Xgeneration of a dense set of Xlinear resonances even for periodic orbits. X XIn order to describe the detailed Xexistence result we need to Xdefine the condition of genuine nonlinearity. XLet $B_0 = \{ x \in \zsquared : |j| + |k| \leq L_0 \} $ Xbe a bounded subdomain of the lattice $\zsquared$, Xand define $\Pi_0$ Xto be the orthogonal projection onto X$\l2(B_0)$. XThis projection commutes with $T_\xi$, Xand furthermore $\Pi_0$ Xcommutes with the lattice involution X\equ(invol) and thus Xpreserves the reality condition. XConsider an approximate problem to X\equ(NLlattice) for a sequence X$U_0 \in \l2(B_0)$ X$$ X \Pi_0 \bigl( W(U_0) + V(\Omega)U_0 \bigr) = 0. X\EQ(P0NLLat) X$$ XIf the sequence $\{ \omega_j \}_{j=1}^\infty$ Xis $(L_0, d_0)$ nonresonant then X$\Pi_0 V(\Omega)$ restricted to $\ell^2(B_0)$ Xhas only a double eigenvalue at X$\Omega = \omega_1$, Xwith eigenvectors supported on the lattice sites X$N = \{ (1,\pm 1) \}$. XWe denote the orthogonal projection onto X$\l2(N)$ by $Q$, and set X$P= (\11 - Q)$. XParametrize a neighborhood of zero in the Xnull space of $\Pi_0 V(\go_1)$ by X$\pp = (p_1,p_2) \rightarrow \varphi(\pp), X\ \|\pp\| < r_0$, Xwith $\varphi (\pp) = X (p_1 + i p_2)\delta_{(1,1)}(x) + X (p_1 - ip_2)\delta_{(1,-1)}(x)$. XEquation \equ(P0NLLat) possesses a branch of nontrivial Xsolutions $(U(\pp),\Omega_0(\pp))$ Xbifurcating from $(\pp,\BGO) = (0,\go_1)$. XDenoting rotations by angle $\xi$ in the plane X$\pp \in \real^2$ also by $T_\xi$, then X$T_\xi \varphi(\pp) = \varphi(T_\xi \pp)$, Xand the above branch of Xsolutions is $T_\xi$ invariant. XThe condition of genuine nonlinearity Xis that for this branch of Xapproximate solutions the frequency parameter X$\Omega_0$ is nondegenerate in $\pp$. X X\CLAIM Definition (twist) XThe problem \equ(NLlattice) is said to Xsatisfy a twist condition Xif the bifurcation surface of the Xapproximate problem X$ \CC_0 = (\pp,\Omega_0(\pp))$ satisfies X$$\eqalign{ X \Omega_0(0) &= \omega_1 \cr X \det \partial^2_{\pp} &\Omega_0 (0) \not= 0 \cr} X\EQ(twistcond) X$$ XSince the branch of solutions is Xinvariant under $T_\xi$ then X$\partial_{\pp} \Omega_0 (0) = 0$ automatically . X X XRecall that $\eta$ is a small positive constant fixed above. X\CLAIM Theorem (NLLat2) XLet the sequence $\{ \go_j \}_{j=1}^\infty$ Xbe $(L_0,d_0)$ nonresonant with $\omega_1$ Xfor some $L_0 > L_*$ with $d_0 > L_0^{-\eta}$. XIf the nonlinear term in \equ(NLlattice) satisfies Xhypotheses {\bf H1}-{\bf H3}, there is a neighborhood X$\Eta_0 = \{ (\pp,\BGO); \|\pp\| < r_0, X |\BGO - \go_1| < r_0^2 \}$ Xin the parameter space and a function X$u(x;\pp,\BGO) \in \HH_{\gs/2}, \, Qu = 0$ Xwhich is $C^\infty$ in the parameters X$(\pp,\BGO) \in \Eta_0$ satisfying: X(i) there is a Cantor set $\Eta \subseteq \Eta_0$ Xin parameter space, invariant under X$T_\xi$, such that for X$(\pp,\BGO) \in \Eta, \, u(x;\pp,\BGO)$ Xis a solution of the first Xbifurcation equation X$$ X P \bigl( W(\varphi(\pp) + u) + V(\BGO)u \bigr) = 0. X\EQ(bifeq1) X$$ X(ii) Set the exponent $0 < \nu < 1 - \eta$. XIf additionally the approximate problem \equ(P0NLLat) on X$\l2(B_0)$ satisfies a twist condition X$$ X \det\partial^2_{\pp} \BGO_0(0) \geq {\Kappa_0^2} > 0 X\EQ(twist2) X$$ Xwith ${\Kappa_0^2} > L_0^{-\nu}$, Xthen there exists a $C^\infty$ surface X$ \CC = (\pp,\BGO(\pp))$ invariant Xunder $T_\xi$ satisfying the second bifurcation equation X$$ X Q \bigl( W(\varphi(\pp) + u(\pp)) + V(\BGO(\pp))u(\pp) \bigr) = 0. X\EQ(bifeq2) X$$ Xwhere the intersection X$C = \{ \pp ; \CC (\pp) \cap \Eta \not= \emptyset \}$ Xhas positive measure. X XThe intersection of $\CC$ with $\Eta$ Xis of course the solution Xset for equation \equ(NLlattice), Xconsisting typically of a XCantor set foliated by invariant circles. XThe measure of the set X$C = \{ \pp ; \CC(\pp) \cap \Eta \not= \emptyset \}$ Xis relatively large, Xon the order of $\pi r_0^2$. X XWe note that even if the non-degeneracy condition Xfails to hold, one may have periodic solutions. XSuch situations are explored in [CW]. X XIt is possible that the actual bifurcation point X$(0,\go_1) \notin \Eta$ due to an exact or Xnear resonance of $\go_1$ and X$\go_j$, with $j$ such that $(j,k) \notin B_0$. XHowever if none of these Xoccur then there is a result on the density of Xthe periodic orbits within radii $0 < \|\pp\| < r_0$. XFix exponents $0 < {\overline \tau}$ and X$ 0 < {\overline \alpha}$ such that X${\overline \alpha} + 1 > {\overline \tau}$. X%%%%%%%%%%%%%%%%%% X X\CLAIM Theorem(density) XAn $(L_0,d_0)$-nonresonant sequence $\{\omega_j \}_{j=1}^\infty $ Xis fully nonresonant Xwith $\omega_1$ if there exist positive constants X$c_1$ and $c_2$ such that for all X$(j,k) \in \ZZ^+ \times \ZZ$, $(j,k) \ne (0,0)$, X$$ X |k \omega_1 - j| > \ { c_1 \over X (|j| + |k|)^{\overline \tau} },~~ X$$ Xand if for $(j,k) \not= (1,\pm 1)$, X$$ X |k^2 \omega_1^2 - \omega_j^2| > \ X { c_2 \over (|j|+|k|)^{\overline \alpha}} ~~. X$$ XIn this case the Cantor set of \clm(NLLat2) has $\pp = 0$ Xas an accumulation point. Furthermore there is an Xestimate of the density of periodic orbits near $\pp = 0$. XThere are constants $\mu > 0, \ C_g$ such that for all X$0 < r_1 < r_0$, X$$ X {\rm meas} \{ r \in (0,r_1); \|\pp\| = r, \ X (\pp,\Omega(\pp)) \in \Eta \} X \geq r_1(1-C_g r_1^\mu). X$$ X X X XBecause it depends upon the details of the induction Xprocess, the proof of this density Xresult is deferred to section 6. In the proof Xestimates of the size of the exponent $\mu$ will be given. X X X X X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%. X\SUBSECTION The Nonlinear Klein-Gordon and sine-Gordon equations X XPrincipal examples of problems of the form \equ(NLW) Xare the nonlinear Klein Gordon equation X$$ X \partial_t^2 u = \partial_x^2 u - m^2u + (m^2/3) u^3 X\EQ(NLKG) X$$ Xand the related sine-Gordon equation X$$ X \partial_t^2 u = \partial_x^2 u - m^2 \sin (u). X\EQ(NLsG) X$$ XBoth equations \equ(NLKG) and \equ(NLsG) Xhave frequency sequences X$$\{ \omega_j \}_{j=1}^\infty = X\{ \sqrt{j^2 + m^2} \}_{j=1}^\infty$$ for the XDirichlet problem, and X$$\{ \sqrt{4[(j+1)/2]^2 + m^2 } \}_{j=0}^\infty$$ Xfor the periodic problem. (The term X$[(j+1)/2]$ inside the square root means the integer Xpart of $(j+1)/2$.) X XWhen posed with periodic boundary conditions on the interval X$[0,2\pi]$, \equ(NLsG) is a Xcompletely integrable Hamiltonian system. XThis is not the case for \equ(NLKG), or for X\equ(NLsG) with Dirichlet conditions posed at X$x=0,\pi$. Traveling wave solutions for \equ(NLKG) and X\equ(NLsG) satisfying Xperiodic boundary conditions on the interval X$[0,\pi]$ are easily Xdescribed using phase plane analysis for Xfunctions $u(x-ct)$. Thus, we will concentrate on the Xcase of Dirichlet boundary conditions. XOn a formal level this problem was Xdiscussed by J.B. Keller \& L. Ting [KT], Xand the curvature of the solution surfaces X$\partial^2_\pp \BGO(0)$ was derived. XThese solutions are related to a class called X`breather solutions' in the literature, Xwhich are spatially localized, time Xperiodic solutions to nonlinear wave Xequations posed on all of X$x \in \real$. X\clm(NLWDir) implies the existence Xof small amplitude time periodic solutions of both X\equ(NLKG) and \equ(NLsG) for a Xset of values of the parameter $m^2$ of full measure. XWe have only to check that the hypotheses of \clm(NLLat2) Xare satisfied for the associated lattice problem. Either Xof these equations could be perturbed Xby an additional nonlinear term, $h(x,u)$, and as long Xas the total nonlinearity satisfied the twist condition, Xthe results discussed below would still hold. X X X X X X X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X X\CLAIM Theorem (nlkgfreqs) XConsider a sequence of constants $d_0,L_0$ Xsuch that $d_0 \log (L_0) \rightarrow 0$. XThere is an open set ${\cal M}$ of full XLebesgue measure such that if $m^2 \in {\cal M}$, Xthen the frequency sequences X$\{ \omega_j \}_{j=1}^\infty$ for the equations X\equ(NLKG) and \equ(NLsG) are X$(d_0,L_0)$ nonresonant with $\omega_1$ for Xsome $(d_0,L_0)$. X X X X\PROOF Fix $d_0,L_0$ and consider an Xarbitrary interval $[a,b]$ of parameters $m^2$. XFor the Dirichlet problem, the first X$(d_0,L_0)$-nonresonance Xcondition is violated for those $m^2$ such that X$| k^2(1+m^2) - (j^2+m^2)| \leq d_0$. That is X$$ X \bigl| \bigl( { j^2-k^2 \over k^2-1 } \bigr) - m^2 \bigr| X \leq { d_0 \over |k^2-1|}, X$$ Xhence by excising a closed interval of length $2d_0/|k^2-1|$ Xabout every point X$(j^2-k^2)/(k^2-1),\ |j|+|k| \leq L_0, \ X (j,k) \not= (1, \pm 1)$ Xwhich falls within the interval $[a,b]$, Xthe remaining values of $m^2$ satisfy the first condition Xof \clm(L-nonresonance). Note that this imposes Xno condition for $k= \pm 1$, and that X$m^2 = 0$ is excised. The diophantine condition Xof \clm(L-nonresonance) is violated for those Xvalues of $m^2$ such that X$|k \sqrt{m^2 + 1} - j| \leq d_0/(|j|+|k|)^\tau$. XExcising a closed interval of length X$2d_0/(|j|+|k|)^\tau$ about every point X$m^2 = (j/k)^2 - 1$ as well, the remaining Xparameters satisfy both conditions in \clm(L-nonresonance). XCall this open set ${\cal M}(d_0,L_0)$. XThe only $(j,k)$ that need to Xbe considered are those for which X$(1+a)k^2 - C_0 \leq j^2 \leq (1+b)k^2 + C_0$. XThe total measure of the excised intervals is estimated by X$$ X \sum_{{|j|+|k| \leq L_0 \atop k\not= \pm 1} \atop X (j^2/k^2) \in (a-C_1,b+C_1) } X {2d_0 \over |k^2-1|} \ + \ X {2d_0 \over (|j|+|k|)^\tau} X \leq C d_0 \log (L_0), X$$ Xas long as $\tau \geq 2$. XThe set ${\cal M} \cap [a,b] = X \cup_{d_0,L_0} {\cal M}(d_0,L_0)$, Xthus if $d_0\log (L_0) \rightarrow 0$, the set ${\cal M}$ is Xof full measure. The proof in the case of periodic Xboundary conditions is similar. X\endproof X XComputing the curvature of the approximate bifurcation branches Xfor the equation \equ(2.6) Xis a straightforward exercise, and it is non-zero and Xindependent of the choice of approximate domain X$B_0$ as long as $L_0 \geq 6$. X\clm(NLLat) implies the following Xexistence result X X\CLAIM Theorem (NLKG+sG) XConsider the equations \equ(NLKG) and \equ(NLsG), Xsatisfying Dirichlet boundary conditions Xon the interval $[0,\pi]$. XThe curvature of the branch bifurcating from the Xfrequency $\go_1$ is X$$ X \partial_{\pp}^2 \BGO(0) = X {-3 m^2 \over 16 \sqrt{1+m^2}}{\bf 1}. X\EQ(KGcurv) X$$ XThus the nonlinear problem satisfies the Xtwist condition, and for $m^2 \in {\cal M}$ Xand $d_0 > L_0^{-\eta}$, there exist small Xperiodic solutions with frequency close to $\go_1$. X XIf the sign of the nonlinear term in the equation X\equ(NLKG) is reversed, that is, if X$g(x,u) = m^2 (u + (1/3)u^3)$ then the curvature X\equ(KGcurv) reverses sign, but still retains a Xtwist and the existence theorem holds. X X X X X END_OF_FILE if test 29844 -ne `wc -c <'sec2.tex'`; then echo shar: \"'sec2.tex'\" unpacked with wrong size! fi # end of 'sec2.tex' fi if test -f 'sec3.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sec3.tex'\" else echo shar: Extracting \"'sec3.tex'\" \(45156 characters\) sed "s/^X//" >'sec3.tex' <<'END_OF_FILE' X X X\SECTION The Induction Argument X X\SUBSECTION Notation X XIn this subsection we introduce notation Xrelating to the function spaces in which we solve X\equ(NLlattice) . XWe wish to show that the solutions of these lattice problems Xdecay exponentially; to measure this Xdecay we introduce a family of Hilbert spaces. XWhen specifying points in $\zsquared$ by a single Xsymbol, we will use $x$, $y$, etc. XWhen we wish to specify their components we will Xuse $(j,k)$, $(l,m)$, etc. XFor points $x=(j,k) \in \zsquared$, Xwe always use the $\ell^1$ norm, X$|x|=|j|+|k|$. X XDefine a family of Hilbert spaces $\HH_{\sigma} X\subset \l2 (\zsquared)$, Xwhich consist of elements of $\l2 (\zsquared)$ for Xwhich the norm X$$ X \|u\|^2_{\sigma} = \sum_{x \in \zsquared} |u(x)|^2 X e^{2\sigma |x|} X$$ Xis finite. We denote the inner product in these Hilbert spaces Xby $\langle \cdot ,\cdot \rangle_{\sigma}$. XIf $\sigma = 0$, X$\HH_{\sigma} = \l2$. We will denote the inner product in $\l2$ Xby either $\langle \cdot ,\cdot \rangle_{0}$ or X$\langle \cdot ,\cdot \rangle$, depending on the circumstances. XSimilarly, $\| \cdot \|$ will mean $\| \cdot \|_0$--$i.e.$ Xthe $\ell^2$ norm. XThis is similar to sequence spaces used in [P2]. X XIf $S : \HH_{\sigma} \to \HH_{\sigma} $ , let $\| S \|_{\sigma}$ Xbe the usual Hilbert space operator norm. The following Xproposition shows that these norms respect the exponential Xweights. X X\CLAIM Proposition(opnorm) The norm is a Banach Xalgebra norm; X$$\parallel ST\parallel_{\sigma} \leq \parallel XS\parallel_{\sigma}\parallel T\parallel_{\sigma}.$$ XIf $\parallel S\parallel_{\sigma} \leq C_{s}$ Xthere is a sup norm estimate of the matrix elements Xof $S$; X$$|S(x,y)|\leq C_{s} e^{-\sigma |x-y|}.$$ XIf for some $\sigma$ the matrix elements of an Xoperator $S$ are bounded by X$$|S(x,y)|\leq C e^{-\sigma |x-y|},$$ Xthen for all $0\leq \gamma <\sigma$ X$$\parallel S\parallel_{\sigma-\gamma}\leq C/\gamma^{2}.$$ X XAn additional simple but important property of these norms Xis embodied in the following lemma. X X\CLAIM Lemma(cutoff) Let $B_{L} = \{(j,k) \in X\ZZ^+ \times \ZZ ; |j|+|k| \le L \}$. Let X$\Pi_{L}$ be orthogonal projection onto X$\ell^2(B_{L})$, and let ${\bf 1} - \Pi_{L}$ Xbe its orthogonal complement. If X$0 < \gamma \le \sigma$ then X$$\eqalign{ X \| \Pi_L f \|_\gs &\leq X e^{\gamma L} \| f \|_{\gs-\gamma} \cr X \| ({\bf 1} - \Pi_{L}) f \|_{\sigma - \gamma} &\leq X e^{-\gamma L} \| f \|_{\sigma}. \cr} X$$ X\PROOF From the definition of the norm, X$$\eqalign{ X \| \Pi_L f \|_\gs \leq& X \sup_{|j|+|k| \leq L} X e^{\gamma (|j|+|k|)} \| f \|_{\gs-\gamma} \cr X \leq& e^{\gamma L} \|f\|_{\gs-\gamma} \cr X \|({\bf 1} - \Pi_{L}) f\|_{\sigma-\gamma} \le & X \sup_{|j|+|k| > L} e^{-\gamma(|j|+|k|)} X \| f \|_{\sigma} \cr X \le & e^{-\gamma L} \|f\|_{\sigma}. \cr} X$$ X\endproof X XWe will also use the analyticity of these solutions Xas functions of parameters. For this purpose, Xif $\NN \subset \real^2 \times \real$, we define complex domains X$D(\NN;\rho) X = \{(\zz,\BGO) \in \complex^2 \times \complex ; X\sqrt{ \| \zz - \pp \|^2 + |\Omega_1 - \Omega|^2 } X< \rho \ {\rm for \ some} \ (\pp,\Omega_1) \in \NN \}$. X X X X\SUBSECTION The bifurcation problem on $B_0$ X XThe induction procedure of this paper is started Xby solving an approximate bifurcation problem X\equ(P0NLLat), consisting of the full Xequation \equ(NLlattice), restricted to the lattice Xsubdomain $B_0$. For $u \in \ell^2(B_0)$, we write X$V_0(\Omega) u = \Pi_0 V(\Omega) u$ and $W_0(u) = \Pi_0 W(u)$. XLinearizing \equ(P0NLLat) around $u \equiv 0$, we obtain X$V_0(\Omega) \phi = 0$. If the sequence X$\{\omega_j \}_{j=1}^{\infty}$ is $(d_0,L_0)$-nonresonant Xwith $\omega_1$, then $V_0(\omega_1)$ has a two-dimensional Xnull space $\ell^2(N)$ parameterized by $\phi(\pp)$, X$\pp \in \real^2$. As we have indicated above, Xthese will be solved using a Lyapunov-Schmidt decomposition; X$$ X P(W_0(\phi(\pp) +u) + V_0(\Omega) u ) = 0~~, X\EQ(firstbifur) X$$ X$$ X Q(W_0(\phi(\pp) +u) + V_0(\Omega) u ) = 0~~. X\EQ(secondbifur) X$$ XThis pair of equations is equivalent to \equ(P0NLLat). X X XWith these definitions in hand, we Xcan construct the first approximation to the Xsolution of \equ(NLlattice). XFor any subset $B \subset \ZZ^2$, we write X$\overline{B} = B \backslash N$, where $N$ is the set Xof lattice points supporting the null space of $V_0(\omega_1)$. XDefine $\NN_0 = \{ (\pp, \Omega) X\in \real^2 \times \real ~|~ X|| \pp || < r_0, |\Omega - \omega_1| < r_0^2 \}$, a neigborhood Xof the point $(0,\omega_1)$ at which bifurcation Xbranches are to be constructed. X X\CLAIM Lemma(firstbifonB0) XSuppose that the sequence $\{ \omega_j \}_{j=1}^\infty$ Xis $(L_0,d_0)$ nonresonant with $\omega_1$. For any X$0 < r_0 < (1/6C_W) d_0/L_0^2$ and X$ \rho_0 < d_0/(3CL_0^2)$ Xthere exists a solution $u_0(x;\pp,\bgo) \in \l2(B_0)$ Xof (3.1) which is analytic in $D(\Eta_0,\rho_0)$. XFurthermore, X$$T_\xi u_0(x;\pp,\bgo) = X u_0(x;T_\xi \pp,\bgo), X$$ Xand $u_0$ satisfies the estimate X$$ X \| u_0 \|_\gs < { 3C_WL_0 \over d_0} \|\pp\|^2 X\EQ (u_0est) X$$ Xfor $\gs \leq \gs_{*}-(1/L_0)$. X XFor $(\pp,\BGO)$ in the complex subdomain X$D(\Eta_0,\rho_0/2)$ the Cauchy estimates implies that X$$\eqalign{ X \| \partial_\BGO^\beta u_0 \|_\gs X \leq& {3C_W L_0 \over d_0} X {\|\pp\|^2 \over (\rho_0/2)^\beta } \cr X \| \partial_{\pp} \partial_\BGO^\beta u_0 \|_\gs X \leq& {3C_W L_0 \over d_0} X \left( {\|\pp\| \over (\rho_0/2)^\beta } \right) X \left( 1 + {r_0 \over (\rho_0/2) } \right), \cr} X\EQ(u0derivs) X$$ Xand in general for $|{\bf \alpha}| \geq 2$, X$$ X \| \partial_{\pp}^\alpha \partial_\BGO^\beta u_0 \|_\gs X \leq {3C_W L_0 \over d_0} X {1 \over (\rho_0/2)^{|\alpha|+\beta-2} } X \bigl( 1 + {r_0^2 \over (\rho_0/2)^2 } \bigr). X$$ X X\PROOF Linearizing the function on the left hand Xside of (3.1) about $u_0=0$ and setting X$(\pp,\Omega) = (0,\omega_1)$ gives X$$ X PV_0(\omega_1) = H_{\overline {B_0}} (\omega_1). X$$ XBy hypothesis this is invertible, with inverse X$G_{\overline {B_0}}(\omega_1)$ bounded by $C/d_0$. XEquation (3.1) is finite dimensional, solvable by Xthe implicit function theorem to give a solution Xanalytic in a complex neighborhood of X$(\pp,\bgo) = (0,\omega_1)$. XThe point of this lemma is to estimate the size Xof this neighborhood. XSolutions of (3.1) are fixed points of the mapping X$$ X u_1 = -G_{\overline {B_0}}(\omega_1) X \bigl( W_0(\phi(\pp) + u_0) X + (H_{\overline {B_0}}(\bgo) X - H_{\overline {B_0}}(\omega_1))u_0 \bigr). X\EQ (mapping) X$$ XThe symmetry of the equation with respect to $T_\xi$ Xassures the covariance property of the solution. X XUnique analytic solutions are assured for parameter Xvalues such that the mapping in \equ(mapping) preserves a Xneighborhood of the origin, on which it is a contraction. XTo find a neighborhood that is mapped to itself, Xwe use hypothesis {\bf H1} to estimate X$$\eqalign{ X \|u_1\|_\gs \leq& {C \over d_0}\|W_0(\phi(\pp) + u_0)\|_\gs X + \| G_{\overline {B_0}}(\omega_1) X \bigl( H_{\overline{B_0}}(\bgo) X - H_{\overline {B_0}}(\omega_1)u_0 \bigr) \|_\gs \cr X \leq& {C_W \over d_0 \gamma} X (\|\pp\|^2 + {1 \over \gamma^2} X \|u_0\|^2_{\gs+\gamma}) X + {{\rm sup} \atop (j,k) \in {\overline {B_0}}} X \biggl| { \omega_j^2 - \bgo^2 k^2 \over X \omega_j^2 - \omega_1^2 k^2} - 1 \biggr| \|u_0\|_\gs \cr} X$$ XUsing \clm(cutoff), X$$\eqalign{ X \|u_1\|_\gs \leq& {C_W \over d_0 \gamma} X (\|\pp\|^2 + X {e^{2\gamma L_0} \over \gamma^2} \|u_0\|_\gs^2) \cr X &+ {C L_0^2 \over d_0} |\bgo-\omega_1| X \|u_0\|_\gs, \cr} X$$ Xand by optimizing over X$\gamma$ such that $\gs + \gamma \leq \gs_{*}$ Xwe find that $\gamma = 1/L_0$ and X$$ X \|u_1\|_\gs \leq C_W { L_0 \over d_0} X (\|\pp\|^2 + L_0^2\|u_0\|_\gs^2) X + C {L_0^2 \over d_0} |\bgo-\omega_1| \|u_0\|_\gs~~. X\EQ (mapest) X$$ XWe require $r_0 = \rho_0 < d_0 / (3(C_W + C) L_0^2)$, Xand define $C_u = 3 C_W L_0/ d_0$. XFor $(\pp, \Omega) \in D(\NN_0 ; \rho_0)$, the complex Xneighborhood of $\NN_0$, if $\Vert u_0 \Vert_{\sigma} X\le C_u \Vert \pp \Vert^2 $ Xthen X$\Vert u_1\Vert_\gs < C_u \Vert \pp \Vert^2$ also. XThus the fixed point of the map will lie in this neighborhood. X XTo obtain a contraction estimate, we use hypothesis {\bf H2} of Xsection 6. X$$\eqalign{ X \|G_{\overline {B_0}}(\omega_1)&(W_0(\phi(\pp) + u_2) - X W_0(\phi(\pp) + u_1)\|_\gs \cr X \leq& {C \over d_0} \| \int_0^1 DW_0((\phi(\pp) + X tu_2+(1-t)u_1) \ dt \ (u_2-u_1)\|_\gs \cr X \leq& { C_W L_0^2 \over d_0} (\|\pp\| + L_0\|u_1\|_\gs X + L_0\|u_2\|_\gs) \|u_2-u_1\|_\gs \cr} X$$ XIn order that this gives a contraction mapping when X$\|\pp\| < r_0 + \rho_0 X$ and $\|u_1\|_\gs, \|u_2\|_\gs < C_u \|\pp\|^2$, Xwe ask that $\rho_0 = r_0 < (1/9C_W)(d_0/L_0^2)$. X XThe second term is easier, X$$ X \|G_{\overline {B_0}}(\omega_1)(H_{\overline {B_0}}(\bgo) - X H_{\overline {B_0}}(\omega_1))(u_2-u_1)\|_\gs X \leq {CL_0^2 \over d_0} |\bgo-\omega_1| \|u_2-u_1\|_\gs X$$ Xwhich gives a contracting estimate if X$|\Omega - \omega_1| < r_0^2 + \rho_0$, X and $2C(r_0^2+\rho_0) < d_0/L_0^2$. X XEstimate \equ(mapest) leads to an a priori estimate on the Xfixed point. X$$ X \|u_0\|_\gs \leq C_u \|\pp\|^2 X = {{3C_W L_0}\over{ d_0} }\|\pp\|^2. X$$ XThis is the upper bound stated in \equ(u_0est). X X\endproof X X XOver the neighborhood $\NN_0$ the problem \equ(P0NLLat) Xis reduced to finding the zero set of the equation X\equ(secondbifur). A trivial solution branch is X$\{(\pp,\Omega) : \pp =0, -r_0^2 < (\omega_1 - X\Omega) < r_0^2 \}$, and we seek an additional family of Xsolutions parametrized by $\{ \pp : \| \pp \| < r_0 \} $. XAs the equations \equ(firstbifur) and \equ(secondbifur) Xand the solution $u_0(x;\pp,\Omega)$ behave covariantly Xwith respect to the translation $T_{\xi}$, the zero Xset of \equ(secondbifur) is invariant Xunder $T_{\xi}$. Properties of this set are given Xin the next result. X X\CLAIM Lemma(secondbifonB0) The mapping $(\pp,\Omega) \to XG_0(\pp,\Omega) = Q\left( W_0 (\phi(\pp) + Xu_0(\pp,\Omega) ) + V_0(\Omega) \phi(\pp) \right)$ Xis analytic on $\NN_0$, zero for X$\{(\pp,\Omega) : \pp = 0, -r_0^2 < (\omega_1 - X\Omega) < r_0^2 \}$, and has a Taylor expansion at $\pp = 0$, X$\Omega = \omega_1$ with the following Taylor coefficients: X$$\eqalign{ X \partial_{\pp} G_0(0,\omega_1) \ =& \ 0 \cr X \partial_{\Omega} G_0(0,\omega_1) \ =& \ 0, \cr X} X\EQ(firstderiv) X$$ Xand X$$\eqalign{ X \partial_{\pp}^2 G_0(0,\omega_1) =& \ 0 \cr X \partial_{\Omega}^2 G_0(0,\omega_1) =& \ 0 \cr X \partial_{\pp} \partial_{\Omega} X G_0(0,\omega_1) =& X \ Q\left( \partial_{\Omega} V_0(\omega_1)) X \partial_{\pp} \phi(0) \right) \cr X =& - 2\omega_1 \partial_{\pp} \phi(0)~~. \cr X} X\EQ(secondderiv) X$$ XIf $\omega_1 \ne 0$, the last term is nonvanishing. X XThus the zero set of $G_0(\pp,\Omega)$ in a neighborhood Xof $(\pp,\Omega) = (0,\omega_1)$ consists of the X$\Omega$ axis union a surface $(\pp,\Omega_0(\pp))$, Xgiven as a graph over a neighborhood of zero in the X$\pp$-plane. X X\PROOF The analyticity follows immediately since Xthe implicit function theorem guarantees that X$u_0(\pp,\Omega)$ is analytic, $\phi(\pp)$ is Xanalytic by construction, and $W$ and $V$ are analytic Xby assumption. XWe next compute the Taylor expansion of the function X$G_0$ at $(\pp,\Omega) = (0,\omega_1)$. Clearly X$Q\left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) + V_0(\Omega) X\phi(\pp) \right)|_{(\pp,\Omega) = (0,\omega_1)} X= 0$, and the first derivatives are X$$\eqalign{ X \partial_{p_j} G_0(0,\omega_1) =& X Q\left( V_0(\omega_1) \partial_{p_j} \phi(0) \right) X = 0~~;~j=1,2~, \cr X \partial_{\Omega} G_0(0,\omega_1) =& X Q\left( \partial_{\Omega} V_0(\omega_1) \phi(0) \right) X = 0 ~~, \cr X} X\EQ(firstderivtwo) X$$ Xwhere we use that $\phi(0) = 0$ and $\partial_{p_j} X\phi(0)$ is in the null space of $V_0(\omega_1)$. XTo compute the Xsecond derivatives we use that $D_u W_0(0) = 0$, X$\| u_0 \|_{\sigma} \le C_u \| \pp \|^2$, and the fact Xthat \equ(secondbifur) is covariant with respect to X$T_{\xi}$. Indeed, X$$\eqalign{ X T_{\pi} Q& \left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) + X V_0(\Omega) \phi(\pp) \right) \cr X =& \ Q\left( W_0 (\phi(-\pp) + u_0(-\pp,\Omega) ) + X V_0(\Omega) \phi(-\pp) \right) \cr X =& - Q\left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) + X V_0(\Omega) \phi(\pp) \right)~~.\cr X} X$$ XThus, $G_0$ is odd in $\pp$, and both X$\partial_{\pp}^2 G_0(0,\omega_1) = 0 $ and X$\partial_{\Omega}^2 G_0(0,\omega_1) = 0 $. XThe mixed partial is nonzero, however, as long as X$\omega_1 \ne 0$. X$$\eqalign{ X \partial_{\Omega} \partial_{p_j} G_0(0,\omega_1) =& \ X Q \big( D^2 W_0(0)(\partial_{p_j} \phi + \partial_{p_j} X u_0 ) (\partial_{\Omega} u_0) \cr X &+ D W_0(0)(\partial_{\Omega} \partial_{p_j} u_0) + X \partial_{\Omega} V_0(\Omega) \partial_{p_j} X \phi \big)|_{(\pp,\Omega)=(0,\omega_1)} \cr X = & Q\left(\partial_{\Omega} V_0(\omega_1) X \partial_{p_j} \phi(0) \right)\cr X = & -2 \omega_1 \partial_{p_j} \phi(0)~~. \cr X} X\EQ(mixedpartial) X$$ X\endproof X X XThe $T_{\xi}$ invariance of the zero set of $G_0(\pp,\Omega)$ Xalso implies that the surface $(\pp,\Omega_0(\pp))$ obeys X$\partial_{p_j} \Omega_0(0) = 0$, that is, Xit is tangent to the plane $\{ (\pp,\Omega) ; X\Omega = \omega_1 \}$. By a further Taylor expansion Xwe will compute $\partial^2_{p_j p_k} \Omega_0(0)$, Xwhich pertains to the twist condition of the nonlinear Xproblem. X X X\CLAIM Lemma(bifsurface) If X$ X\sqrt{r_0^2 + \rho_0^2}(3C_W L_0 / d_0) < C, X$ Xand X$ X(r_0^2 + \rho_0^2) C_W L_0 / (d_0 \rho_0) << 1 $, Xthe solution surface $(\pp,\BGO_0(\pp))$ is defined Xover the full neighborhood X$\{ \pp ; \| {\rm Re} \ \pp \| < r_0, X\| {\rm Im} \ \pp \| < \rho_0 \}$. XThe surface satisfies X$$\eqalign{ X \BGO_0(0) &= \omega_1 \cr X \partial_{p_j} \BGO_0(0) &= 0 \cr X \partial^2_{p_j p_k} \BGO_0(0) &= \Kappa_0 \delta_{jk}, X \cr} X\EQ(OmegaTaylor) X$$ Xwhere X$$\eqalign{ X \Kappa_0 =& { 1 \over 2\omega_1} \biggl( X {1 \over 6}\langle \partial_\pp \phi, X (D^3W_0(0)(\partial_\pp \phi)^3) \rangle \cr X &- {1 \over 2} \langle (D^2W_0(0)(\partial_\pp \phi)^2), X G_{\overline {B_0}}(\omega_1) X (D^2W_0(0)(\partial_\pp \phi)^2) \rangle X \biggr). \cr} X\EQ(curvature) X$$ X X\PROOF The expression for the curvature Xis derived from the Taylor expansion of the mapping X$G_0(\pp,\BGO)$ at $(0,\omega_1)$. Differentiating (3.1) Xwith respect to $\pp$, we easily find that X$$\eqalign{ X u_0(0,\omega_1) &= 0 \cr X \partial_{p_j} u_0(0,\omega_1) &= X \partial_\bgo u_0(0,\omega_1) = 0 \cr} X\EQ(taylor12) X$$ Xand $\partial^2_{p_j p_k} u_0(0,\omega_1)$ satisfies X$$ X P(D^2W_0(0)[\partial_{p_j} \phi,\partial_{p_k} \phi] X + V_0(\omega_1)\partial^2_{p_j p_k} u_0 ) = 0. X$$ XSince $PV_0(\omega_1)$ has inverse $G_{\overline {B_0}}$, X$$ X \partial^2_{p_j p_k} u_0(0,\omega_1) X = -G_{\overline {B_0}}(\omega_1)(D^2W_0(0) X [\partial_{p_j} \phi,\partial_{p_k} \phi] ). X$$ XIn order to compute the curvature of the surface, Xthe relevant third order term in the Taylor expansion Xof the mapping $G_0(\pp,\bgo)$ is X$\partial^3_\pp G_0(0,\omega_1)$. X$$ X \partial^3_\pp G_0(0,\omega_1) = X Q((D^3W_0(0)(\partial_\pp \phi)^3) X + 3(D^2W_0(0)[\partial_\pp \phi, X \partial_\pp^2 u_0]))~~. X\EQ(taylor3) X$$ X X XUsing the expression for $\partial^2_\pp u_0$ Xand \equ(taylor12) Xthis gives \equ(curvature) for the curvature of the Xnontrivial zero set $(\pp,\BGO_0(\pp))$ Xof the mapping $G_0$ at $\pp=0$. XIncidentally we may easily deduce that X$\partial_\BGO \partial^2_\pp G_0(0,\omega_1) = 0, \ X\partial^3_\bgo G_0(0,\omega_1) = 0$ Xsince the mapping is odd. X XWe also establish that the surface $(\pp,\BGO_0(\pp))$ Xis defined and analytic throughout the complex neighborhood X$\{ \pp; \Vert \pp - \pp_1 \Vert < \rho_0 ~~{\rm for~~ Xsome}~~ \pp_1 \in \real~~{\rm with}~~ \Vert \pp_1 \Vert X< r_0 \}$. XSince the branch is simple it suffices to show that X$\partial_\pp \BGO_0(\pp)$ is bounded. XUsing that $G_0(\pp,\BGO_0(\pp)) = 0$ and Xdifferentiating with respect to $\pp$, we find X$$ X Q \bigl( V_0(\BGO) \partial_{p_j} \phi X + \partial_\bgo V_0(\BGO) \phi \partial_{p_j}\BGO X + DW_0(\phi(\pp) + u_0)(\partial_{p_j} \phi X +\partial_{p_j} u_0 X + \partial_\BGO u_0 \partial_{p_j} \BGO) X \bigr) = 0. X$$ XThus X$$ X Q \bigl( \partial_\BGO V_0(\BGO) \phi X + (DW_0) \partial_\BGO u_0) \bigr) \partial_{p_j} \BGO X = -Q \bigl( V_0(\BGO) \partial_{p_j} \phi X + DW_0 \partial_{p_j}(\phi + u_0) \bigr). X$$ XQuite clearly $\| Q(V_0(\BGO) \partial_{p_j} \phi + XDW_0 \partial_{p_j}( \phi + u_0) ) \|_0 < C$, Xso that the only possible singularities occur when X$Q (\partial_\BGO V_0(\BGO) \phi + DW_0 \partial_\bgo u_0)$ Xvanishes. The first term is explicit -- X$ Q \partial_\BGO V_0 \phi(\pp) = 2\BGO \phi(\pp)$, Xwhich is bounded below by $c \|\pp\|$ for X$\BGO$ bounded away from zero. XUsing \clm(firstbifonB0), Xand a Cauchy estimate for the second term, X$$\eqalign{ X \| Q DW_0 \partial_\BGO u_0 \|_0 &\leq X C_W ( \|\pp\| + \|u_0\|_\gs) X \|\partial_\BGO u_0 \|_\gs \cr X &\leq C_W( \|\pp\| + X {3C_W L_0 \over d_0} \|\pp\|^2) X { 6C_W L_0 \over \rho_0 d_0 }\|\pp\|^2 \cr X &\leq {c \over 2} \|\pp\|~~, \cr X} X$$ Xas long as constants are chosen as in the hypotheses of the Xlemma. X X X\endproof X X X X\SUBSECTION The Induction Hypotheses X XStarting from the approximate solution of the previous Xsubsection we will inductively construct Xbetter approximate solutions in ever larger Xregions of the lattice and prove that in the limit Xthey converge to a solution of the original problem X\equ(NLlattice). In this subsection, we will state Xthe inductive hypotheses, and in the next, Xwe verify that they suffice to prove \clm(NLLat2). X XWe begin the statement of the induction scheme Xwith the definition of the constants which appear in Xthe induction and a description of their role which we Xhope will help the readers orient themselves. X X{\bf Inductive Constants}: X X X(1) At the ${n}^{\rm th}$ step in the iteration Xwe solve a finite dimensional Xproblem in a lattice box X$B_{n} = \{(j,k) : |j|+|k| \le L_n \}$, Xwhose size is given by X$$ XL_n = 2^n L_0\qquad n \ge 1~~. X$$ X X(2) The amount by which our ${n}^{\rm th}$ Xapproximate solution fails to be a Xtrue solution is measured by X$$ X\epsilon_n = \epsilon_0^{{\kappa}^n}~~, X~{\rm with}~~\kappa > 1 X~{\rm and}~~ n\ge 1~~. X$$ XThe constants $\epsilon_0$ and X$\kappa$ are fixed in section 3.5. X X(3) As the iteration proceeds we encounter Xworse and worse small denominators Xwhose effects are estimated by X$$ X\delta_n = {1 \over {L_n^{\alpha}}}~~, X$$ Xwhere $\alpha$ is a fixed Xconstant. X X X(4) There is a length scale $\ell_n$ which estimates the distance Xover which the effects of a resonance $\delta_n$ can be felt. Set X$$ X\ell_n = L_n^{\beta}~~, X$$ Xwith $\beta$ is a small constant. We will need to estimate Xthe spectra of the local Hamiltonian operators, X$H_{C_{\ell_n}(S)} = (V(\Omega) + D_u W)|_{C_{\ell_n}(S)}$, Xwhich are defined on disks X$C_{\ell_n}(S) \equiv X\{ x\in \ZZ^+ \times \ZZ : \dist(x,S) < \ell_n \}$, XHence we require Xthat $\ell_n << L_n$. X X(5) The approximate solutions will decay exponentially Xin the size of the indices $|(j,k)|$. XThe exponential decay rate will change during Xthe iteration. This rate is determined by X$$ X\sigma_{n+1} = \sigma_n - 6 \gamma_n~~,~\gamma_n X= {{\sigma_0}\over{64 (n+2)^2}}~~, ~n\ge 0~~. X$$ XHere, $\sigma_0 < \gs_* $ is defined in Section 6, Xand is related to ${\overline \gs}$, Xthe width of the strip in which the Xcoefficients of (2.1) are analytic. Note that for all X$n \geq 0, \ \gs_n > {\overline \gs}/2$. X X(6) The solutions will depend on parameters X(such as the frequency $\Omega$ and $\pp$, the Xamplitude and phase of the periodic solution). XWe require them to be analytic in a Xcomplex neighborhood, whose size is governed by X$$ X\rho_n = {{\rho_0 \delta_n}\over{L_n^2}}~~,~n\ge 1~, X$$ Xwhere $\rho_0$ is the size of our original analyticity Xdomain. X XWe are now in a position to state the induction Xhypotheses. XThe equation \equ(NLlattice) is equivalent to two Xequations obtained by the Lyapunov Schmidt decomposition Xof the full lattice problem X$$ X F(\pp,\BGO,u) \equiv P X \bigl( W(\phi(\pp) + u) + V(\BGO)u \bigr) = 0 X\EQ(Peqn) X$$ X$$ X G(\pp,\BGO,u) \equiv Q X \bigl( W(\phi(\pp) + u) + V(\BGO)\phi(\pp) X \bigr) = 0~~~. X\EQ(Qeqn) X$$ XThere exists a constant $C_G$ and a positive Xexponent $\mu$ such that, provided Xthe conditions stated in section 3.5 Xare satisfied, the following induction statements Xare true. X X X\item{(n.1)} There is a function $u_n(\pp,\BGO ;x) = Xu_{n-1}(\pp,\BGO ;x) + v_{n-1}(\pp,\BGO ;x) = Xu_0(\pp,\BGO ;x) + \sum_{j=0}^{n-1} v_j(\pp,\BGO ;x)$, Xdefined on $\Eta_0 \times \Bn$, and analytic on X$D(\NN_n,\rho_n)$ satisfying: X\itemitem{(i)} For any $(\pp,\Omega) \in D(\NN_j,\rho_j)$, X$$ X \|v_{j}(\pp,\BGO ;\cdot)\|_{\sigma_j-\gamma_j} \le X \|\pp\|^2 \gre_{j} C_G^{j+1}/\delta_{j+1} \gamma_{j}^{12}~~, X~~ j= 0, \dots , n-1~~, X$$ Xand on $\NN_0$, one has X$$\eqalign{ X \|\partial_\BGO^\beta X v_j(\pp,\BGO;\cdot ) \|_{\gs_j - \ga_j} \leq& X {C_G^{j+1}\epsilon_j \over \gd_{j+1} \ga_j^{12} } X (\beta + 3)! X {\|\pp\|^2 \over (\rho_{j+1}/2)^\beta} \cr X \|\partial_\pp \partial_\BGO^\beta v_j (\pp,\BGO;\cdot ) X \|_{\gs_j-\ga_j} X \leq& {C_G^{j+1} \epsilon_j \over \gd_{j+1}\ga_j^{12} } X (\beta + 4)! X {\|\pp\| \over (\rho_{j+1}/2)^\beta } X \bigl( 1 + {r_0 \over \rho_0} \bigr). X\cr} X$$ XFor higher $\pp$ derivatives there is a general estimate X$$ X \|\partial_{\pp}^{\bf \alpha} \partial_{\Omega}^{\beta} X v_{j}(\pp,\BGO ;\cdot)\|_{\sigma_j-\gamma_j} X \le {C_G^{j+1} \gre_j \over \delta_{j+1} \gamma_{j}^{12}} X {(|{\bf \alpha}|+\beta +3)! X \over \rho_{j+1}^{|{\bf \alpha}| + \beta -2}}~~. X$$ XThis estimate quantifies our control of the X$C^k$ norms of $v_j$. X\itemitem{(ii)} The function $u_n(x,\pp,\BGO)$ is an Xapproximate solution of the first bifurcation equation X\equ(Peqn). For $(\pp,\Omega) \in D(\NN_n,\rho_n/2)$, X$$ X \|F(u_n)\|_{\sigma_n} \le \|\pp\|^2 \epsilon_n~~. X$$ X X\item{(n.2)} There exists a closed set of parameters, X$(\pp, \BGO)$ $\in \Eta_{n+1} \subset \Eta_{n} \subset X\dots \subset \Eta_0$, with the following properties: X\itemitem{(i)} If $(\pp, \BGO)\in \Eta_{n+1}$, Xand if X$x_i$ and $x_j$ are any two singular sites Xin $B_{n}^c$, which are not in the same singular region, Xthen the distance between $x_i$ and $x_j$ is greater Xthan $2 \ell_{n+1}$. X\itemitem{(ii)} If $S$ is a singular region in $B_{n+1}\backslash XB_{n}$, Xthen for any $(\pp,\BGO) \in D(\Eta_{n+1}, \rho_{n+1})$, X$$ X {\rm dist}({\rm spec} X (H_{S}(\pp, \BGO; u_n)),0) > \delta_{n+1} X$$ X$$ X{\rm dist}({\rm spec} X(H_{C_{\ell_{n+1}}(S)}(\pp, \BGO;u_n)),0) > \delta_{n+1} X$$ X\itemitem{(iii)} For any $C^\infty$ curve, $\BGO(\pp) X= \Kappa \|\pp\|^2(1+C(\|\pp\|)) + \go_1$, with X$|\Kappa| \geq L_0^{-\nu}$ and X$|C(\cdot)|_{C^1} < 1/2$, then X$$ X {\rm meas} \left\{ \|\pp\| \in (-r_0,r_0); X (\pp,\BGO(\pp))\in \Eta_{n+1} \right\} \ge X r_0(1-Cr_0^\mu)~~. X$$ X(Recall that the exponent $\nu$ was defined in X\clm(NLLat).) X XThe induction hypotheses $(n.1)$ and $(n.2)$ suffice Xto prove \clm(NLLat2). We will verify this in the next Xsubsection. The inductive estimates imply in particular Xthat throughout the parameter domain $\Eta_0$ there is Xa bound on the derivatives of $u_n$. X$$ X \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_n X \|_{\gs_n-\ga_n} \ X \leq \ X \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_0 X \|_{\gs_n-\ga_n} X + \| \sum_{j=0}^n \ \partial_\pp^{\bf \alpha} X \partial_\BGO^\beta v_j \|_{\gs_n-\ga_n}. X$$ X Thus for ${\bf \alpha} = 0$, we estimate X$$ X \| \partial_\BGO^\beta u_n \|_{\gs_n-\ga_n} \ X \leq \ {CL_0 \over d_0} X \bigl( 1 + {C\epsilon_0^{1/ \kappa} \over \gd_0} X ( \beta !)^b \bigr) X {{\|\pp\|^2 }\over {(\rho_0/2)^\beta}}~~. X$$ XFor $|{\bf \alpha}|=1$, X$$ X \| \partial_\pp \partial_\BGO^\beta u_n X \|_{\gs_n-\ga_m} \ X \leq \ {CL_0 \over d_0} X \bigl( 1 + {C\epsilon_0^{1/\kappa} \over \gd_0} X (( \beta+1 )!)^b \bigr) X {\|\pp\| \over (\rho_0/2)^\beta} X \bigl( 1 + {r_0 \over (\rho_0/2)} \bigr)~~. X$$ Xand in general X$$ X \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_n X \|_{\gs_n-\ga_m} \ X \leq \ {C L_0 \over d_0}{1 \over X (\rho_0/2)^{|{\bf \alpha}| + \beta - 2} } \bigl( X 1 + {C\epsilon_0^{1/\kappa} \over \gd_0} X ((|{\bf \alpha}|+\beta)!)^b X (1+ {r_0^2 \over (\rho_0/2)^2}) \bigr)~~. X\EQ(allderivsu) X$$ XThese are obtained using elementary bounds and the Xabove choices of inductive constants. The exponent X$b = 1/\log \kappa + 1$. The approximate solution $u_n$ Xis analytic in $D(\NN_n,\rho_n/2)$, and an estimate similar Xto \equ(allderivsu) holds over the complex neighborhood X$D(\NN_n,\rho_n/4)$, where in fact the exponent $b= 1/\kappa$. X X XThe induction hypotheses also allow us to prove the following Ximportant result about the inverse of the linearized Xoperator which appears in Newton's method. X X\CLAIM Theorem(Greens) Suppose that the induction Xhypotheses $(j.1)$ and X$(j.2)$ hold for $j=0,1,\dots,n$. XFor any non-singular lattice Xregion $A$ and X$E_{n+1} \subset \overline{B_{n+1}} \cup A$, the XGreen's function is analytic on X$D(\NN_{n+1},\rho_{n+1})$, and satisfies X$$ X \|G_{E_{n+1}}(\pp,\Omega,u_n)\|_{\sigma_n-\gamma_n} \le X {{C_G^{n+1}}\over{ \delta_{n+1} \gamma_{n}^{12}}}~~. X\EQ(redGreen1) X$$ XUnder perturbations of $u_n$ a similar estimate holds. XIf X$$ X \| u - u_n \|_{\sigma_n -\gamma_n} \le X \| \pp \|^2 \epsilon_n C_G^{n+1} X /\delta_{n+1} \gamma_{n}^{12}~~, X$$ Xthe estimate holds, X$$ X\|G_{E_{n+1}}(\pp,\Omega,u)\|_{\sigma_n-2\gamma_n} \le X{{2 C_G^{n+1}}\over{ \delta_{n+1} \gamma_{n}^{12}}}~~. X\EQ(redGreen2) X$$ X X XWe delay the proof of this theorem until section 5, Xwhere a detailed analysis of the linearized operators Xis presented. X X\SUBSECTION Proof of \clm(NLLat2). X XIn this section we demonstrate that the Xinduction hypotheses lead Xto a proof of \clm(NLLat2). X XSet X$\NN = \cap_{n\ge0} \NN_n$, the parameter domain Xfor which we obtain a solution of the first Xbifurcation equation \equ(Peqn) . XThe parameters $\ga_n$ governing loss of exponential decay Xin the induction process satisfy X$\sigma_n \ge X {\overline \gs}/2$, for all $n$. XThus the induction Xhypothesis $(n.1)(i)$ implies that $u_n(\pp,\BGO;x)$ Xconverges to a sequence X$u(\pp,\BGO;x) \in \HH_{{\overline \gs}/2}$, X$$ X u(\pp,\Omega;x) \equiv u_0(\pp,\Omega;x) X + \sum_{j=0}^{\infty} v_j(\pp,\Omega;x), X$$ Xwhich is $C^\infty$ as a function of X$(\pp,\Omega)$ on the parameter domain $\NN_0$. XInduction hypothesis $(n.1)(ii)$ implies that for X$(\pp,\Omega) \in \NN \subseteq \Eta_0$ the sequence X$u$ satisfies the first bifurcation equation \equ(Peqn). X XTo obtain a solution to the full nonlinear lattice problem, Xit remains to solve the second bifurcation equation X\equ(bifeq2). This is a finite Xdimensional problem, recovering the zero set of the mapping X$G$. Not surprisingly, we will show that the solution Xis close to the approximate solution that was Xdiscussed in Section 3.2. X X X\CLAIM Lemma(map) The mapping $(\pp,\Omega) \to XG(\pp,\Omega)$ $= Q ( W(\phi(\pp) + u(\pp,\Omega) ) X$ $+ V(\Omega) \phi(\pp))$ is $C^{\infty}$ on $\NN_0$, Xzero for $\{ (\pp,\Omega) ; \pp =0,$ X$ -r_0^2 < (\omega_1 - \Omega) < r_0^2 \}$ and Xhas a Taylor expansion at $\pp = 0$, $\Omega = \omega_1$ Xwith the following Taylor coefficients: X$$ X\partial_{\pp} G (0,\omega_1) = 0~~,~~ X\partial_{\Omega} G (0,\omega_1) = 0~~, X$$ Xand X$$\eqalign{ X\partial_{\pp}^2 G (0,\omega_1) = 0~~, &~~ X\partial_{\Omega}^2 G (0,\omega_1) = 0~~,\cr X\partial_{\pp} \partial_{\Omega} G (0,\omega_1) X= Q(\partial_{\Omega} V(\omega_1) ) & \partial_{\pp}\phi(0) X= -2 \omega_1 \partial_{\pp} \phi(0)~~.\cr X} X$$ X X\PROOF The fact that $G$ is $C^{\infty}$ follows Xfrom the fact that $u$ is $C^{\infty}$ on $\NN_0$, which Xfollows from the uniform bounds in \equ(allderivsu). XThe Taylor coefficients are computed as in the proof Xof \clm(secondbifonB0). X X\endproof X XThe zero set of the mapping $G(\pp,\BGO)$ consists of the X$\BGO$-axis union a surface $(\pp,\BGO(\pp))$ which is Xinvariant under the translations $T_\xi$, and is given as Xa graph over the neighborhood X$\{ \pp \in \real^2; \|\pp\| < r_0 \}$. This is the surface Xof solutions of the second bifurcation equation X\equ(bifeq2). XIt is close to the approximate solution surface X$(\pp,\BGO_0(\pp))$. Intersections of $(\pp,\BGO(\pp))$ with X$\NN$ correspond to solutions of the first bifurcation Xequation as well, thus are solutions of the full problem X\equ(NLlattice). Such intersections are guaranteed if the Xsurface $(\pp,\BGO(\pp))$ has nonzero curvature at zero. X X X\CLAIM Lemma(bend) The solution surface $(\pp,\Omega(\pp))$ Xof \equ(bifeq2) is defined in X$\{\pp \in \real^2; \| \pp \| < r_0\}$. XThe surface satisfies X$$ X \Omega(0) = \omega_1~~,~\partial_{p_j} \Omega(0) = 0~~, X \partial_{p_j p_k}\Omega(0) = \Kappa \delta_{j k}~~, X$$ Xwhere X$$ X \Kappa \ = \ {1 \over 6\omega_1} X \langle \partial_\pp \phi, (D_u^3 W(0) X (\partial_\pp \phi)^3) \rangle X + {1 \over 2\omega_1} X \langle \partial_\pp \phi, D_u^2 W(0) X [\partial_\pp \phi,(\partial_\pp^2u)] \rangle. X$$ XThe difference $|\Kappa - \Kappa_0| < X C \bigl( e^{-({\overline \gs} L_0)/2} X + \epsilon_0^{1/\kappa} L_0/d_0 \bigr) X /(\gd_0 \rho_0^2)$. X X X\PROOF The proof is essentially the same as that for X\clm(bifsurface), using the solution $u$ in place of Xthe approximate solution $u_0$. The estimates \equ(u0derivs) Xon $u_0$ from the bifurcation theory on $B_0$ are Xreplaced by \equ(allderivsu) from the induction Xargument. The expression for the curvature can Xbe rewritten as; X$$\eqalign{ X \Kappa \ =& \ \Kappa_0 + {1 \over 6\omega_1} X \langle \partial_\pp \phi, (\11-\Pi_0) X D_u^3 W(0)(\partial_\pp \phi)^3 \rangle \cr X &+ \ {1 \over 2\omega_1} \langle \partial_\pp \phi, X (\11-\Pi_0) D_u^2 W(0) [\partial_\pp \phi, X \partial_\pp^2u] \rangle \cr X &+ {1 \over 2\omega_1} \langle \partial_\pp \phi, X D_u^2 W_0(0) [\partial_\pp \phi, X (\partial_\pp^2 u - \partial_\pp^2 u_0)] \rangle X\cr} X$$ XThe difference is estimated using \equ(allderivsu). X$$\eqalign{ X |\Kappa - \Kappa_0| \ \leq& \ C\bigl( \|(\11-\Pi_0) X D_u^3 W(0)(\partial_\pp \phi)^3 \|_0 X + \| (\11-\Pi_0)D_u^2 W(0)[\partial_\pp \phi, X \partial_\pp^2 u ] \|_0 \cr X &+ \ \|D_u^2 W_0(0)[\partial_\pp \phi, X (\partial_\pp^2 u - \partial_\pp^2 u_0)] \|_0 \bigr) X \cr X \leq& \ Ce^{-({\overline \gs}L_0)/2} X (1 + \|\partial_\pp^2 u \|_{{\overline \gs}/2}) X + {C_W \over ({\overline \gs}/2)^2} X \|\partial_\pp^2(u - u_0)\|_0 \cr X \leq& \ Ce^{-({\overline \gs}L_0)/2} X \bigl( 1 + {1 \over \gd_0\rho_0^2} \bigr) X + {C_W \over ({\overline \gs}/2)^2} X {L_0 \epsilon_0^{1/\kappa} \over d_0 \delta_0} X \left( 1 + {r_0^2 \over \rho_0^2} \right) X. \cr} X\EQ(curvediff) X$$ XAssuming that $|\Kappa_0| > L_0^{-\nu}$, if X$$ X \bigl( e^{-({\overline \gs} L_0 / 2)} + X \epsilon_0^{1/\kappa} L_0/d_0 \bigr) X /(\delta_0 \rho^2_0) << L_0^{-\nu}, X$$ Xthe curvature Xof the surface $(\pp,\BGO(\pp))$ satisfies X$|\Kappa| > cL_0^{-\nu}$, and thus by the induction Xhypothesis $(n.2)(iii)$ the set X$(\pp,\BGO(\pp)) \cap \Eta$ is nonempty. This inequality will Xbe verified in Section 3.5 which follows X X\endproof X XThe Klein-Gordon case $G(x,u) = m^2(u - u^3)$ Xis somewhat special in these considerations of Xcurvature of the bifurcation branches. Firstly X$D_u^2 W(0) = 0$, since the nonlinearity appears only at Xcubic order. Additionally for X$L_0 \geq 6$, $ (\11-\Pi_0) D_u^3 X W(0)(\partial_\pp \phi)^3 = 0$, thus the Xexpression \equ(curvediff) gives that X$\Kappa = \Kappa_0$, and the twist condition is Xsatisfied automatically. X X X X X X X X\SUBSECTION Final Reckoning X XThe proof of convergence of this induction is based on a Xproper choice of constants that appear in the analysis Xthroughout the paper. There are roughly three requirements Xto be satisfied. First the inductive constants $\alpha, X\beta, \tau$ and $\kappa$ must be chosen so that the Newton iteration Xmethod is convergent. This involves estimates of the Xtruncation in approximating the nonlinearity, the excision Xprocess in the parameter domain, and the iterative Xconstruction of the Green's function. Secondly, the initial Xapproximate bifurcation problem and parameter neighborhood Xmust be sufficiently large so that, even after the excisions Xof the induction steps, the remaining solution set has Xlarge measure. Finally, we must be able to carry out Xthe analysis of convergence for an open dense set of Xnonlinearities $g(x,u)$ for the wave equation, thus Xrequirements of Proposition 2.4 and Theorems 2.8 and 2.9 Xmust be satisfied. In this subsection we show that all Xthe requirements of the paper can be simultaneously satisfied. XWe accordingly make choices of the Xinductive parameters $\alpha,\beta, \tau$ and $\kappa$, Xand show that the principal requirements on the parameters X$r_0, \rho_0$ and $d,d_0$ and $d_s$ reduce to a condition Xthat $L_0$, the initial radius of an approximating lattice domain, Xbe sufficiently large. As technical aspects of the induction Xverification appear in sections 4 and 5, this subsection Xoccasionally refers ahead to to formulae in these sections. X XBefore the induction starts, a bifurcation analysis is Xperformed on the initial domain $B_0$ of radius $L_0$. XThe apriori estimates on the initial domain require that Xthe induction starts with X$u_0 \in \HH_{\gs_0}, \ \gs_0 + \ga_0 < \gs_* - 1/L_0$. XSetting $d_0 = L_0^{-\eta}$ for some $1/2 < \eta < 1$, Xand $r_0 = \rho_0 = c L_0^{-(2+\eta)}$, for $c$ some Xsmall positive number, we are able to Xsatisfy the hypotheses of \clm(firstbifonB0) and X\clm(bifsurface), X$$\eqalign{ X r_0 < & C { d_0 \over L_0^2} \cr X \rho_0 < & C{d_0 \over L_0^2} \cr X r_0^2 {L_0 \over d_0 \rho_0} & << 1 \cr} X\EQ(FR1) X$$ X X XWe next turn to the choice of the inductive constants Xand the question of convergence. The sequence X$u_0 \in \ell^2(B_0) \cap \HH_{\gs_0}$ is an Xexact solution of the restricted equation $\Pi_0 F(u_0) = 0$, Xthus there exists $c_0 > 0$ such that X$$\eqalign{ X \| F(u_0(\pp,\BGO) ) \|_{\gs_0} X & = \| (\11 - \Pi_0) W(\phi(\pp) + u_0) \|_{\gs_0} \cr X & \leq {C_W \over \ga_0} e^{-\ga_0 L_0} X \bigl( \|\pp\|^2 X + {1 \over \ga_0^2}\|u_0\|^2_{\gs_0+\ga_0} \bigr) \cr X & \leq e^{-c_0L_0} \|\pp\|^2 \cr} X$$ Xas long as $L_0 \geq L_*$, is chosen sufficiently large so that X$\gs_*-1/L_0 - \ga_0 \geq \gs_0 > {\overline \gs}/2$. We have Xused \equ(FR1) and {\bf H1} to estimate the nonlinear term. XMake the choice X$$ X \epsilon_0 = e^{-c_0L_0}. X$$ XFor $n>0$ the iteration will have a supergeometric rate Xof convergence as long as the hypotheses of Proposition X4.1 are satisfied; X$$ X \epsilon_{n-1} {C C_G^{n+1} \over \gd_n \ga_{n-1}^{14} } X \left( e^{-\ga_{n-1}L_n} X + {r_0^2 \epsilon_{n-1} C_G^{n+1} X \over \gd_n \ga_{n-1}^{13} } \right) X \leq \epsilon_n. X\EQ(FR2) X$$ XUsing the definition of the inductive constants, \equ(FR2) Xwill follow from the pair of inequalities X$$ X C_G^{n+1} L_0^\alpha e^{\alpha \log(2)n} X \bigl( {c \over |n|^2} \bigr)^{14} X e^{-L_0 c2^n/|n|^2} \leq X {1 \over 2} \epsilon_0^{\kappa^{n-1}(\kappa-1)} X\EQ(FR3) X$$ Xand X$$ X L_0^{-2(2+\eta)} C_G^{2(n+1)} L_0^{2\alpha} X e^{2\alpha \log (2) n} X \bigl( {c \over |n|^2} \bigr)^{27} X \leq {1 \over 2} X \epsilon_0^{\kappa^{n-1}(\kappa-2)} X\EQ(FR4) X$$ XThe governing factor of the left hand side of \equ(FR3) is X$e^{-cL_0 2^n/|n|^2}$, while from the definition of X$\epsilon_0$ that of the right hand side is X$e^{-c_0L_0(1-(1/\kappa))\kappa^n}$. XIf necessary decrease $c_0$ so that X$c_0(1-(1/\kappa)) < 2c$. For $L_*$ sufficiently large Xthe estimate \equ(FR3) will hold for all $n>0$, uniformly Xin $L_0 > L_*$. X XA similar discussion verifies that \equ(FR4) will hold Xfor all $n > 0$ if $L_0 > L_*$ is sufficiently large. XThe right hand side is governed by the factor X$e^{(\ga_0L_0(2-\kappa)\kappa^{n-1})}$, while the Xleft hand exponent grows at most linearly in $n$ and Xlogarithmically in $L_0$. For $L_0 > L_*$, $L_*$ chosen Xsufficiently large, it too will hold for all $n > 0$. X XIn order to make the excision process work, relations (4.4)--(4.7) Xof Proposition 4.6 will have to hold. Set X$$ X 2d = 2d_s = d_0 = L_0^{-\eta} X$$ XIf we use the definitions above, we see that if X$(1-\tau\beta) > 0, \ \beta(1 +c_0 \log (2)) < 1$ and X$L_0 > L_*$ is chosen large, then the first two relations Xof (4.4)-(4.7) are satisfied. The following three relations Xwill have already been satisfied in \equ(FR1). Without yet Xdiscussing the measure of the remaining parameter Xregion, we will verify the rest of the requirements Xof section 4. In order to separate the localizing Xneighborhoods $C_{\ell_n}(S)$, the hypothesis of XLemma 4.8 is that X$$ X d > L_0^{\tau\beta}(L_0^{-1} + r_0^2L_0^\beta ) X = L_0^{\tau\beta-1} + L_0^{\beta(\tau+1)-2(2+\eta)}~~, X$$ Xwhich will hold for $\eta < (1 - \tau\beta)$ Xand $\beta(\tau+1) < \eta + 4$, which can be Xsatisfied by taking $\beta$ small. The extension of Xthese estimates to the complex domain will go through Xif (4.17) holds. By definition X$L_n \rho_n < \gd_n/2$, and one checks readily that X$(1 + L_0/d_0(1+(r_0/\rho_0)))\rho_{n+1} = X (1 + 2L_0^{(1+\eta)}) L_0^{-(2+\eta)} X \gd_{n+1} /L_{n+1}^2 \leq \gd_{n+1}$. X XIn order to construct the Green's function using the Xprocedure of section 5, define $2d_s = d_0 = L_0^{-\eta}$, Xand ask that $L_0 > L_*$ is sufficiently large so that X$C_W r_0/ ({\overline \gs}^2 d_s) \leq C\gs_*$. XThis permits the construction of the Green's Xfunction on non-singular domains. If we choose$$ X \gs_0 + 5 \gamma \equiv ({{35}\over{32}}) < \gs_* - X 2\log (1+4\sqrt{C_W r_0/({\overline \gs}^2 d_s)})~~, X$$ Xthen the restrictions on $\sigma_0$ in both Theorem 5.1 and XProposition 5.3 can be satisfied. X X XThis is the second of two conditions to be satisfied Xin defining $\gs_0$, the starting decay rate of the Xinduction. The requirements of the extension lemmas X5.3 and 5.4 are that $r_0 < Cd_0$ and $r_0 < \ga_0^4 d_s$, Xwhich are both satisfied as in \equ(FR1) by an increase Xin $L_*$ if necessary. The more intricate patching techniques Xof Theorem 5.6 require additionally that X$$ X { Cr_0 e^{-\ga_{n-1}\ell_{n+1}/2} X \over \gd_{n+1}^2 \ga_{n-1}^{34} } X \leq 1. X\EQ(FR5) X$$ XUsing the definitions above, \equ(FR5) reads X$$ X CL_0^{-(2+\eta)} L_0^{2\alpha} X e^{\alpha \log (2) (n+1)} X \times C|n|^{68} X \times e^{-(c/2|n|^2) L_0^\beta}{ X e^{\beta \log (2) (n+1)} } < 1~~~. X$$ XAs before, for any choices of $\alpha, \beta$ there is an X$L_*$ such that this will hold for all $n\geq 1$, uniformly Xin $L_0 > L_*$. Using \equ(FR5), the hypotheses of Lemmas X5.9 and 5.10 follow accordingly. XThe second type of extension of the Green's function is Xaddressed in Theorem 5.12, where it is asked that X$$ X r_0^2 \epsilon_n X \leq { \gd_{n+1}^2 \ga_n^{27} \over 2C_W C_G^{2n}}. X\EQ(FR6) X$$ XUsing the definitions, it is clearly tantamount to X$$ X L_0^{-2(2+\eta)} \epsilon_0^{\kappa^n} X \leq L_0^{-2\alpha} e^{-2\alpha \log (2) (n+1)} X \bigl( {c \over |n+1|^2 } \bigr)^{27} X {1 \over 2C_W C_G^{2n} }, X$$ Xwhich can be uniformly satisfied for $L_0 > L_*$, for all X$n \geq 0$ as long as $L_*$ is sufficiently large. XCorollary 5.13 follows from Theorem 5.12 if the correction Xterm $v_j$ is small in norm. That is, given \equ(FR2) Xand \equ(FR3), we have X$$ X \|v_j\|_{\gs_j-\ga_j} \leq \|\pp\|^2 X { \epsilon_j C_G^j \over \gd_{j+1} \ga_j^{12} } X \leq {1 \over 4^{j+1}} X { \ga_{j+1} \ga_0^4 d_s \over 2C_W C }, X$$ Xwhich is even easier to satisfy than \equ(FR6). X XThe remaining requirements that are placed on the Xinductive constants come from Proposition 4.6, Xwhere an estimate is made of the measure of the Xparameter set after the ${n}^{\rm th}$ excision. This will Xinvolve the choice of $\alpha$ and $\tau$. To satisfy X(4.5), X$$\eqalign{ X c \sum_{j=0}^n \ {d \over L_j^\tau} & X ( 1 + 2r_0^2 L_{j+1}) \cr X = CL_0^{-(\tau+\eta)} & X \sum_{j=0}^n e^{-\tau \log (2) j} X + 4Cr_0^2 L_0^{-(\tau+\eta-1)} X \sum_{j=0}^n e^{-\tau \log (2) j} \cr X \leq C {L_0^{-(\tau + \eta)} \over \tau} X &\leq r_0^2 \cr} X\EQ(FR7) X$$ Xwhich will hold, uniformly in $n$, as long as X$\tau > 4 + \eta$ and $L_*$ is large. We will construct Xthe $\Eta_{n+1}$ by first constructing two intermediate Xsets $\Eta_{n+1}^{1}$ and $\Eta_{n+1}^{2}$. XThe estimate of \equ(FR7) bounds the measure of the excisions that Xare made in constructing $\Eta_{n+1}^{(1)}$. XTo control the size of excisions in constructing X$\Eta_{n+1}^{(2)}$, we require X$$\eqalign{ X \sum_{j=0}^n & \ C L_j(1 + r_0^2 L_{j+1}) X \ell_{j+1}^2 {\gd_{j+1} \over \Kappa L_j^2} \cr X \leq & C L_0^{-(\alpha +1 -2\beta -\nu)} X \sum_{j=0}^n \ e^{-(\alpha +1 - 2\beta) \log (2)j} \cr X \qquad &+ CL_0^{-(\alpha + 2(2+\eta) - 2\beta - \nu)} X \sum_{j=0}^n \ e^{-(\alpha - 2\beta) \log (2) j} \cr X \leq & r_0^2~~~~. X\cr} X\EQ(FR8) X$$ XAs long as we satisfy the requirements that X$$\eqalign{ X \alpha & > 2\beta + \nu \cr X \alpha + 1 - 2\beta & > 2(2+\eta) \cr X \alpha + 4 +2\eta - 2\beta & > 2(2+\eta) \cr} X$$ Xthen \equ(FR8) will hold uniformly in $L_0 > L_*$ and $n>0$. X X XAddressing (4.6), we see that the inequalities will be Xsatisfied if the curvature $\Kappa$ of the surface is Xrestricted from being too small. Using the above choices, X$$ X C({1 \over d_s} + X {L_0 \over d_0}(1+{r_0 \over \rho_0}) ) X = C(L_0^\eta + 2L_0^{1+\eta}) X \leq \Kappa L_0^2 2^{2n}. X$$ XThis will hold for all $n \geq 0$, uniformly in $L_0 > L_*$ Xif the curvature satisfies X$$ X \Kappa > L_0^{-\nu}, \qquad 0 \leq \nu < 1-\eta X\EQ(FR9) X$$ XWe define $\nu$ in the twist condition of Theorem 2.7, for Xa bifurcation surface must intersect the sets X$\Eta_{n+1} = \Eta_{n+1}^{(1)} \cap \Eta_{n+1}^{(2)}$ Xto exhibit a solution of the equation. To estimate the Xmeasure of this intersection, we show that for $\alpha$ Xand $\tau$ reasonably large, (4.7) is satisfied for some Xexponent $\mu$. The first inequality is X$$\eqalign{ X \sum_{j=0}^n & \sqrt{ {C \over \Kappa} {d \over L_j^\tau} X (1 + r_0^2 L_0^{j+1}) } \cr X & \qquad \le \sum_{j=0}^n \sqrt{C \over \Kappa} \bigl( X L_0^{-(\eta + \tau)/2} e^ {-\tau \log (2) j/2} \cr X & \qquad \qquad \qquad+ L_0^{-(\eta + \tau -1)/2} L_0^{-(2+\eta)} X e^{-(\tau - 1) \log (2) (j+1)/2} \bigr) \cr X & \qquad \le C \sqrt{1 \over \Kappa} X \bigl( L_0^{-(\tau + \eta)/2} + X L_0^{-(\tau + \eta +3)/2} \bigr) \cr} X\EQ(FR10) X$$ Xwhere the exponential sums are bounded uniformly Xin $n$ as long as $\tau > 1$. This sum is dominated Xby $r_0^{(1+\mu)} = L_0^{-(2+\eta)(1+\mu)}$ if X$\Kappa > L_0^{-\nu}$ Xand $0 < \mu < (\tau - (\eta + \nu + 4))/2$. XConsiderations of the second inequality are similar. X$$\eqalign{ X \sum_{j=0}^n & \sqrt{CL_j(1+r_0^2 L_0^{j+1}) X \times \ell_{j+1}^2 \times {1 \over \Kappa } X \times {\gd_{j+1} \over L_j^2} } \cr X & \leq \sum_{j=0}^n \sqrt{C \over \Kappa} \bigl( X L_0^{-(\alpha + 1 -2\beta)/2} X e^{-(\alpha + 1 - 2\beta) \log (2) j/2} \cr X & \qquad \qquad + L_0^{-(\alpha +2(2+\eta) - 2\beta)/2} X e^{-(\alpha -2\beta) \log (2) j/2} \bigr) \cr X & \leq \sqrt{C \over \Kappa} X \bigl( L_0^{-(\alpha+1-2\beta)/2} X + L_0^{-(\alpha + 2(2+\eta) - 2\beta)/2} \bigr), \cr} X\EQ(FR11) X$$ Xwhere we must require that $\alpha > 2\beta$. Allowing X$\Kappa > L_0^{-\nu}$, this expression will be dominated Xby $r_0^{(1+\mu)}$ for X$$ X 0 < \mu < {\alpha - (\nu +2(\beta + \eta) + 3) X \over 2(2+\eta) }. X$$ XClearly a choice of $\alpha$ and $\tau$ large, Xand $\beta$ small is available so that these Xinequalities will hold throughout the induction. X XThe last relations that should be verified pertain Xto the the density of the nonlinearities to which Xthe results apply. We require that $d_0 = o(L_0^{-1/2})$ Xin order to satisfy the hypotheses of Proposition 2.4. X This gives a genericity result for the coefficients $g_1$. XWe thus restrict $1/2 < \eta < 1$ in order to do this. XSecondly, we ask for bounds on the curvature in the Xtwist condition. Setting $|\Kappa| \geq L_0^{-\nu}$ with X$\nu > 0$ gives a bound which is decreasing in $L_0$, and again Xthere is an open dense set of nonlinearities possessing a Xsufficiently large twist for Theorem 2.7 to apply. X X X X X X X X X X X X X X X X X X X X X END_OF_FILE if test 45156 -ne `wc -c <'sec3.tex'`; then echo shar: \"'sec3.tex'\" unpacked with wrong size! fi # end of 'sec3.tex' fi if test -f 'sec4.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sec4.tex'\" else echo shar: Extracting \"'sec4.tex'\" \(47949 characters\) sed "s/^X//" >'sec4.tex' <<'END_OF_FILE' X X X\SECTION Verification Of The Induction Hypotheses X XIn this section we begin the verification of the Xinduction hypotheses. There are two main subsections; Xin the first, we show that if $(j.1)$ Xand $(j.2)$ Xhold for $j=0,1, \dots , n-1$ then we can prove X$(n.1)$, and in the second we construct sets $\NN_{n+1}$ Xwhich satisfy $(n.2)$. X X\SUBSECTION Verification Of $(n.1)$ X XAssume that $(j.1)$ and $(j.2)$ Xhold $j=0,1, \dots , n-1$. Then $(n.1)$ Xis a consequence of the following estimate on the XNewton iteration. X X\CLAIM Proposition(newton) XIf the inductive parameters satisfy X$$ X\epsilon_{n-1} X{{C C_G^{n+1} }\over{\delta_n \gamma_{n-1}^{14} }} X\left[ e^{-\gamma_{n-1} L_n }+ X{{r_0^2 \epsilon_{n-1} C_G^{n+1} }\over{\delta_n X\gamma_{n-1}^{13} }} \right] \le \epsilon_{n}~, X\EQ(4.1) X$$ Xthen there is a function $u_{n}(\var) X$ $= u_0(\var) + \sum_{j=0}^{n-1} Xv_j(\var)$, defined on $\NN_0 \times \Bn$, Xwhich obeys Xthe following estimates: X\item{(1)} For any $j=0,\dots ,n-1$, $v_{j}$ is analytic Xon $D(\NN_{j};\rho_{j}/2)$ and for any $(\pp, \Omega)$ Xin this set it satisfies X$$ X \|v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j} X - \gamma_{j}} \le \epsilon_{j}\|\pp\|^2 C_G^{j+1} / X \delta_{j+1} \gamma_{j}^{12}~~. X$$ XFurthermore, there exists a positive constant $C_3$ such that X$v_{j}$ is a $C^{\infty}$ function of X$(\pp,\BGO)$ on all of $\NN_0$, satisfying the Xfollowing estimates. X$$\eqalign{ X \| \partial_{\Omega}^{\beta} v_{j}(\pp,\Omega;\cdot) X \|_{\sigma_{j}-\gamma_{j}} X \le& C_3 \epsilon_{j} C_G^{j+1} X { (\beta + 3)! \over X (\delta_{j+1} \gamma_{j}^{12} X \rho_{j+1}^{\beta}) } \|\pp\|^2~~, \cr X \|\partial_{\pp} \partial_{\Omega}^{\beta} X v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j}-\gamma_{j}} X \le& C_3 \epsilon_{j} C_G^{j+1} X { (\beta + 4)! X \over (\delta_{j+1} \gamma_{j}^{12} X \rho_{j+1}^{\beta}) } X \bigl( 1 + { r_0 \over \rho_0} \bigr) \|\pp\|~~. \cr} X$$ Xand for $|{\bf \alpha}| \geq 2$, X$$ X \|\partial_{\pp}^{{\bf \alpha}} \partial_{\Omega}^{\beta} X v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j}-\gamma_{j}} \le X C_3 \epsilon_{j} C_G^{j+1} X { (|{\bf \alpha}| + \beta + 3)! X \over (\delta_{j+1} \gamma_{j}^{12} X \rho_{j+1}^{|{\bf \alpha}| + \beta -2}) }~~. X$$ XThe usual multi-index notation for Xderivatives is used in this estimate. X\item{(2)} For every $(\pp,\Omega) \in D(\NN_n;\rho_n/2)$, Xthe function $u_{n}$ satisfies X$$ X \| F(u_n(\pp,\Omega)) \|_{\sigma_{n}} X \le \|\pp\|^2 \epsilon_n~~. X$$ X X\PROOF The idea of this proposition is to construct X$v_{n-1}$ using Newton's method. XWe expect the solution to be small Xfar from the origin, so rather than trying Xto solve $F(u_{n-1}+v_{n-1}) = 0$, we study Xthe approximation X$\Pi_{n} F(u_{n-1}+v_{n-1}) = 0$, where X$\Pi_n$ is the orthogonal projection onto $\l2(B_n)$. XSince X$$ X \Pi_{n} F(u_{n-1}+v_{n-1}) \approx X \Pi_{n} \left[ F(u_{n-1}) + DF(u_{n-1}) v_{n-1} \right]~~, X$$ Xa better approximation of a solution to the problem Xis given by performing an iteration step based on XNewton's method. This entails inverting the Xlinearized operator X$ \Pi_{n}(DF(u_{n-1}))^{-1} \Pi_{n}, \ X = G_{\overline{B_{n}}}(u_{n-1}) $. This Green's function Xis not defined on all of $\NN_0$, but on the subset X$D(\NN_n,\rho_n)$ it is boundedly invertible. XOn this set we define the correction to $u_{n-1}$, X$$ X \tilde{v}_{n-1} = - \Pi_{n} X (DF(u_{n-1}))^{-1} \Pi_{n} F(u_{n-1}). X$$ X From the estimates of X\clm(Greens), $G_{\overline{B_{n}}}$ is analytic Xon $D(\NN_{n};\rho_{n})$, and satisfies X\equ(redGreen2). XFurthermore by $(n-1.1)(ii)$, we know that X$\|P F(u_{n-1}( \pp,\Omega))\|_{\sigma_{n-1}} X \le \|\pp\|^2 \epsilon_{n-1}$, Xon $D(\NN_{n-1},\rho_{n-1}/2)$. XThus, on $D(\NN_{n};\rho_{n}/2)$, X$G_{\overline{B_{n}}}$ and X$\Pi_{n}F(u_{n-1})$ are analytic, and X$$\eqalign{ X \|\tilde{v}_{n-1}\|_{\sigma_{n-1}-\gamma_{n-1}} \le& X \| G_{\overline{B_{n}}} \|_{\sigma_{n-1}-\gamma_{n-1}} X \| \Pi_{n}F(u_{n-1}) \|_{\sigma_{n-1}-\gamma_{n-1}} \cr X \le& \epsilon_{n-1}\|\pp\|^2 C_G^{n} / X (\delta_{n} \gamma_{n-1}^{12}) ~~, \cr} X$$ Xwhere we used \equ(redGreen2) to estimate X$\| G_{\overline{B_{n}}} X \|_{\sigma_{n-1}-\gamma_{n-1}}$. XOne has X X\CLAIM Lemma(vn) $\tilde{v}_{n-1}(\pp,\Omega;x)$ Xis analytic on X$D(\NN_{n}; \rho_{n})$ and for any $(\pp, \BGO)$ in Xthis domain it satisfies X$$ X \| \tilde{v}_{n-1}(\pp,\Omega;\cdot)\|_{ X \sigma_{n-1}-\gamma_{n-1}} X \le \epsilon_{n-1}\|\pp\|^2 X {{C_G^{n}}\over{ \delta_{n} \gamma_{n-1}^{12}}} ~~. X\EQ(vntilde) X$$ X X XWe now construct $v_{n-1}$ by smoothly Xextending $\tilde{v}_{n-1}(\pp,\Omega;x)$ Xto all of $\NN_0$, Xdoing this essentially by setting $\tilde{v}_{n-1}=0$ on X$\NN_{0} \backslash \NN_n$, Xhowever using some care in order to obtain in the Xlimit a $C^{\infty}$ function. We use the Xfollowing from the appendix of [C]. X X\CLAIM Lemma(chierchia) For every $R>0$ and for every Xcompact set $\Delta \subset \complex^d$, there exists Xa $\chi \in C^0(\complex^d) \cap C^{\infty}(\real^d): X\complex^d \to [0,1]$. $\chi$ has support in X$Y_R(\Delta) \equiv \cup_{\eta_0 \in \Delta} X\{ \eta \in \complex^d ; \|\eta - \eta_0 \| \le R \}$ Xand $\chi(\eta) = 1$ for every $\eta \in Y_{R/2}(\Delta)$. XFinally, for every positive integer $k$, X$$ X\sup_{\real^d} \| \partial_{\eta}^k \chi \| \le X{{|k| (|k|+2)!}\over{R^{|k|} }}~~. X$$ X XDefine $\chi_{n-1}$ to be a function satisfying Xthe hypotheses of the lemma with $d=3$, X$\Delta = \overline{\Eta_n}$, Xand $R= \rho_n/2$, and set $v_{n-1} = \chi_{n-1} X\tilde{v}_{n-1}$. X X X\CLAIM Lemma(vntwo) XWithin the domain of analyticity $D(\Eta_n;\rho_n/2)$ Xthe Cauchy estimate applied to \equ(vntilde) controls Xall derivatives X$\partial_{\Omega}^{\beta} \partial_{\pp}^{{\bf \alpha}} X v_{n-1}( \pp,\Omega;\cdot) $. XUsing \clm(chierchia), the estimate on all of $\Eta_0$ is, X$$\eqalign{ X \| \partial_{\Omega}^{\beta} v_{n-1}(\pp,\Omega;\cdot) X \|_{\sigma_{n-1}-\gamma_{n-1}} X \le& C_3 \epsilon_{n-1} C_G^{n} X { (\beta + 3)! \over X (\delta_{n} \gamma_{n-1}^{12} X \rho_{n+1}^{\beta}) } \|\pp\|^2~~, \cr X \|\partial_{\pp} \partial_{\Omega}^{\beta} X v_{n-1}(\pp,\Omega;\cdot)\|_{\sigma_{n-1}-\gamma_{n-1}} X \le& C_3 \epsilon_{n-1} C_G^{n} X { (\beta + 4)! X \over (\delta_{n} \gamma_{n-1}^{12} X \rho_{n+1}^{\beta}) } X \bigl( 1 + { r_0 \over \rho_0} \bigr) \|\pp\|~~. \cr} X$$ Xand for $|{\bf \alpha}| \geq 2$, X$$ X \|\partial_{\pp}^{{\bf \alpha}} \partial_{\Omega}^{\beta} X v_{n-1}(\pp,\Omega;\cdot)\|_{\sigma_{n-1}-\gamma_{n-1}} \le X C_3 \epsilon_{n-1} C_G^{n} X { (|{\bf \alpha}| + \beta + 3)! X \over (\delta_{n} \gamma_{n-1}^{12} X \rho_{n+1}^{|{\bf \alpha}| + \beta-2}) }~~. X$$ X X\clm(vn) and \clm(vntwo), together with the remark that Xfor $\dist(( \pp,\Omega),\NN_{n}) \le \half \rho_{n}$, X$v_{n-1} = \tilde{v}_{n-1}$, imply the first assertion of X\clm(newton). X XWe complete the proof of \clm(newton) Xby verifying that $u_{n} \equiv u_{n-1} + v_{n-1}$ Xis an approximate solution of the first bifurcation equation X$ F(u) = 0$. XNote that X$$\eqalign{ X F(u_n) =& F(u_{n-1}+v_{n-1}) = \{ (\11 - \Pi_{n}) F(u_{n}) \cr X &+ \Pi_n \left( F(u_{n-1}+v_{n-1}) - [F(u_{n-1}) + X DF(u_{n-1}) v_{n-1}] \right) \}~~, \cr} X\EQ(newton) X$$ Xwhere we used the fact that $v_{n-1}$ was constructed so that X\hfill \break $\Pi_{n}[F(u_{n-1})+ DF(u_{n-1}) v_{n-1}] = 0$, X(on $D(\NN_n;\rho_n)$.) X XWe can bound the various terms in \equ(newton) by using Xthe following observations: X\item{(i)} From hypotheses ${\bf H2}$ and ${\bf H3}$ of Section 6, Xthere exist positive Xconstants, $C_W$ and $\sigma$, (with $\sigma < \sigma_0$) Xsuch that the Xoperator norm satisfies $\| D_u W(u)\|_{\sigma -\gamma} X\le C_W/\gamma^2$ and X$\| D^2_u W(u)[v,w] \|_{\sigma -\gamma} X \le C_W/\gamma^3 \|v\|_\gs \|w\|_{\gs-\ga}$, X for all $u$ with $\| u \|_{\sigma} \le 1$. X\item{(ii)} Using the explicit form of $V(\BGO)$, we see Xthat $|V(\Omega)(j,k)| \le (\Omega_m^2 + 1)(j^2+k^2)$, thus X$$ X\|V(\Omega) v\|_{\sigma - \gamma} \le X(\Omega_m^2+1) \sup_{j,k} (j^2+k^2)e^{-\ga(|j|+|k|)} X\|v\|_{\sigma} X\le {{4 (\Omega_m^2 + 1)}\over{\ga^2}} \|v\|_{\sigma}~. X$$ Xwhere $\Omega_m = $ the maximum value of $\Omega = X\omega_1 + \sqrt{ r_0^2 + \rho_0^2}$. X\item{(iii)} By \clm(cutoff) X$\|(\11 - \Pi_{n})w\|_{\sigma - \ga} X \le e^{-\ga L_{n}} \|w\|_{\sigma}~~.$ X XPoint $(iii)$ implies that X$$ X\|(\11 - \Pi_{n})F(u_{n})\|_{\sigma_{n}} \le Xe^{-\ga_{n-1} L_{n}} \|F(u_{n})\|_{ X\sigma_{n-1}-2\gamma_{n-1}}~~. X$$ XThe fundamental theorem of calculus implies that X$$\eqalign{ X \|F(u_{n})\|_{\sigma_{n-1}-2\gamma_{n-1}} X\le& ||F(u_{n-1})||_{\sigma_{n-1}-2\gamma_{n-1}} \cr X &+ \| \int_0^1 (DF(u_{n-1}+tv_{n-1}) v_{n-1})dt X \|_{\sigma_{n-1}-2\gamma_{n-1}} \cr} X$$ XBy $(n-1.1)$ the first term is bounded Xby $\epsilon_{n-1} \|\pp\|^2$. XPoints $(i)$ and $(ii)$ imply that X$$ X \| \int_0^1 (DF(u_{n-1}+tv_{n-1}) v_{n-1})dt \ X \|_{\sigma_{n-1}-2\gamma_{n-1}} \le X \left[ {{C_W +4(\Omega_m^2 +1)} X \over{\gamma_{n-1}^2}} \right] X \|v_{n-1}\|_{\sigma_{n-1}-\gamma_{n-1}}~~, X$$ Xcompleting the estimate of the truncation error. X XWe now estimate $[F(u_{n-1}+v_{n-1}) X- F(u_{n-1}) - DF(u_{n-1}) Xv_{n-1}]$, Xusing the fundamental theorem of calculus to rewrite it as Xas \relax $\int_0^1 \int_0^t D^2F(u_{n-1}+sv_{n-1}) Xv_{n-1} v_{n-1} ds dt X$. This is quadratic in $v_{n-1}$. X{}From the first of the observations above, X$$ X \| \int_0^1 \int_0^t D^2F(u_{n-1}+sv_{n-1}) X v_{n-1} v_{n-1} ds dt X \|_{\sigma_{n-1}-2\ga_{n-1}} \le X {C_W \over \ga_{n-1}^3} X \|v_{n-1}\|_{\sigma_{n-1} - \ga_{n-1}}^2~~. X$$ XIn deriving Xthis estimate we used the fact that X$D_u^2 F(u_{n-1}) = D_u^2 W(u_{n-1})$. XCombining these three estimates, we obtain the lemma: X X\CLAIM Lemma(L0estimate) The exists a constant $C$, Xdepending on $C_W$ and $\Omega_m$, Xsuch that if X$$ X\epsilon_{n-1} X{{C C_G^{n+1} }\over{\delta_n \gamma_{n-1}^{14} }} X\left[ e^{-\gamma_{n-1} L_n }+ X{{r_0^2 \epsilon_{n-1} C_G^{n+1} }\over{\delta_n X\gamma_{n-1}^{13} }} \right] \le \epsilon_{n} X$$ Xthen X$$ X \|F(u_{n})\|_{\sigma_{n}} X \le \|\pp\|^2 \epsilon_n~~, X$$ Xfor $(\rho, \Omega) \in D(\NN_n, \rho_n/2)$. X X\PROOF The estimates above imply that the truncation Xerror is X$$\eqalign{ X \|(\11 - \Pi_n) F(u_n) \|_{\sigma_n} X &\le e^{-\gamma_{n-1} L_n} X \| F(u_n) \|_{\sigma_{n-1} X - 2\gamma_{n-1} } \cr X &\le \|\pp \|^2 \epsilon_{n-1} X e^{-\gamma_{n-1} L_n} X \left[1 + {{(C_W + 4(\Omega_m^2+1) X C_G^{n} }\over{\delta_n X \gamma_{n-1}^{14} }} \right] \cr X &\le \|\pp \|^2 \epsilon_{n-1} X e^{-\gamma_{n-1} L_n} X {{C C_G^{n} }\over X {\delta_n \gamma_{n-1}^{14} }} ~~.\cr X} X$$ X XThe contribution from the quadratic error is bounded by X$$ X \| \Pi_n \left\{ F(u_{n-1}+v_{n-1})-[F(u_{n-1}) X +DF(u_{n-1}) v_{n-1}]\right\} \|_{\sigma_n} \le X {{ \epsilon_{n-1}^2 \|\pp \|^4 C_W C_G^{2n}} X \over{ \delta_n^2 \gamma_{n-1}^{27} }}~~. X$$ X XThe proposition follows if the sum of Xthese two terms to be less than $\epsilon_n$. XCombining these two estimates and using Xthe fact that $\| \pp \|^2 \le r_0^2$, this follows from Xthe hypothesis of \clm(L0estimate). X X X X\SUBSECTION Verification of $(n.2)$ X XWe continue now by showing how one constructs the set X$\NN_{n+1}$, described in the induction hypothesis X$(n.2)$. We will Xconstruct $\NN_{n+1}$ using the properties of $u_n$ given Xby $(n.1)$. XRecall that we constructed X$\NN_0$ in Section 2.3. We denote by $C$ a constant Xindependent of $n$ and of the inductive constants. X The following proposition not only implies X$(n.2)$, but includes some other useful information as well. X X X X\CLAIM Proposition(parameters) Assume that the following Xrelationships hold between the inductive constants. X$$\eqalign{ X {Cd \over 2^{\tau + 1}} < L_n^{(1-\tau \beta)}, X & \quad \tau \beta < 1 \cr X\beta(1+ c_0 \log(2) ) & < 1 \cr X {C_W L_0 r_0^2 \over X {\overline \gs}^3 d_0 \rho_0} <& \ 1 X < \BGO_{min} L_n^2 \cr X {16 C_W \over {\overline \gs}^2} r_0 X <& \ d_s \cr X (1 + {L_0 \over d_0} r_0) <& \ C \cr} X\EQ(indconstconds) X$$ XThen there exists a closed set $\Eta_{n+1} X\subseteq \Eta_n$ such that: X\item{(a)} If $(\pp,\BGO) \in \Eta_{n+1}$, then Xany two singular sites in $B_{n}^c$, which Xare not in the same singular region, are separated by Xa distance of at least $2 \ell_{n+1}$. X\item{(b)} If $S$ is a singular region Xin $B_{n+1} \backslash B_{n}$ and $(\pp, \BGO ) X\in D(\NN_{n+1},\rho_{n+1}) $ then X$$\eqalign{ X &{\rm dist}({\rm spec}(H_{C_{\ell_{n+1}}(S)} X (\pp, \BGO; u_n)),0) > \delta_{n+1} \cr X {\rm and}~~~& \cr X &{\rm dist}({\rm spec}(H_{S} X (\pp, \BGO ; u_n)),0) > \delta_{n+1} \cr} X$$ XIf the inductive constants satisfy X$$\eqalign{ X \sum_{j=0}^n& \ C {d \over L_j^\tau } X (1 + 2r_0^2 L_{j+1}) < r_0^2 \cr X \sum_{j=0}^n& \ CL_j (1 + r_0^2 L_{j+1}) X \times \ell_{j+1}^2 X \times {\gd_{j+1} \over L_j^2 } X < r_0^2 \cr} X\EQ(4.4) X$$ Xthen $\Eta_{n+1}$ has positive measure. If Xadditionally, X$$ X C\bigl( {1 \over d_s} + {L_0 \over d_0} X (1 + {r_0 \over \rho_0}) \bigr) X < {\Kappa L_n^2 \over 4} X\EQ(4.5) X$$ Xand for some $\mu > 0$, X$$\eqalign{ X \sum_{j=0}^n& \ X \sqrt{ {Cd \over \Kappa L_j^\tau } X (1 + r_0^2 L_{j+1}) } < r_0^{1+\mu} \cr X \sum_{j=0}^n& \ \sqrt{ C L_j (1 + r_0^2L_{j+1}) X \times \ell_{j+1}^2 X \times {\gd_{j+1} \over \Kappa L_j^2} } X < r_0^{1+\mu} \cr} X\EQ(4.6) X$$ Xthen the set $\NN_{n+1}$ satisfies the Xfollowing intersection condition: X\item{(c)} Let $\BGO(\pp) =\omega_1 + \Kappa X\| \pp \|^2(1+C(\|\pp \|)) $, be a surface with Xnon-degenerate quadratic contact at $\pp =0$. XIf $|\Kappa| > L_0^{-\nu}$, Xand $|C(\cdot )|_{C^{1}} \le 1/2$, Xthen the set $ \{ 0 \leq r < r_0 ; \ r = \|\pp\|, \ X(\pp, \BGO(\pp) ) \in \NN_{n+1} \}$ Xhas measure greater than or equal to X$r_0(1 - Cr_0^\mu)$. X X\REMARK In fact the set $\NN_{n+1}$ is invariant under Xthe rotations $\pp \to T_{\xi} \pp$. This invariance is a Xconsequence of the invariance of the Xspectrum of the linearized Xoperator $H_{B_{n+1}}(u)$, under the rotations $u \to XT_{\xi} u$. X X\CLAIM Lemma(invariance) The spectrum of the operator X$H_{E_{n+1}}(T_{\xi} u)$ is independent of $\xi$. X X\PROOF The properties of the nonlinear function Xin \equ(NLlattice) are that $T_{\xi} F(u) = F(T_{\xi} u)$, Xand furthermore $T_{\xi}$ commutes with orthogonal Xprojection onto $\ell^2(E)$ for any $E \subset X\zsquared$. Differentiating with respect to $u$ we find X$T_{\xi} DF(u) v = DF(T_{\xi} u) T_{\xi}v$, thus X$DF(u)$ and $DF(T_\xi u)$ are unitarily equivalent, and Xthe result follows. X X\endproof X X XPart $(a)$ of \clm(parameters) can be proven Xwithout restricting $\pp$--we Xneed only place conditions on $\Omega$. X{}From the diophantine condition on $\omega_1$ Xthere is an integer $N_0$ such that for the Xinitial induction steps $0 \leq n \leq N_0$, Xthe separation condition $(a)$ is satisfied Xfor all $(\pp,\BGO) \in \Eta_0$. This is the Xresult of the next lemma, whose hypotheses form Xa subset of those of \clm(parameters). X X\CLAIM Lemma(separation) Assume that X$\tau \beta < 1$, $\beta(1+c_0 \log(2) ) <1$, Xand that $d > L_0^{\tau \beta} X(1/L_0 + r_0^2 L_0^\beta)$. There exists a Xconstant $c_0 = c_0(\tau,\beta)$ such that, Xif $L_0$ is sufficiently large and X$N_0 = c_0 \log (L_0)$, then for all $n \le N_0$ Xand $(\pp,\BGO) \in \Eta_n$, if X$x = (k,j)$, $x' = (k',j')$ are a pair of singular Xsites in $B_n^c$ which are not in the same singular Xregion, they are well separated; $i.e.$ X$$ X\dist (x,x') \ge 2 \ell_{n+1}~~. X$$ X X X X\PROOF Assume that $|x - x'| < 2 \ell_{n+1}$. We will derive a Xcontradiction. If $x$ is a singular site then X$|j^2 \Omega^2 - \omega_k^2| < d_s$. Factoring the left Xhand side of this inequality and assuming that $j$ and X$k$ are non-negative this implies X$|j \Omega - \omega_k| < d_s/|j\Omega + \omega_k| X\le C(\Omega_m) d_s /(|j|+|k|)$, for some constant $C(\Omega_m)$. XSimilar estimates hold for $|j' \Omega - \omega_{k'}|$, Xand for the cases when $j$ and $j'$ are negative. XWe now use the asymptotics of the Sturm-Liouville operator. XFor the case of Dirichlet boundary conditions there exists Xa constant $C_{g_1}$ such that $|\omega_k - k| < C_{g_1}/k$, for all $k \ge 1$. X (For a review of the properties of these Xeigenvalues see, for example [PT].) For periodic boundary Xconditions, if $k = 2m$ or $k = 2m -1$, $m \ge 1$, then X$|\omega_k - 2m| < C_{g_1}/m$. XNote that since $x$ and $x'$ are Xin $B_n^c$, there exists a constant $C(\Omega_m)$ such that X$\min (|k|,|k'|) \ge C(\Omega_m) L_n$. In the case of XDirichlet boundary conditions we have X$$\eqalign{ X |(j-j') \Omega - (k-k')| \ =& \ X |(j-j') \Omega - (\omega_k - \omega_{k'}) X + (\omega_k - \omega_{k'}) - (k-k')| \cr X \le& \ {{2 C(\Omega_m) d_s}\over{L_n}} + X {{2 C(\Omega_m) C_{g_u}}\over{L_n}} \ \le \ X {{C(g_1,\Omega)}\over{L_n}}~~. \cr X} X\EQ(upperbnd) X$$ XSince $x \in B_n^c$ we have $|j|+|k| \ge L_n$, with a Xsimilar estimate for $x'$. We are assuming that X$\dist (x,x') = |j-j'|+|k-k'| \le 2 \ell_{n+1}$ $= X2 L_{n+1}^{\beta} = 2 (2^{n+1} L_0)^{\beta}$. But X$n \le N_0 = c_0 \log (L_0)$ so X$\dist (x,x') \le 2(2^{c_0 \log(L_0) }L_0)^{\beta} X= 2(L_0^{\beta(1+c_0 \log 2)}) < L_0$, if X$ \beta(1+c_0 \log 2) < 1$ and $L_0$ is sufficiently Xlarge. The principal frequency $\omega_1$ is X$(d_0,L_0)$-nonresonant, thus it satisfies a Xfinite diophantine condition over the lattice Xpoints within $B_0$. Hence as long as X$2 \ell_{n+1} < L_0$ and $|\BGO-\omega_1| < r_0^2$, X$(x - x') \in B_0$ and the components satisfy X$$\eqalign{ X |(j-j') \Omega - (k-k')| \ \ge & X \ |(j-j') \omega_1 - (k-k')| X - |(j-j')(\Omega - \omega_1)| \cr X \ge & \ {{d}\over{(2 \ell_{n+1})^{\tau}}} - X 2 \ell_{n+1} r_0^2~~. \cr X} X\EQ(lowerbnd) X$$ XBy the hypotheses of the lemma, $4 \ell_{n+1} r_0^2 X< \half {{d}\over{(2 \ell_{n+1})^{\tau}}}$, so we find X$|(j-j') \Omega - (k-k')| \ge X \half {{d}\over{(2 \ell_{n+1})^{\tau}}}$. Combining this with X\equ(upperbnd) gives X$$ X \half {{d}\over{(2 \ell_{n+1})^{\tau}}} \le X {{C(g_1,\Omega_m)}\over{L_n}} X$$ Xor, using $\ell_{n+1} = L_{n+1}^{\beta}$, X$L_{n+1}^{(1-\tau \beta)} \le 2^{\tau(1+\beta)} C(g_1,\Omega)/d$. XIf $\tau \beta < 1$, and $L_0$ is sufficiently Xlarge, this inequality Xcannot be true and the lemma follows. X X\endproof X XIn the case of periodic boundary conditions, the argument Xneeds to be modified only slightly to accound for the different Xasymptotics of the eigenvalues. The estimate in X\equ(upperbnd) is replaced by X$$ X |(j-j') \Omega - 2([k/2]-[k'/2])| \le X {{C(g_1,\Omega_m)}\over{L_n}}~~, X\EQ(pupperbnd) X$$ Xwhere $[\cdot]$ denotes the integer part. Combining Xthis estimate with the lower bound coming from Xthe fact that $\omega_1$ is $(d_0, L_0)$ Xnonresonant, we find that $\dist (x,x')$ Xcannot be less than $2 \ell_{n+1}$ except in one special Xcase. This special case occurs when $k=2m$ Xand $k' = 2m-1$ or vice-versa. XThen \equ(pupperbnd) becomes X$|(j-j') \Omega| \le C(g_1,\Omega_m)/L_n$. XBut if, in addition, $j=j'$ we cannot apply the diophantine Xestimate to obtain the contradiction. Thus, if X$(j,k)$ and $(j',k')$ are any pair of singular sites with either X$j \not= j'$ or $\{ k,k' \} \not= \{ 2m, 2m-1 \}$, then they are Xseparated by a distance of at least $2 \ell_{n+1}$. X X\REMARK As a corollary of the proof of this Xlemma, we see that in the case of Dirichlet Xboundary conditions, the singular Xregions consist of isolated singular sites, while in Xthe case of periodic boundary conditions the singular Xregions consist either of isolated sites, Xor pairs of adjacent sites, Xwith $j=j'$, and $\{k,k'\} = \{2m, 2m+1 \}$. X X XSingular sites are not necessarily well separated Xfor all $(\pp,\BGO) \in \Eta_0$, for induction steps X$n > N_0$. In order to insure that part $(a)$ of X\clm(parameters) will hold when $n > N_0$, we delete Xfrom $(\omega_1 - r_0^2, \omega_1 + r_0^2)$ Xthose values of $\Omega$ for which it would Xfail to be true. For this purpose we Xuse the following elementary result in Diophantine Xapproximation. X X\CLAIM Lemma(excise) For any bounded interval Xof the real line, $I$, Xof length $|I|$, X$$\eqalign{ X \meas \{ \Omega \in I ;& |j \Omega -k| < X d(|j|+|k|)^{-\tau}, ~~{\rm for}~~ X L_n \le |j|+|k| \le 4 L_{n+1} \} \cr X &\le {Cd \over L_n^\tau} ( 1 + |I|L_n)~~. \cr} X\EQ(dio) X$$ X XFor $n > N_0$ define $\NN_{n+1}^{(1)}$ Xinductively by deleting Xfrom $\NN_n$ all points $(\pp ,\Omega)$ Xsuch that $|j \Omega -k| < X d (|j|+|k|)^{-\tau}$, Xfor some $(j,k)$ with $ L_n \le |j|+|k| \le 4 L_{n+1}$. XThis will delete finitely many open disks from X$\NN_n^{(1)}$. XWe assert that for $\BGO \in \Eta_{n+1}^{(1)}$, Xthen any two singular sites $x,x' \in B_n^c$ which are Xnot in the same singular region satisfy X${\rm dist}(x,x') \geq 2\ell_{n+1}$. XIndeed the argument used in \clm(separation) Xcan be modified to prove this degree of separation. XThe one change is that in the Xestimate leading to \equ(lowerbnd), if X$\Omega \in \Eta_{n+1}^{(1)}$, Xthen $|j \Omega -k| \ge Xd (|j|+|k|)^{-\tau}$ for all $0 < |j|+|k| \le 4 L_{n+1}$. XThus, \equ(lowerbnd) is replaced by X$$ X |(j-j')\Omega - (k-k')| \ge X {{d}\over{(2\ell_n)^{\tau} }}~~. X\EQ(lowerbndtwo) X$$ XThus we have obtained the following result. X X\CLAIM Lemma(separationtwo) For any $n \ge 0$, X$(\pp,\Omega) \in \NN_{n+1}^{(1)}$, and any Xpair of singular sites in $B_n^c$, which are not Xin the same singular region, X$$ X\dist(x,x') \ge 2 \ell_{n+1}~~. X$$ X XSince we will define $\NN_{n+1}$ to be a subset Xof $\NN_{n+1}^{(1)}$, this lemma implies part $(a)$ Xof \clm(parameters). X XNext consider part $(b)$ of \clm(parameters), where Xwe must address the dependence of the local hamiltonians Xand their spectra upon the parameters $(\pp,\BGO)$. XIn order to belong to a singular region $S$, a lattice Xpoint $x=(j,k)$ must satisfy X$(\omega_1 -r^2_0)|k| - C_0 \leq |j| X \leq (\omega_1 + r_0^2)|k| + C_0$. Furthermore, Xin either of the above problems a singular region Xconsists of either one site alone, or a pair of adjacent Xsites of the form $x_1=(j,2m-1), \ x_2=(j,2m)$. Suppose X$S \subseteq B_{n+1} \backslash B_n$ is a set of this type Xso it has the possibility for being a singular region Xfor some $(\pp,\BGO) \in \Eta_0$. The local hamiltonian Xfor the surrounding neighborhood $C_{\ell_{n+1}}(S)$ is X$$ X H_{C_{\ell_{n+1}}(S)}(\pp,\BGO;u_n) = X \bigl( D_u W(\phi(\pp)+u_n) X + V(\BGO) \bigr)|_{C_{\ell_{n+1}}(S)}. X$$ XThis matrix has at most $4 \ell_{n+1}^2$ many eigenvalues, Xwhich we list in order; X$e_i(\pp ,\BGO )$, for $i = 1, 2, \dots $. XIn this subsection we will study the behavior of the Xeigenvalues of $H_S$ and $H_{C_{\ell_{n+1}}(S)}$ with Xrespect to parameters. The goal is to control the subsets Xof $\Eta_0$ on which either $H_S$ or X$H_{C_{\ell_{n+1}}(S)}$ has a small eigenvalue, of size X$|e_i(\pp,\BGO)| < \gd_{n+1}$. (To save space, unless Xthe meaning is unclear we will write $H_{C_{\ell_{n+1}}(S)}$ Xas $H_{C(S)}$.) X X\CLAIM Lemma(4.monotone) For $S$ fixed, each eigenvalue X$e_i(\pp,\BGO)$ of $H_{C(S)}$ is continuous in X$(\pp,\BGO) \in \Eta_0$, differentiable almost everywhere, Xand is strictly monotone decreasing in $\BGO$ if X$S \subseteq \{ (j,k) \in \ZZ^+ \times \ZZ, k > 0 \}$. X(They are strictly increasing in the opposite case X$S \subseteq \{ (j,k) \in \ZZ^+ \times \ZZ, k < 0 \}$.) X X\noindent{\bf Proof:} XFor $S \subseteq \{ (j,k) \in \ZZ^+ \times \ZZ, Xk > 0 \} \cap B_n^c$, then as long as X$\ell_{n+1} < L_n/(2\omega_1) + C_0$, every lattice site X$x = (j,k) \in C(S)$ satisfies $k>L_n/2$, that is, all of X$C(S)$ is contained within the positive $k$ quadrant. XThe local hamiltonian $H_{C(S)}$ is monotone decreasing Xin $\BGO$. Indeed, X$$ X \partial_\BGO H_{C(S)} (\pp,\BGO;u_n) = X \bigl( \partial_\BGO V(\BGO) + D_u^2 W(\phi(\pp) + u_n) X (\partial_\BGO u_n) \bigr)_{C(S)}, X$$ Xand since for every $x=(j,k) \in C(S), \ |k| > L_n/2$, X$$ X \bigl( \partial_\BGO V(\BGO) \bigr)_{C(S)} (x,y) X = -2\BGO (k^2 \gd(x,y)) X < - \BGO_{min} L_n^2/2 (\gd(x,y)). X$$ XThe second term is bounded by X$$\eqalign{ X\|D_u^2 W(\phi(\pp)+u_n)[\partial_\bgo u_n,w] \|_0& X \leq {C_W \over (\gs_n - \ga_n)^3 } X \|\partial_\BGO u_n \|_{\gs_n-\ga_n} \Vert w\Vert_0,\cr X& \le {{8 C_W L_0 r_0^2}\over{\overline{\sigma}^3 d_0 \rho_0}} X\Vert w \Vert_0 \cr} X\EQ(4.bound) X$$ XFor $(C_W L_0 r_0^2 / ({\overline \gs}^3 d_0 \rho_0)) X\leq 1 \leq \BGO_{min} L_n^2$, the quadratic form X$\langle \psi, H_{C(S)} \psi \rangle$ is monotone Xdecreasing in $\BGO$, and thus for fixed $\pp$ the Xeigenvalues $e_i(\pp,\BGO)$ are monotone Xdecreasing in $\BGO$. X XNote that if there are no quadratic terms in the Xnonlinearity $W$, then X$$ X \| D_u^2 W[\partial_\BGO u_n, w] \|_0 X \leq {8 C_W \over {\overline \gs}^3} X {L_0 r_0^3 \over d_0 \rho_0 } \|w \|, X\EQ(4.cubic) X$$ Xa better estimate than \equ(4.bound). X XThe eigenvalues vary continuously with respect to X$(\pp,\BGO) \in \Eta_0$, however they are not Xnecessarily smooth in $\Eta_0$. Examples of Rellich, X[R], show that even a two by two matrix depending Xanalytically on several parameters may not have Xdifferentiable eigenvalues. This lack of smoothness Xoccurs at points of collision between eigenvalues. XHowever on sets where either all eigenvalues are Xdistinct, or the multiplicity of all eigenvalues Xdoes not change, it is possible to choose eigenvalues and Xeigenvectors to be smooth functions of $(\pp,\BGO)$, [K]. XThese sets of constancy for eigenvalue multiplicity of X$H_{C(S)}$ are semianalytic sets, thus $e_i(\pp,\BGO)$ Xare differentiable almost everywhere on $\Eta_0$, and Xthe eigenvectors can be chosen differentiably Xalmost everywhere as well. \endproof X XLet us fix a set $S$ which has the possibility of being Xsingular for some $(\pp,\BGO) \in \Eta_0$, and Xassume that for $x \in C(S), k > 0$. We are concerned Xwith the location within $\Eta_0$ of the zero sets X$Z_i \equiv \{ (\pp,\BGO) \in \Eta_0; Xe_i(\pp,\BGO) = 0 \} $. XFor $\pp = 0, e_i(0,\BGO) =- (\BGO^2 m^2 - \omega_\ell^2)$ Xfor some $(\ell,m) \in C(S)$, so that it vanishes only for X$\BGO = \omega_\ell / m$. For each X$\pp \in \{ \|\pp\| < r_0 \}$, $e_i(\pp,\BGO)$ is Xmonotone decreasing, thus $Z_i \cap \{ \pp = {\rm const} \}$ Xconsists of at most one point. By the continuity and Xstrict monotonicity of $e_i(\pp,\BGO)$, $Z_i$ is given by Xa surface $(\pp, \BGO_i(\pp))$, where $\BGO_i(\pp)$ is Xcontinuous but not necessarily smooth, and X$\BGO_i(0) = \omega_\ell / m$ for some $(\ell,m) \in C(S)$. XUsing \clm(invariance), the sets X$Z_i$ are invariant under $T_\xi$. X XSmall neighborhoods of these sets $Z_i$ contain all the Xparameter values in $\Eta_0$ for which the local Xhamiltonian $H_{C(S)}$ has a small eigenvalue. X X\CLAIM Lemma(4.smallev) XLet $e_i(\pp,\BGO)$ be an eigenvalue of $H_{C(S)}$ such Xthat $|e_i(\pp,\BGO)| < \gd_{n+1}$. Then there exists X$\BGO_1$ with $|\BGO - \BGO_1| < C \gd_{n+1} / L_n^2$ Xsuch that $e_i(\pp,\BGO_1) = 0$. X XThus by excising all $(\pp,\BGO) \in \Eta_0$ with X${\rm dist}(\BGO,\BGO_1) < C \gd_{n+1} /L_n^2$, for Xany $(\pp,\BGO_1) \in Z_i, \ Xi= 1, 2, \dots$, the remaining parameters Xavoid small eigenvalues for $H_{C(S)}$. If this excision Xis performed for all regions $S \subseteq B_{n+1} X\backslash B_n$ which are potentially singular regions, Xthe remaining set of parameters will satisfy X\clm(parameters), $(b)$. Our goal will be to estimate Xthe measure, and the geometry of the remaining parameter Xregion. X X\PROOF XFor fixed $\pp$, $e_i(\pp,\BGO)$ is a decreasing function Xof $\BGO$, thus it is differentiable almost everywhere, and Xby Fatou's lemma X$$ X e_i(\pp,\BGO_1) - e_i(\pp,\BGO_2) \geq X \int_{\BGO_1}^{\BGO_2} \ X - \partial_\BGO e_i(\pp,\omega) \ d\omega X$$ Xfor any $\BGO_1 \leq \BGO_2$. Given $e_i(\pp,\BGO)$ Xand associated eigenvector $\psi_i$, X$$ X e_i = \langle \psi_i, H_{C(S)} \psi_i \rangle, X$$ Xthus where $e_i$ and $\psi_i$ are differentiable, X$$\eqalign{ X \partial_\BGO e_i =& \langle \psi_i, X \partial_\BGO H_{C(S)} \psi_i \rangle X + 2\langle \partial_\BGO \psi_i, X H_{C(S)} \psi_i \rangle \cr X =& \langle \psi_i, X \partial_\BGO H_{C(S)} \psi_i \rangle X + 2 e_i \langle \partial_\BGO \psi_i, X \psi_i \rangle \cr}. X$$ XThe eigenvectors are normalized, so that X$$\eqalign{ X 1 =& |\psi_i|^2 = \langle \psi_i, \psi_i \rangle \cr X 0 =& 2 \langle \partial_\BGO \psi_i, X \psi_i \rangle, \cr} X$$ Xhence X$$ X \partial_\BGO e_i = \langle \psi_i, X \partial_\BGO H_{C(S)} \psi_i \rangle . X\EQ(feynman-hellman) X$$ XThis is known as the Feynman-Hellman formula. XWe can estimate the derivative, X$$\eqalign{ X -\partial_\BGO e_i \geq& \ 2 \BGO_{min} L_n^2 X - \| D_u^2 W[\partial_\BGO u_n, \psi_i] \|_0 \cr X \geq& \ \BGO_{min} L_n^2, \cr} X\EQ(FH2) X$$ Xas long as $8 C_W L_0 r_0^2 / d_0 \rho_0 \overline{\sigma}^3 X< \BGO_{min} L_n^2$. XNow suppose that X$0 \leq e_i(\pp,\BGO_1) \leq \gd_{n+1}.$ Then X$$\eqalign{ X (\BGO - \BGO_1) (\BGO_{min}/2) L_n^2 \leq& X \int_{\BGO_1}^{\BGO} X -\partial_\BGO e_i(\pp,\omega) \ d\omega, \cr X \leq& \ e_i(\pp,\BGO_1) - e_i(\pp,\BGO) \ X \leq \ \gd_{n+1} - e_i(\pp,\BGO), \cr} X$$ Xthus $e_i$ must vanish for a nearby value $\BGO = \BGO_2$, Xand furthermore $|\BGO_2 - \BGO_1| < C \gd_{n+1} / L_n^2$. X X\endproof X XTo obtain the subset $\Eta_{n+1}^{(2)} X\subseteq \Eta_n \subseteq \Eta_0$ Xon which all local hamiltonians avoid small Xeigenvalues, delete from $\Eta_n^{(2)}$ Xall points $(\pp,\BGO)$ Xsuch that there is some $S \subseteq \{ (j,k) \in B_{n+1} X\backslash B_n; \ (\omega_1 - r_0^2)|k| - C_0 < j < X(\omega_1 + r_0^2)|k| + C_0 \}$, Xand some $(\pp,\BGO_1) \in Z_i$ Xa zero set for an eigenvalue of the operators $H_S$ or X$H_{C_{\ell_{n+1}}(S)}$ such that X$|\BGO_1 - \BGO| < C\gd_{n+1} /L_n^2$. XThe remaining parameters Xin $\Eta_{n+1}^{(2)}$ will now satisfy X$$ X |e_i(\pp,\BGO)| > 2 \gd_{n+1}. X$$ X XThese lower bounds on the size of eigenvalues are now Xextended to the complex domain X$D(\NN_{n+1}, \rho_{n+1})$. XFor $(\pp,\BGO) \in D(\NN_{n+1}, \rho_{n+1})$, Xthere are real $(\pp_1,\BGO_1) \in \NN_{n+1}$ Xwith $\sqrt{ \|\pp - \pp_1\|^2 + |\BGO - \BGO_1|^2 } < X\rho_{n+1}$. Compute the operator norm of the Xdifference of the local hamiltonians, X$(H_{C(S)}(\pp,\BGO;u_n) X - H_{C(S)}(\pp_1,\BGO_1;u_n))$ ; X$$\eqalign{ X \| H_{C(S)} &(\pp,\BGO;u_n) X - H_{C(S)}(\pp_1,\BGO_1;u_n) \|_0 \cr X \leq& \ \| (V(\BGO) - V(\BGO_1)){C(S)} \|_0 + X \| (D_u W(\phi(\pp) + u_n) X - D_u W(\phi(\pp_1) + u_n))_{C(S)} \|_0 \cr X \leq& \ CL_{n+1}^2 \rho_{n+1} + X {C_W \over {\overline \gs}^3 } X \bigl( \|\partial_\pp (\phi + u_n)\|_ X {\gs_n-\ga_n} \|\pp-\pp_1\| X + \|\partial_\BGO u_n\|_{\gs_n-\ga_n} X |\BGO - \BGO_1| \bigr) \cr X \leq& \ CL_{n+1}^2 \rho_{n+1} X + {C_W \over {\overline \gs}^3} X \bigl( 1 + {L_0 \over d_0}(1 + {r_0 \over \rho_0}) X + { L_0 r_0^2 \over d_0 \rho_0 } \bigr) X \rho_{n+1} \cr X \leq& \ \gd_{n+1} \cr} X$$ Xas long as both X$$\eqalign{ X C &L_{n+1} \rho_{n+1} < \gd_{n+1}/ 2, \cr X C(1 + &(L_0/d_0)(1 + (r_0/\rho_0))\rho_{n+1} X < \gd_{n+1}. \cr} X\EQ("after lemma 4.12") X$$ X XThe remaining parameters in $D(\NN_{n+1}, \rho_{n+1})$ Xwill now satisfy $(b)$ of \clm(parameters). XWe can estimate the total measure of the region $\Eta_0$ that Xis excised by this process. The number of possible singular Xregions $S$ is bounded by $CL_n(1+r_0^2L_{n+1})$. For each Xregion $S$ and surrounding neighborhood $C_{\ell_{n+1}}(S)$ Xwe excise $C\gd_{n+1}/L_n^2$ of zero sets $Z_i$. XThere are one or two small eigenvalues for $S$ and at most X$4 \ell_{n+1}^2$ for $C_{\ell_{n+1}}(S)$. XEach of these neighborhoods is invariant under $T_\xi$, thus Xthe total measure of $\Eta_0$ that is excised is bounded Xby X$$ X (C L_n(1 + r_0^2 L_{n+1}) \ell_{n+1}^2 ) X ( \gd_{n+1} / L_n^2 )\pi r_0^2. X\EQ(4.n2b) X$$ XDefine $\Eta_{n+1} \equiv \Eta_{n+1}^{(1)} X\cap \Eta_{n+1}^{(2)}$, the good parameter region for the X$(n+1)^{st}$ induction step. Conditions \equ(4.4) Xinsure that the sum over $n$ of the measure of Xthese excisions will be small when compared to X${\rm meas} \ \Eta_0 = 2\pi r_0^4$. X XThis is still not enough to prove statement $(c)$ of X\clm(parameters), which is the guarantee that a smooth Xsurface $(\pp,\BGO(\pp))$ with nonzero curvature will Xintersect $\Eta_{n+1}$ for a set of $\pp$ of large Xmeasure. This will rely on more detailed knowledge of the Xgeometry of the sets $Z_i$. X X X\CLAIM Lemma(zi-cones) XIf $L_0 r_0^2 / d_0\rho_0 < c_0$, then there is a constant X$C_1$ such that the set $Z_i$ lies between the cones X$$ X {\omega_\ell \over m} \pm {C_1 \over L_n^2} \|\pp\| X$$ Xfor some $(\ell , m) \in C_{\ell_{n+1}}(S)$. X X X\PROOF We have chosen $e_i(\pp ,\BGO)$ Xso that $e_i(0,\omega_\ell / m) = 0$ Xfor some $x \in (\ell , m) \in C(S)$. XThe operator $H_{C(S)}$ is Xmonotone decreasing along any line $(\pp (s),\BGO (s)) = X(s\pp_0, \omega_\ell / m + s C_1 / L_n^2) \in \Eta_0$. XIndeed, using the arguments of \clm(4.smallev), we have X$$ X {d \over ds} H_{C(S)}(\pp (s),\BGO (s);u_n) = X {C_1 \over L_n^2} \partial_\BGO H_{C(S)} + X \pp_0 \cdot \partial_\pp H_{C(S)}. X$$ XThe $\pp$ derivative is bounded by X$$\eqalign{ X \|\pp_0\cdot\partial_\pp H_{C(S)} \|_0 \leq& X \| D_u^2 W (\phi(\pp) + u_n) X (\partial_\pp (\phi + u_n)) \|_0 \|\pp_0\| \cr X \leq& \ {C_W \over (\gs_n-\ga_n)^3} \|\pp\| X \|\partial_\pp(\phi + u_n)\|_{(\gs_n-\ga_n)} \cr X \leq& \ {C_W \over {\overline \gs}^3 } \|\pp_0\| X (1 + {CL_0 \over d_0}(1+{r_0 \over \rho_0})). \cr} X$$ XUnder the hypotheses of the lemma there is a constant X$C_1$ such that the monotonicity of X$\partial_\BGO H_{C(S)}$ dominates the variations Xgiven by $\pp \cdot \partial_\pp H_{C(S)}$, thus the Xoperator is monotone decreasing along $(\pp (s),\BGO (s))$, Xthus so are its eigenvalues. X XTo show that $Z_i$ lies to the right of the cone X$(\omega_\ell / m) + (C_1 / L_n^2) \|\pp\|$, Xconsider X$$\eqalign{ X e_i(\pp (s),\BGO (s)) &= X e_i(\pp (s),\BGO (s)) - e_i(0,\omega_\ell / m) \cr X &\leq \int_0^1 \ {d \over ds } X e_i(\pp (t),\BGO (t)) \ dt \leq 0. \cr} X$$ XThe cone $(\omega_\ell / m) - (C_1 / L_n^2) \|\pp\|$ Xis shown to be a lower bound with a similar argument. X X\endproof X X\REMARK XIn the case where the nonlinear term $W$ contains no Xquadratic terms, there is a better estimate of both X$\partial_\BGO W$ from \equ(4.cubic), and $\partial_\pp W$; X$$ X \| D_u^2 W (\phi + u_n)(\partial_\pp(\phi + u_n) \|_0 X \leq {C_W \over {\overline \gs}^3} \biggl( 1 + X {L_0 r_0^2 \over d_0\rho_0} \biggr) \|\pp\|. X\EQ(4.cubic2) X$$ XThe sets $Z_i$ can then be shown to lie between the Xparaboloids X$$ X {\omega_\ell \over m} \pm {C_1 \over L_n^2} \|\pp\|^2. X$$ X XThe more subtle control over the behavior of the sets $Z_i$ Xthat we require Xis that within the region $\Eta_{n+1}^{(1)}$, there is Xa form of parabolic estimate even for the general nonlinearity. X X\CLAIM Lemma(zi-paraboloids) XChoose $C_1 = \hat{C}(1/d_s + (L_0/d_0)(1+(r_0/\rho_0))$. XIf $(\pp_1,\BGO_1),(\pp_2,\BGO_2) \in Z_i$ also lie within Xthe same component of $\Eta_{n+1}^{(1)}$, then X$$ X |\BGO_2 - \BGO_1| < {C_1 \over L_n^2} X \bigl| \|\pp_2\|^2 - \|\pp_1\|^2 \bigr|. X\EQ(parab-est) X$$ X XThat is, the sets $Z_i$ are restricted by paraboloids Xrather than cones, whenever they intersect X$\Eta_{n+1}^{(1)}$. The proof of this lemma Xdepends upon several facts. First, for $(\pp,\BGO) \in X\Eta_{n+1}^{(1)}$ there can be at most two eigenvalues Xof the local hamiltonians $H_S(\pp,\BGO;u_n)$ and X$H_{C(S)}(\pp,\BGO;u_n)$ which are smaller than X$3d_s/4$. Certainly both domains $S$ and $C(S)$ have Xonly one singular region, so they contain at most Xtwo singular sites at which $|V(\BGO)(x)| < d_s$. XThen the operator norm of $D_u W(\phi(\pp)+u_n)$ is Xbounded; X$$\eqalign{ X \|D_u W(\phi(\pp)+u_n)\|_0 <& X {C_W \over (\gs_n-\ga_n)^2} \Bigl( X \|\pp\| + {CL_0 \over d_0}\|\pp\|^2 \Bigr) \cr X \leq& {4C_W \over {\overline \gs}^2 } X \|\pp\|~~. \cr} X$$ XThus if $16 C_W r_0 / {\overline \gs}^2 < d_s$, the Xoperator norm of $D_uW$ is bounded by $d_s/4$, and the Xeigenvalues of $V(\BGO)$ can be perturbed by at Xmost this much. X XSecondly, the eigenvectors associated with the small Xeigenvalues of $H_{C(S)}$ have special structure. They Xare supported principally on the singular region $S$, Xwith estimates of their extent over the other sites of X$C(S)$. XDenote as usual $P_S$ the orthogonal projection onto X$\ell^2(S)$. Let $(\psi_i,e_i)$ be Xeigenvector-eigenvalue pairs for the eigenvalues such that $|e_i| < 3d_s/4$. X X\CLAIM Proposition(struct-e-vectors) XUnder the hypotheses of \clm(parameters), if X$(\pp,\BGO) \in \Eta_{n+1}^{(1)}$, then X$$ X \|(\11-P_S)\psi_i \|_0 \leq X {C \|\pp\| \over d_s {\overline \gs}^{10}} X$$ X X\PROOF XThis result is based upon Corollary 5.5 on the Xstructure of the Green's function for a region of the Xcharacter of $C(S)$. There is a contour $\CC$ Xsurrounding all eigenvalues of $H_{C(S)}$ with X$|e_i| \leq d_s/2$ such that X${\rm dist}(\CC,{\rm spec} H_{C(S)})~~> d_s/8$. XFor $C_W \|\pp\|/ {\overline \gs}^2 < Cd_s$, Xwhenever $\zeta \in \CC$ the hypotheses of XCorollary 5.5 are satisfied, and can Xrepresent $G_{C(S)} = G_D \oplus G_S + R$, Xwhere $D = C(S) \backslash S$, $G_D$ is analytic Xover $|\zeta| < 3d_s/4$, and $R(x,y)$ satisfies Xthe estimate X$$ X |R(x,y)| \leq {C \|\pp\| \over d_s^2 {\overline \gs}^{10}} X e^{-(\gs_n-3\ga_n) X [{\rm dist}(x,S) + {\rm dist}(S,y)]}. X$$ XIn particular X$\|R\|_0 \leq C \|\pp\|/(d_s^2 {\overline \gs}^{10})$. X XDefine the orthogonal projection $P_E$ related to the Xsmall eigenvalues X$$ X P_E = {1 \over 2\pi i} \oint_\CC \ X {\bf 0} \oplus G_S(\zeta) \ d\zeta X$$ Xwhich has range within $\ell^2(S)$. Since $G_D$ is Xanalytic in a $3d_s/4$ neighborhood of the origin, X$$\eqalign{ X P_E \ =& \ {1 \over 2\pi i} \oint_\CC \ X G_D(\zeta) \oplus G_S(\zeta) \ d\zeta \cr X =& {1 \over 2\pi i} \oint_\CC \ X G_{C(S)}(\zeta) \ d\zeta - X {1 \over 2\pi i} \oint_\CC \ X R \ d\zeta \cr} X$$ XWhen applied to an eigenvector $\psi_i$ of a small Xeigenvalue $e_i$ lying within the contour $\CC$, X$$\eqalign{ X (\11 - P_E)\psi_i \ =& \ X \Bigl( \psi_i - {1 \over 2\pi i} \oint_\CC \ X G_{C(S)}\psi_i \ d\zeta \Bigr) X - {1 \over 2\pi i} \oint_\CC \ R \psi_i \ d\zeta \cr X =& \Bigl(\psi_i - {1 \over 2\pi i} \oint_\CC \ X {1 \over \zeta - e_i} \psi_i d\zeta \Bigr) X - {1 \over 2\pi i} \oint_\CC \ R \psi_i \ d\zeta \cr X =& - {1 \over 2\pi i} \oint_\CC \ R \psi_i \ d\zeta \cr} X$$ XEstimating the norm of the remainder, X$$\eqalign{ X \|(\11 - P_S)\psi_i\|_0 \leq& \ X \|(\11 - P_E)\psi_i\|_0 \cr X \leq& {1\over 2\pi} X \| \int_\CC \ R \psi_i \ d\zeta \|_0 X \leq {C\|\pp\| \over d_s {\overline \gs}^8}, \cr} X$$ Xwhere we have used that the contour has circumference Xproportional to $d_s$. X X\endproof X XFor $(\pp,\BGO) \in \Eta_{n+1}^{(1)}$ we are now prepared to Xderive better estimates on the behavior of the sets $Z_i$. XKeeping in mind the Feynman Hellman formula X\equ(feynman-hellman) for perturbation of the eigenvalues X$e_i(\pp,\BGO)$, we compute X$\langle \psi_i, \partial_\pp H_{C(S)} (\pp,\BGO;u_n) X \psi_i \rangle$ Xand estimate its size. X X\CLAIM Proposition(N1) XLet $C_2 = C(1/d_s + L_0/d_0 (1 + r_0/\rho_0))$. XFor $(\pp,\BGO) \in \Eta_{n+1}^{(1)}$, there is an Ximproved estimate X$$ X |\langle \psi_i, \partial_\pp H_{C(S)} (\pp,\BGO;u_n) X \psi_i \rangle| X = |\langle \psi_i, D_u^2 W(\phi(\pp) + u_n) X [\partial_\pp(\phi + u_n), \psi_i] \rangle| X \leq C_2 \|\pp\|. X\EQ(p_grad) X$$ X X\PROOF XThe gradient with respect to $\pp$ of $H_{C(S)}$ is X$D_u^2 W(\phi(\pp) + u_n)(\partial_\pp(\phi + u_n))$. XUsing hypothesis {\bf H3} this is described X$(D_u^2 W(\phi + u_n))_{C(S)} = (C + \Delta)_{C(S)}$. XUsing this along with the structure result for Xthe eigenfunctions of $H_{C(S)}$, X$\langle \psi_i, D_u^2 W(\phi(\pp) + u_n) X [\partial_\pp(\phi + u_n), \psi_i] \rangle$ Xwill be shown to be proportional to $\|\pp\|$. X$$\eqalign{ X & \langle \psi_i, D_u^2 W(\phi(\pp) + u_n) X [\partial_\pp(\phi + u_n), \psi_i] \rangle X = \langle P_S \psi_i, C [\partial_\pp \phi, P_S\psi_i] X \rangle \cr X &+ 2\langle (\11-P_S)\psi_i, C [\partial_\pp \phi, P_S\psi_i] X \rangle + \langle (\11-P_S)\psi_i, X C [\partial_\pp\phi , (\11-P_S) \psi_i] \rangle \cr X &+ \langle \psi_i, \Delta[\partial_\pp \phi, \psi_i] X \rangle + \langle \psi_i, X (C+\Delta)[\partial_\pp u_n, \psi_i] \rangle. \cr} X\EQ(H3_decomp) X$$ XReferring to {\bf H3} (see (6.5)), $\langle P_S \psi_i, XC [\partial_\pp \phi, P_S\psi_i] \rangle = 0$, and we estimate Xthe remaining terms. XThe second term of \equ(H3_decomp) is X$$ X |\langle (\11-P_S)\psi_i, C [\partial_\pp \phi, P_S\psi_i] X \rangle| \leq X \|\pp\| {C \over d_s} {C_W \over ({\overline \gs}/2)^4 } X$$ Xand the third term is even better behaved; X$$ X |\langle (\11-P_S)\psi_i, X C [\partial_\pp \phi, (\11-P_S) \psi_i] \rangle| X < \|\pp\|^2 {c^2 C_W \over (d_s^2 {\overline \gs}^4)}. X$$ XNext, X$$ X |\langle \psi_i, \Delta[\partial_\pp \phi, \psi_i] X \rangle| \leq { C_W \over X ({\overline \gs} / 2)^4 } \|\pp\|, X$$ Xand finally, the most significant term is X$$\eqalign{ X |\langle \psi_i, X (C+\Delta)[\partial_\pp u_n, \psi_i] \rangle| X &\leq { C_W \over X ({\overline \gs} / 2)^4 } X \|\partial_\pp u_n\|_{\gs_n-\ga_n} \cr X &\leq {C_W \over ({\overline \gs}/2)^4 } X C{L_0 \over d_0} X (1 + {r_0 \over \rho_0}) \|\pp\| \cr} X$$ Xwhere we have used the induction hypothesis $(n.1)$ Xto control $\|\partial_\pp u_n\|_{\gs_n-\ga_n}$. XAssuming that $\|\pp\| \leq r_0 < d_s$ Xand collecting terms, the result follows. X X\endproof X XThis is enough information to prove the parabolic Xestimates on the sets $Z_i \cap \Eta_{n+1}^{(1)}$. XLet $(\pp_1,\BGO_1) \in Z_i \cap \Eta_{n+1}^{(1)}$, Xand consider a path X$\bigl( \pp (s), \BGO_\pm (s) \bigr) = X \bigl( \pp_1 + s (\pp_1/\|\pp_1\|) , \BGO_1 \pm X C_1 ( \|\pp (s) \|^2 - \|\pp_1\|^2) X / 2L_n^2 \bigr)$. XAt points on this path at which $e_i(\pp,\BGO)$ Xis differentiable, X$$\eqalign{ X {d \over ds} e_i &= \ \partial_\BGO e_i X {d \BGO_\pm \over ds} + \partial_\pp e_i X { d\pp \over ds } \cr X &= \ \pm \langle \psi_i, \partial_\BGO H_{C(S)} X \psi_i \rangle {C_1 \|\pp (s)\| \over L_n^2} X + \langle \psi_i, X \partial_\pp H_{C(S)} \psi_i \rangle X \cdot {\pp (s) \over \|\pp (s) \| } \cr} X$$ XChoosing the plus sign to illustrate one case, X$$ X {d \over ds} e_i \leq - \BGO_{min} X {C_1 \over L_n^2} \times L_n^2 \|\pp (s)\| X + C_2 \|\pp (s) \|, X$$ Xwhere \equ(feynman-hellman), \equ(FH2), and \equ(p_grad) are used to Xestimate the inner products. By choosing the constant $\hat{C}$ Xin \clm(zi-paraboloids) sufficiently large, we can insure tha X$C_1 \BGO_{min} > C_2$ which implies that X$d e_i / ds \leq 0$ at points of Xdifferentiability along the curve X$(\pp (s), \BGO_+ (s))$. Hence X$$ X e_i(\pp (s),\BGO_+ (s)) - e_i(\pp_1, \BGO_1) X \leq \int_0^s \ {d \over ds} X e_i(\pp (t), \BGO_+ (t)) \ dt \leq 0, X$$ Xand $Z_i$ must remain to the left of X$(\pp (s) ,\BGO_+ (s))$ as long as this curve Xremains within $\Eta_{n+1}^{(1)}$. A similar Xargument shows that $Z_i$ lies to the right Xof the curve $(\pp (s), \BGO_- (s))$. XWith this knowledge we can prove X\clm(zi-paraboloids). Let $(\pp_1,\BGO_1)$ and X$(\pp_2,\BGO_2) \in Z_i$, lying within the same Xcomponent of $\Eta_{n+1}^{(1)}$, with X$\|\pp_1\| \leq \|\pp_2\|$. Since $Z_i$ is Xinvariant under $T_\xi$ we may also assume Xthat $\pp_1$ and $\pp_2$ are parallel. XConsider the curves $(\pp (s),\BGO_\pm (s))$, Xwith $\pp (t) = \pp_2$ for some $t$. If X$(\pp (t),\BGO_\pm (t)) \in \Eta_{n+1}^{(1)}$ Xthen by our analysis X$$ X \BGO_-(t) \leq \BGO_2 \leq \BGO_+ (t) X\EQ(BGO_pm) X$$ Xfrom which we conclude that X$$ X |\BGO_2 - \BGO_1| \leq X {C_1 \over L_n^2} X \bigl| \|\pp_2\|^2 - \|\pp_1\|^2 \bigr| X$$ XRecall that $\Eta_{n+1}^{(1)}$ consists of Xopen disks that are constant in the $\pp$ Xdirection, thus if either of $(\pp (t), X\BGO_\pm (t))$ is not within X$\Eta_{n+1}^{(1)}$, then automatically X\equ(BGO_pm) holds. This finishes the proof Xof \clm(zi-paraboloids). X XWe now are in position to prove part $(c)$ of X\clm(parameters). Consider an arbitrary $C^\infty$ Xsurface $\CC = (\pp,\BGO(\pp))$ which is invariant Xunder $T_\xi$; $\BGO(\pp) = \omega_1 + \Kappa X\|\pp\|^2 (1+\|\pp\| C(\|\pp\|))$, where X$\Kappa \not= 0$. We will see that this surface Xintersects the good set of parameters X$\Eta_{n+1} = \Eta_{n+1}^{(1)} \cap X\Eta_{n+1}^{(2)}$ in a large set. In practice Xthese surfaces are solution sets of the Xsecond bifurcation equation (3.14)(b), and Xintersections with $\cup_{n=0}^\infty \Eta_n$ Xcorrespond to solutions of the full equations X\equ(NLlattice). X XThe total measure of $\BGO$ within X$\{ \omega_1-r_0^2 < \BGO < \omega_1 + r_0^2 \}$ Xwhich has been excised in order to construct X$\Eta_{n+1}^{(1)}$ from $\Eta_n^{(1)}$ is Xestimated in \clm(excise) by X$Cd/L_n^\tau(1+r_0^2 L_n)$. From the parabolic Xnature of $\CC$ an estimate of the new excisions Xis given by, X$$ X {\rm meas} \{ 0 \leq r < r_0; (\pp,\BGO(\pp)) X \in \Eta_n^{(1)} \backslash (\Eta_{n+1}^{(1)}) X ~{\rm with}~ \|\pp\| = r \} \leq X \sqrt{{Cd \over |\Kappa| L_n^\tau}(1+r_0^2 L_{n+1})} X\EQ(n1-excision) X$$ X XThe parabolic estimates \clm(zi-paraboloids) are used to Xcontrol the effect of the excisions in order to Xsatisfy condition $(b)$ on the surface $\CC$. XSuppose that $(\pp_1,\BGO(\pp_1)) \in \CC \cap XZ_i \cap \Eta_{n+1}^{(1)}$, and let $(\pp,\BGO) X\in \CC$ be another point in $\Eta_{n+1}^{(1)}$, Xwith $|\BGO - \BGO_1| < \gd_{n+1}/L_n^2$. We have Xan estimate on the quantity $|\|\pp\|^2 - \|\pp_1\|^2|$. XWithout loss of generality consider $\pp$ Xparallel to $\pp_1$. On $Z_i \cap \Eta_{n+1}^{(1)}$, X$|\BGO_i(\pp) - \BGO_i(\pp_1)| \leq X (C_1/L_n^2)|\|\pp\|^2 - \|\pp_1\|^2|$, Xwhile on $\CC$, $| \BGO(\pp) - \BGO(\pp_1) | > X|\Kappa/2| |\|\pp\|^2 - \|\pp_1\|^2|$. XThus in forming $\Eta_{n+1}^{(2)}$ by deletion of X$\gd_{n+1}/L_n^2$ neighborhoods of each $Z_i$, Xany other points of $\Eta_0$ that are excised satisfy X$$\eqalign{ X{\gd_{n+1} \over L_n^2} \geq & |\BGO(\pp) - \BGO_i(\pp)| X \geq |\BGO(\pp) - \BGO(\pp_1)| X - |\BGO_i(\pp_1) - \BGO_i(\pp)| \cr X \geq & ({|\Kappa| \over 2 } - {C_1 \over L_n^2}) X |\|\pp\|^2 - \|\pp_1\|^2|. \cr} X$$ XAs long as $C_1/L_n^2 < |\Kappa| / 4$, we have X$|\|\pp\|^2 - \|\pp_1\|^2| < (4/ |\Kappa|) X\gd_{n+1} / L_n^2$. XEliminating such $\pp$ for each possible singular Xsite and each resulting eigenvalue set $Z_i$, the Xtotal measure of the parameters $\pp$ excluded Xin constructing $\Eta_{n+1}^{(2)}$ from X$\Eta_n^{(2)}$ is controlled by X$$\eqalign{ X {\rm meas}& \{ 0 \leq r < r_0; \|\pp\| = r, \ X (\pp,\BGO(\pp)) \in \Eta_{n+1}^{(1)} \cap X (\Eta_n^{(2)} \backslash \Eta_{n+1}^{(2)} )\} \cr X \leq & X \sqrt{CL_n (1+r_0^2L_{n+1}) X \times \ell_{n+1}^2 X \times (1/|\Kappa|) (\gd_{n+1}/L_n^2) } \cr} X\EQ(n2excision) X$$ XWith this we have finished the proof of X\clm(parameters). X X\endproof X X X X X END_OF_FILE if test 47949 -ne `wc -c <'sec4.tex'`; then echo shar: \"'sec4.tex'\" unpacked with wrong size! fi # end of 'sec4.tex' fi if test -f 'sec5.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sec5.tex'\" else echo shar: Extracting \"'sec5.tex'\" \(40593 characters\) sed "s/^X//" >'sec5.tex' <<'END_OF_FILE' X X X\SECTION The Green's Function X XThe principal goals of this section are to Xprove estimates X on solutions to the linearized first bifurcation equation X \equ(bifeq1), where the linearization is about an X approximate solution obtained in the induction. Actually, X solutions are obtained for the linearized equations X restricted to finite subdomains $\l2 (B_{n})$ of X ${\l2 (\zsquared)}$; this restriction is both the analog of X the smoothing procedure of the classical Nash-Moser method, X and it allows us to consider only finitely many possibly X singular sites at each induction step. In the induction X the domains $B_{n}\subseteq B_{n+1}$ on which we X approximately solve \equ(bifeq1) grow. In addition, Xthe tolerance for X small denominators, or small spectra, increases, quantified X by the sequence $\delta_{n}$. The full nonlinear equations X \equ(NLlattice) also have small or zero eigenvalues, stemming X from the null space of the operator X$H_{B_{n}}(0,\omega_{1})$; Xhowever the Lyapounov Schmidt decomposition Xallows us to work Xindependently of these small eigenvalues. XIn the induction the linearized operators $H_{B_{n}}$ X and the Green's functions $G_{B_{n}}$ depend upon the X parameters in the problem, the frequency $\Omega$ and the X parametrization $\pp $ Xof the null space of $H_{B_{0}}(0,\omega_{1}), X $. The first step of the induction defines X $G_{B_{0}}(\pp,\Omega)$ for $(\pp,\Omega) \in X \Eta_{0}$ a full neighborhood of the bifurcation Xpoint X $(0,\omega_{1})$. In subsequent induction steps we only Xobtain X control of $G_{B_{n}}(\pp,\Omega)$ in subsets X ${\Eta}_{n}\subseteq {\Eta}_{n-1}\subseteq X\cdots {\Eta}_{0}$, on X which the tolerance of the ${ n}^{\rm th}$ linearized operator X $H_{B_{n}}(\pp,\Omega,u_{n})$ for small spectra are satisfied. X X\SUBSECTION The Green's function on nonsingular domains X XThe first bounds on the Green's function concern Xthe nonresonant case. We denote $N=\{n_{j}\}$ the lattice Xsites which support the null vectors of X$H_{B_{0}}(0,\omega_{1})$ Xand $\overline{B}_{0}=B_{0}-N$. For $(\pp,\Omega)$ in a Xneighborhood $\tilde{\NN}$ of $(0,\omega_{1})$ the diagonal Xelements X$|V(\Omega)(n_j,n_j)|$ can be very small. However, Xsince we assumed that our initial frequencies were X$(d_0,L_0)$ nonrenonant, Xall of the Xsites of $\overline{B}_{0}$ have X$|V(\omega_1)(x,x)|>d_{0}$. On the full lattice X${\zsquared}$ we call a site $x$ {\bf singular}, Xif $|V(\Omega)(x,x)| < d_s$, where $d_s$ is a parameter Xwe defined in Section 3.5. XWhen restricted to nonsingular sites, there is a X$\sigma_{0}$ such that the operator $H(\pp,\Omega)$ is Xinvertible on ${\cal{H}}_{\sigma}$ for all $0\leq X\sigma<\sigma_{0}$. XThis is the conclusion of the following. X X\CLAIM Theorem(T4.2) Let $H_A=V(\Omega)+DW$ be a Xlinear operator defined on $\ell^{2}(A)$, for any Xdomain $A \subseteq \zsquared$ Xconsisting entirely of sites where $|V(x)| > \delta$ . XSuppose Xthat $DW(x,y)$ satisfies an estimate of the form X$$ X |DW(x,y)| \leq \epsilon e^{-\sigma|x-y|}~~. X\EQ(4.3) X$$ XThen for X$$ X{{\epsilon}\over{\delta}} < \min\left[ X{{\sigma^2}\over{64}},{{\sigma}\over{8}} \right]~~, X$$ Xand for all $|z|\leq \half \delta $ the Green's Xfunction $G_A(z)=(H_A-z \11)^{-1}$ exists and decays Xexponentially off diagonally X$$ X|G_A(z)(x,y)| \leq {32 \over \delta} e^{-\sigma^{'}|x-y|} X\EQ(4.4) X$$ Xwhere X$\sigma^{'} = X \sigma - 4\log (1+4\sqrt{(\epsilon/\gd) })$. XFurthermore if X$H_A$ depends analytically on parameters, then so does X$G_A(z)$. In operator norms, for any $0 < \gamma X \le \sigma^{'}$ there is a constant $C_{0}$ Xsuch that X$$ X\parallel G_A(z)\parallel_{\sigma' - \gamma} X \leq {C_0 \over \delta \gamma^2}~~. X\EQ(4.5) X$$ X X X X\PROOF This proof is neatest via path expansion of Xthe resolvent $(H_A-z{\bf{1}})^{-1}$. XFirst subtract the diagonal from $DW$, X$$ X H _A= V(\Omega) + DW = V_{1}+DW_{1} X$$ Xwhere $V_{1}=V + {\rm{diag}} \ DW$. XFor $\epsilon \leq (1/4) \delta $ Xthe elements of the diagonal matrix X $|V_{1}(x,x)|~\geq~{{3}\over{4}}~\delta $, Xwhile diag $DW_{1}=0$, and otherwise X$DW_{1}(x,y) \leq \epsilon e^{-\sigma |x-y|}$. XCertainly $|V^{-1}_{1}(x,x)|~\leq~4~\delta^{-1}$. XConsider the Neumann series for the XGreen's function X$G_A(z)(x,y) = (H_A-z{\bf{1}})^{-1}(x,y) X= (V_{1}~-~z{\bf{1}}~+~DW_{1})^{-1}(x,y)$, Xconsidering $DW_{1}$ as the perturbation, X$$ X(H_A-z{\bf{1}})^{-1} = X (V_{1}-z{\bf{1}}+DW_{1})^{-1} = X (V_{1}-z{\bf{1}})^{-1} X (1+DW_{1}(V_{1} - z{\bf{1}})^{-1})^{-1}. X$$ XSet $T=-DW_{1}(V_{1}-z{\bf{1}})^{-1}$, Xwhich also vanishes on Xthe diagonal, so that at least formally, X$$ XG_A(z) = (H_A-z{\bf{1}})^{-1} X = ( V_{1}-z{\bf{1}} )^{-1} X \sum^{\infty}_{n=0} T^{n}. X\EQ(4.1.1) X$$ XWe of course have X$|T(x,y)| \leq 4 \epsilon X\delta^{-1} e^{-\sigma|x-y|}$. XWe will sum \equ(4.1.1) over paths, Xthereby preserving an exponential decay estimate for the XGreen's function. XConsider a path $\beta =(x,\beta_{1},\beta_{2},\cdots X\beta_{n-1},y)$ from $x$ to $y$ in the domain $A$, Xwith the number of steps taken $\#\beta=n$, with total Xlength $l(\beta)=\sum_{j}|\beta_{j}-\beta_{j-1}|$. Then X$$ XT^{n}(x,y) = \sum_ X {\scriptstyle {\rm paths} \, X \beta:x\rightarrow y \atop \scriptstyle \# \beta = n} \, X \prod_{j=1}^{n} \, X T(\beta_{j-1},\beta_{j})~~, X\EQ(4.1.2) X$$ Xwhere we make the identification $\beta_0 = 0$ and $\beta_n = y$. XInserting the estimate on $|T(\beta_{j-1},\beta_j)|$ from Xabove we find, X$$ X|T^{n}(x,y)| \leq \sum_ X {\scriptstyle {\rm paths} \, X \beta: x \rightarrow y \atop \scriptstyle \# \beta = n} X (4 \epsilon / \delta)^{n} X \prod^{n}_{j=1} X e^{-\sigma |\beta_{j}-\beta_{j-1}|} X\EQ(4.1.3) X$$ X$$\qquad = \sum_{\scriptstyle {\rm{paths}}\, X \beta: x\rightarrow y \atop \scriptstyle \# \beta = n} X (4 \epsilon / \delta)^{n} X e^{-\sigma l(\beta)}. X$$ XThe full Neumann series is now considered. X$$ X|\sum_{n=0}^{\infty} T^{n}(x,y)| X \leq \sum_{n=0}^{\infty} \sum_{L=|x-y|}^{\infty} X \sum_{\scriptstyle {\rm paths} \, X \beta: x\rightarrow y \atop \scriptstyle \ell X (\beta)=L, \, \# \beta =n} X (4 { \gre}\delta^{-1})^n X e^{-\sigma L}. X\EQ(4.1.4) X$$ XThis gives an estimate of the Green's function in terms of Xthe number of paths from $x$ to $y$ of length $L$ with $n$ Xsteps. XFor our two dimensional lattice a sufficiently Xgood upper bound on the number of paths from $x$ to $y$ is X$\#\{\beta : l(\beta)=L, \# \beta =n\} X \leq 4^{n} {L-1 \choose n-1}^2$. XThe resulting estimate of \equ(4.1.4) is X$$\eqalign{ X \delta(x,y) + |\displaystyle{\sum_{n=1}^{\infty}} T^n(x,y)| X & \leq \delta(x,y) + \displaystyle{\sum_{L=|x-y|}^{\infty}} X \displaystyle{\sum_{n=1}^L} X {(4^2 \gre \delta^{-1}) }^n X {L-1 \choose n-1}^2 e^{-\sigma l} \cr X & \leq \delta(x,y) +\displaystyle{\sum_{L=|x-y|}^{\infty}} \, X \left( \sum_{n=1}^L({ 4^2 \gre} X \delta^{-1})^{n/2} X {L-1 \choose n-1} \right)^2 X e^{-\sigma L} \cr X & \leq \delta(x,y) + \displaystyle{\sum_{L=|x-y|}^{\infty}} X ({ 4^2 \gre} \delta^{-1}) X (1 + 4 \sqrt{ (\epsilon / \delta)} )^{2(L-1)} e^{-\sigma L}\cr X} X$$ XThis sum converges if X${{\epsilon}\over{\delta}} < {{\sigma^2}\over{64}}$ Xgiving the estimate X$$ X {{32 \epsilon}\over{\sigma \delta}} e^{-(\sigma - X 2\log (1+ 4\sqrt{({ \gre} X / \delta})))|x-y|}~~. X$$ XCombining this estimate with \equ(4.1.1) yields \equ(4.4). XEstimate \equ(4.5) then follows Xfrom \equ(4.4) by way of \clm(opnorm). XThese estimates are uniform in parameters, as long as the Xhypotheses of the theorem remain valid, therefore Xanalyticity properties of $H_A$ are preserved in $G_A(z)$. X X\endproof X X\noindent X\CLAIM Corollary(stronger) XIn fact one sees from the proof that a Xstronger estimate is true X$$ X |G_A (x,y;z)| \leq {{4}\over{\delta}} X\left\{ \delta(x,y) + {{32 \epsilon}\over{\sigma \delta}} X e^{-\sigma' |x-y|} \right\}~~. X\EQ(GA) X$$ X XLet $A$ be any non-singular domain and let $E_0 X\subset \overline{B_0} \cup A$. Consider the operator X$H_{E_0} = V(\Omega) + D W(u_0)$. If X$(r_0^2 + \rho_0) L_0^2 \le d_0/2$, then Xfor every $\Omega \in D(\NN_0,\rho_0)$, X$|V(\Omega)(x,x)| > d_0/2$, for every point X$x \in \overline{B_0}$. This follows from the Xform of $V(\Omega)$, and the Xfact that $|V(\omega_1)(x,x)| > d_0$ by the $(d_0,L_0)$ Xnon-resonance hypothesis. If X$$ Xd_s \le d_0/2 ~~, X\EQ(dsize) X$$ Xall sites of $E_0$ are nonsingular. The operator X$D W(u_0)$ is bounded from $\HH_{\sigma+ \gamma_0}$ to X$\HH_{\sigma}$ provided $\sigma < \sigma_* - L_0^{-1}$ Xas in \clm(firstbifonB0). Indeed hypothesis ${\bf H2}$ of Section 6 Ximplies that X$$|(D W(u_0))(x,y)| \le ((C_W \| \pp \| ) /\gamma_0^3) Xe^{-\sigma_0 |x-y|}~~. X$$ XIn order to make the labeling of the indices consistent, Xdefine $\sigma_{-1} \equiv \sigma_0 + 6 \gamma_{-1}$, Xwith $\gamma_{-1} \equiv \sigma_0/64$. Note Xthat both of these definitions are consistent with Xthe original inductive definitions in Section 3.3. XThus, we can apply \clm(T4.2) and we obtain Xthe following estimate. X\CLAIM Proposition(GB0) Let $A$ be any non-resonant region and Xlet $B_0$ satisfy the hypotheses of \equ(bifeq1). If in addition X$$ X{{C_W r_0}\over{d_s \gamma_0^3}} < ({{\sigma_0}\over{8}})^2~~, X\EQ(r0hypoth) X$$ X$$ X(r_0^2 + \rho_0) L_0^2 \le d_0/2~~, ~~d_s \le d_0/2~~, X\EQ(r_0hypoth2) X$$ X$$ X\sigma_{-1} \equiv ({{35}\over{32}}) \sigma_0 < \sigma_* X- 2 \log(1+4 \sqrt{\epsilon/\delta})~~, X$$ Xand if $E_0 \subset \overline{B_0} \cup A$ Xthen X$$ X\parallel G_{E_0} X \parallel_{\gs_{-1} - \ga_{-1}} X \le {{C_0} \over {d_s\ga_0^3}}~~. X$$ X X\REMARK If the nonlinearity in \equ(NLlattice) has no Xquadratic term, the estimate on $DW(u_0)$ can Xbe improved to X$$ X|(D W(u_0))(x,y)| \le {{C_W \| \pp \|^2} \over{\gamma_0^3}} Xe^{-\sigma_0 |x-y|}~~. X$$ XThis allows us to replace hypothesis \equ(r0hypoth) Xby X$$ X{{C_W r_0^2}\over{d_s \gamma_0^3}} < ({{\sigma_0}\over{8}})^2~~, X\EQ(newr0hypoth) X$$ Xand still obtain the same estimate on the Green's function. X X\REMARK This proves \equ(redGreen1) when $n=0$. X X Note that the hypothesis is independent of the domains X$E_0$. This emphasizes the fact that it is small values X on the diagonal $V(\Omega)$, and the associated singular X sites which play a major role in determining Xthe decay of the Green's functions for X $H_{B_{n}}(\pp,\Omega,u_{n})$. X XAt each level of the induction there are two steps to X perform. The domain under consideration is extended from X $B_{n}$ to $B_{n+1} \subseteq \zsquared$, and the Green's Xfunction X $G_{\overline{B_{n+1}}}$ on the larger domain must be X constructed and estimated. Secondly, a Newton step is X performed to construct the next correction X $v_n(x,\pp,\Omega)$ to the approximate solution X $u_{n}(x;\pp,\Omega)$. These are the analog of the X smoothing step and the Newton iteration of the classical X Nash-Moser method. In the first of these steps we X construct $G_{\overline{B_{n+1}}}(\pp,\Omega,u_{n};z)$ from X $G_{\overline{B_{n}}}$ for $(\pp,\Omega) \in \NN_{n+1}$. XThis is the most detailed construction X as there may be many resonant sites Xin the region $B_{n+1}\backslash B_{n}$. XThe second step also requires us to X estimate the inverse of the linearized operator X $H_{\overline{B_{n+1}}}(\pp,\Omega;u)$, for $u=u_{n}+v_{n}$. XDue to X the rapid convergence of Newton's method the adjustment X $v_{n}(x;\pp,\Omega)$ is very small and Xonce one has control of X$G_{\overline{B_{n+1}}}(\pp,\Omega,u_{n};z)$ the construction X of $G_{\overline{B_{n+1}}}(\pp,\Omega,u_{n}+v_{n};z)$ is Xrelatively painless. X X\SUBSECTION Domain Extension: X XThe induction hypotheses Xproduce a set of Xparameters $(\pp,\Omega)\in \NN_{n}$ such that Xfor any $E_n \subset \overline{B_n} \cup A$, where X$A$ is a nonsingular region, and for any $|z|<\delta_{n}/2$ Xthe Green's function has the estimate X$$ X\parallel G_{E_n} X \parallel_{\sigma_{n-1} - \gamma_{n-1}} X \leq {{C_G^n}\over{\delta_{n} \gamma_{n-1}^{12}}}. X\EQ(4.5a) X$$ XTake $L_0$ large enough that X$\delta_0 < d_s /C_0$. \clm(GB0) implies that this estimate also Xholds for $n=0$. XIn this subsection we will extend this induction estimate to Xthe domains $E_{n+1} \subset X\overline{B_{n+1}} \cup A$, with tolerance X$\delta_{n+1}$ as long as $A$ remains nonsingular, and all Xsites within $B_{n+1}\backslash B_{n} \equiv A_{n+1}$ obey the Xinduction hypotheses $(j.1)$ and $(j.2)$ X$j=1, \dots , n$. XIn this process we will see that the geometry of the resonant Xsites is important, as well as the severity of the Xresonance. XIn fact this process of domain extension can be Xcarried out in any situation in which both the geometry Xand severity of the resonances are under control. X XLet $\{S_{k}\}$ be the set of singular regions of X$H_{B_{n+1}}(\pp,\Omega,u_{n})$ within $A_{n+1}$, and Xdefine $C_{\ell_{n+1}}(S_{k})$ to be balls of radius $\ell_{n+1}$ about X$S_{k}$, isolating neighborhoods of the singular regions from Xthe rest of $B_{n+1}$. XBy assumption $(n.2)$, for all parameter values X$(\pp,\Omega)\in \NN_{n+1}$ these regions are well Xseparated, and $C(S_{k})$ are disjoint. XThe regions $S_{k}$, of course, Xvary with $\Omega$. The inductive estimate on the XGreen's function will be proved by decomposing the domain X$B_{n+1}$ into subdomains, constructing Green's functions Xon the subdomains, and then patching the full Green's Xfunction together using a resolvent expansion. Related Xestimates and resolvent expansions have appeared in work of X[FS]. First we consider the Green's Xfunction on the neighborhoods $C_{\ell_{n+1}}(S_{k})$. X X\CLAIM Lemma(L4.4) Suppose that $S \subset A_{n+1}$ is a Xsingular region, and $(\pp,\Omega) \in \NN_{n+1}$ so that X$(j.1)$ and $(j.2)$ $j=1, 2, \dots$ hold, Xthen for $|z| < \half \delta_{n+1}$, Xand X$$ Xr_0 < \delta_0 \gamma_0^3~~, X\EQ(4.7.1) X$$ Xthen X$$ X |G_{C_{\ell_{n+1}}(S)}(x,y;z)| \leq X {{C}\over{\delta_{n+1} \gamma_{n-1}^{10} }} X e^{-(\sigma_{n-1} - 3 \gamma_{n-1} )|x-y| }~~. X\EQ(4.7) X$$ XThis implies X$$ X\| G_{C_{\ell_{n+1}}(S)} \|_{\sigma_{n-1} - 4\gamma_{n-1} } X\le {{C}\over{\delta_{n+1} \gamma_{n-1}^{12} }}~~. X$$ X X\REMARK In the case where the nonlinearity in \equ(NLlattice) Xcontains no quadratic term, condition \equ(4.7.1) can Xbe replaced by X$$ Xr_0 ^2< \delta_0 \gamma_0^3~~. X\EQ(4.7.1) X$$ X X X X\PROOF The proof uses resolvent identities Xto decompose the Green's function into several parts, Xallowing matrix elements to be estimated. Let $S$ be the Xsingular region itself, and $D=C_{\ell_{n+1}}(S)-S$ the annulus Xsurrounding this region. Induction hypothesis X$(n.2)$ implies that the singular regons are so Xfar apart that the annulus $D$ contains no other Xsingular sites. The linear operator X$H_{C_{\ell_{n+1}}(S)}$ is decomposed into block diagonal and off Xdiagonal parts, X$$ X H_{C_{\ell_{n+1}}(S)} = H_{D} \oplus H_{S} + \Gamma_{DS}. X\EQ(4.8) X$$ XThe operators $H_{D}$ and $H_{S}$ are the restrictions of X$H_{C_{\ell_{n+1}}(S)}$ to the subdomains $D$ and $S$, and the term X$\Gamma_{DS}$ accounts for the interactions between these Xdomains. There is accordingly a decomposition of the Xresolvent, giving the first resolvent identity, X$$ XG_{C_{\ell_{n+1}}(S)} = G_{D} \oplus X G_{S} + G_{D} \oplus X G_{S} \Gamma_{DS} G_{C(S)} X\EQ(4.9) X$$ X(Again, to save space, we will write $G_{C(S)}$ Xrather than $G_{C_{\ell_{n+1}}(S)}$.) XIterating this and its adjoint we obtain the second Xresolvent identity X$$\eqalign{ X G_{C_{\ell_{n+1}}(S)} = G_{D} & X \oplus G_{S} + X G_{D} \oplus G_{S} \Gamma_{DS} X G_{D}\oplus G_{S} \cr X & + G_{D} \oplus X G_{S} \Gamma_{DS} G_{C(S)} \Gamma_{DS} X G_{D}\oplus G_{S} \cr} X\EQ(4.10) X$$ XThe Hamiltonian $H_{D}$ has diagonal Xelements which are nonsingular, Xand satisfies the Xhypotheses of Corollary 5.2. Thus X$$|G_{D}(x,y;z)| \leq X {{2 C_0}\over{\delta_0 \gamma_0^3}} X e^{-(\sigma_{n-1} - 2\gamma_{n-1})|x-y|}~~. X$$ XThe set $S$ consists of only one or two sites. XThe interaction term Xhas zero diagonal, and decays exponentially off diagonal; X$$ X|\Gamma_{DS}(x,y)| \leq {{C_W \|\pp\|}\over{\gamma_{n-1}^3}} X e^{-(\sigma_{n}-2\gamma_{n-1})|x-y|}~~. X$$ XWe Xwill treat estimate \equ(4.7) in three cases. X XIf both $x,y \in S$, then $|G_{C_{\ell_{n+1}}(S)}(x,y;z)| \leq X2/\delta_{n+1}$, since by the induction hypothesis X$\dist (\spec (H_{C_{\ell_{n+1}}(S)}),0) > \delta_{n+1}$ XThe estimate \equ(4.7) follows since X$|x-y| \le 1$. XSecondly, if $x\in D$ and X$y \in S$ (or vice versa), the first resolvent identity is used X$$\eqalign{ X |G_{C(S)} & (x,y;z)| \leq X |\displaystyle{\sum_ X {{\scriptstyle p \in D \atop \scriptstyle q \in S}}} X G_{D}(x,p) X \Gamma_{DS}(p,q) G_{C(S)}(q,y)| \cr X & \leq \sum_{p\in D} {{2 C_0}\over{\delta_0 \gamma_0^3}} X e^{-(\sigma_{n-1} - 2\gamma_{n-1})|x-p|} X {{C_W \|\pp\|}\over{\gamma_{n-1}^3}} X e^{-(\sigma_{n}-2\gamma_{n-1})|p-q|} {{2}\over{\delta_{n+1}}} \cr X & \leq {{C \|\pp\|}\over{ X\delta_0 \gamma_0^3 \delta_{n+1}\gamma_{n-1}^{5} }} X e^{-(\sigma_{n-1}- 3\gamma_{n-1})|x-y|}. \cr} X\EQ(4.11) X$$ XNote that $G_{D} \oplus G_{S}(x,y)=0$ in this case Xand we have used the fact that $|q-y| \le 1$.. XFinally, consider the case when both $x,y \in D$. XUsing the second resolvent identity, X$$\eqalign{ X |G_{C_{\ell_{n+1}}(S)} & (x,y)| \leq |G_{D}(x,y)| + X |\sum_{ {{\scriptstyle p,t \in D }\atop{ X \scriptstyle r,q,s\in S}} } G_{D}(x,p) X \Gamma_{DS}(p,q)\times \cr X & ~~~~~~~~~~~~~~~~~~~~~~~~~\times G_{C_{\ell_{n+1}}(S)}(q,r) X \Gamma_{DS}(r,t) G_{D}(t,y)| \cr X & \leq {{2 C_0}\over{\delta_0 \gamma_0^3}} X e^{-(\sigma_{n-1} - 2\gamma_{n-1})|x-y|} \cr X & + \sum_{ {{\scriptstyle p,t \in D }\atop{ X \scriptstyle q,r \in S}} } X {{2 C_0}\over{\delta_0 \gamma_0^3}} X e^{-(\sigma_{n-1} - 2\gamma_{n-1})|x-p|} X{{C_W \|\pp\|}\over{\gamma_{n-1}^3}} X e^{-(\sigma_{n-1}-2\gamma_{n-1})|p-q|} {{2}\over{\delta_{n+1}}}\times \cr X & ~~~~~~~~~~~~~~~~~~~~~~~~~\times X{{C_W \|\pp\|}\over{\gamma_{n-1}^3}} X e^{-(\sigma_{n-1}-2\gamma_{n-1})|r-t|} X {{2 C_0}\over{\delta_0 \gamma_0^3}} X e^{-(\sigma_{n-1} - 2\gamma_{n-1})|t-y|}~~. X\cr} X\EQ(4.12) X$$ XWe have used the fact that X$\Gamma_{DS}(p,q)=0$ Xif both of its arguments are in $D$ to eliminate the Xterm linear in $\Gamma$ from \equ(4.12). XWe estimate this exponential sum with a loss of decay X$\gamma_{n-1}$. This results in a bound on \equ(4.12) of the Xform X$$ X\left\{ \left( {{2 C_0}\over{\delta_0 \gamma_0^3}} \right) X+ \left( {{2 C_0}\over{\delta_0 \gamma_0^3}} \right)^2 X\left( {{C_W \|\pp \| }\over{\gamma_{n-1}^3 }} \right)^2 X{{2}\over{\delta_{n+1} }} {{C}\over{\gamma_{n-1}^4}} \right\} Xe^{-(\sigma_{n-1} - 3 \gamma_{n-1}) |x-y|}~~. X\EQ(5.17.2) X$$ XThus inequality \equ(4.7) follows from \equ(4.11) Xand \equ(4.12) using $\| \pp \| < r_0 < \delta_0 \gamma_0^3$. X X\endproof X X XIn fact a useful corollary of the proof describes in more detail Xthe structure of the local Green's function in neighborhoods Xcontaining only one singular region. X X X\CLAIM Corollary(efunct-form) Assume that X$C_W r_0/d_s < {\overline \gs}^2/16$. XSuppose that $S \subseteq A_{n+1}$ is a singular Xregion such that $C_{\ell_{n+1}}(S) \backslash S \equiv D$ Xis nonsingular. Then X$$ X G_{C_{\ell_{n+1}}(S)}(x,y;\zeta) = G_S \oplus G_D + R X\EQ(Greensfunctform) X$$ Xwhere $G_D(x,y;\zeta)$ is analytic for $|\zeta| < 3d_s/4$. XFor any $\zeta, \ |\zeta| < 3d_s/4$ such that X${\rm dist}({\rm spec}(H_{C_{\ell_{n+1}}(S)}), \zeta) > d_s/8$, Xthen X$$\eqalign{ X \|R\|_0 & \leq {C\|\pp\| \over d_s^2}~~, \cr X |R(x,y)| & \leq {C\|\pp\| \over d_s^2 \ga_{n-1}^{10} } X e^{-(\gs_{n-1}-3 \ga_{n-1}) X [{\rm dist}(x,S) + {\rm dist}(S,y)]}~~, \cr X \|R\|_{\gs_{n-1}-4\ga_{n-1}} X & \leq { C\|\pp\| \over d_s^2 \ga_{n-1}^{12} } \cr} X\EQ(formest) X$$ X X X X X XContinuing the proof of the induction step, the X neighborhoods $C_{\ell_{n+1}}(S_{j})$ are X\hfill \break Xpatched back into the region X $B_{n+1}$. This process is performed using a Neumann X series based on the resolvent identities. In X fact this expansion has superior convergence properties if X two sets of domains are used in alternation: the domains X $B_{n+1} - C_{\ell_{n+1}}(S_{k})$, and $C_{\ell_{n+1}} X(S_{k})$, and the same sort X of domains made from $B_{n+1}$ by excising only Xthe resonant regions $\{S_k \}$,rather than the disks X$C(S_k)$. Of course the Green's Xfunctions X $G_{\{S_k\}}$ also satisfy \equ(4.7), by Xvirtue Xof induction hypothesis $(n.2)(ii)$. X X\CLAIM Theorem(T4.5) Suppose that the induction Xhypotheses $(j.1)$ and $(j.2)$, X$j=1, \dots , n$, Xare satisfied. XLet $A$ be any additional nonresonant domain. XThere exist constants $C$ and $C_G$ independent Xof $n$ such that if X$$ X{{r_0}\over{\delta_0 \gamma_0^3}} < 1~~,~~ X{{r_0 C e^{-\gamma_{n-1} \ell_{n+1} /2} }\over{ X\delta_{n+1} \gamma_{n-1}^{15} }} < 1~~, X$$ Xand X$$ X{{r_0 C C_G^n e^{-\gamma_{n-1} \ell_{n+1} /2} }\over{ X\delta_{n+1}^2 \gamma_{n-1}^{34} }} < 1~~~~, X$$ Xand if $E_{n+1} \subset \overline{B_{n+1}} \cup A$, then Xthe Green's function $G_{E_{n+1}} (x,y;z)$ Xsatisfies X$$ X\parallel G_{E_{n+1}} X\parallel_{\sigma_{n-1}- 5\gamma_{n-1}} \leq X{{ C_G^{n+1} }\over{\delta_{n+1} \gamma_{n-1}^{12}}}. X\EQ(4.14) X$$ X XA central tool in the proof of this theorem Xis an estimate of the many exponential sums that arise in Xresolvent expansions. XIn particular there arise many sums over classes Xof paths which, by Xthe arrangement of the decomposition of domains, are forced Xto contain a ``long step". XIn the induction the size of this step will typically be X$\ell_{n+1} /2 $. XThe following elementary lemma concerns such an Xestimate. X X\CLAIM Lemma (L4.6) Let $\{\beta\}_{n,\ell} $ be the collection Xof all paths from $x$ to $y$ in a domain X$B \subseteq \zsquared$, with a fixed number Xof steps, $\# \beta = n$, and a length $\ell(\beta)$ Xof at least $\ell$. Then for any $0<\gamma<\sigma$, X$$ X\sum_{ \{\beta\}_{n,\ell}} Xe^{-\sigma \ell(\beta)} \le {{C^{2n} }\over{\gamma^{2n}}} Xe^{-(\sigma - \gamma)\ell } X$$ X X\PROOF X$$ X\sum_{ \{\beta\}_{n,\ell} } Xe^{-\sigma \ell(\beta)} \le e^{- (\sigma - \gamma)\ell} X\sum_{ \{\beta\}_{n,\ell} } Xe^{-\gamma \ell(\beta)}~~. X$$ XSince X$\ell(\beta) = |x-\beta_1| + |\beta_1-\beta_2| + \dots X+ |\beta_{n-1}-y|$, we interpret X$((x-\beta_1),(\beta_1-\beta_2),\dots,(\beta_{n-1}-y))$ Xas a vector in $\ZZ^{2n}$. Then, X$$ X\sum_{ \{\beta\}_{n,\ell} } Xe^{-\gamma \ell(\beta)} \le X\sum_{ {{v \in \ZZ^{2n} }\atop{ |v| \ge \ell}} } Xe^{-\gamma |v| } \le {{C^{2n} }\over{\gamma^{2n}}} X~~, X$$ Xby an elementary exponential estimate, and the lemma Xfollows. X X\endproof X X X X\PROOF (of \clm(T4.5)). We estimate $G_{E_{n+1}}$, Xfor an arbitrary subset X$E_{n+1} \subset \overline{B_{n+1}} \cup A$ Xby a path expansion. The paths appearing in the Xexpansion of $G_{E_{n+1}}$ are a subset of those appearing Xin the expansion of $G_{\overline{B_{n+1}} \cup A}$, Xso the estimate on $G_{E_{n+1}}$ follows from that Xon $G_{\overline{B_{n+1}} \cup A}$. XThe Xprocedure is to isolate each singular X region $S_{k}\subset A_{n+1}$ with neighborhoods $C_{\ell_{n+1}}(S_{k})$. X Since the radius of $C_{\ell_{n+1}}(S_{n})$ is X $\ell_{n+1}$ the resulting alternating X expansions will be composed of elements whose path X expansions consist of long paths of Xlength $\OO(\ell_{n+1})$. The result is a X convergent Neumann expansion and good estimates of X exponential decay. X XThe resolvent expansion for X$G_{\overline{B_{n+1}} \cup A}(x,y;z)$ Xis the Neumann series constructed from Xresolvent identities Xwhere two sets of Xdomains are used alternately. XLet $B = E \cup_j D_j = \overline{E} X\cup_j \overline{D}_j$, Xwhere $E , \, \{ D_j \}$ and X$\overline{E} , \, \{ \overline{D}_j \}$ Xare collections of disjoint domains of the lattice. XThen we can express X$$ X H_{B} = H_{E} \oplus_j H_{D_j} + \Gamma_{ED} X = H_{\overline{E}} \oplus_j H_{\overline{D}_j} + X \Gamma_{\overline{ED}}. X\EQ(4.17) X$$ XApplying the first resolvent identity to both of these Xrelations X$$\eqalign{ X G_{B} &= G_{E} \oplus_j G_{D_j} + X G_{E} \oplus_j G_{D_j} \Gamma_{ED} G_{B} \cr X & = G_{\overline{E}} \oplus_j G_{\overline{D}_j} X + G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}} G_{B}. X \cr} X\EQ(4.18) X$$ XAlternately using the first and second of these formulas, a Xformal expansion of the Green's function is obtained. X$$\eqalign{ X G_{B} = G_{E} \oplus_j G_{D_j} X &+ G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D_j}} \cr X & + \sum^{\infty}_{\ell=1} X (G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}})^{\ell} \times \cr X & \times \bigl( G_{E} \oplus_j G_{D_j} + X G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j X G_{\overline{D_j}} \bigr). X \cr} X\EQ(4.19) X$$ XWe will use the expansion in this paper with the Xcollections $\{ D_j \}$ and $ \{ \overline{D_j} \}$ Xthe set of singular regions, $S_k$, or their neighborhoods X$C_{\ell_{n+1}}(S_k)$. X X XFor situations in which all operators in this expression Xhave exponential off diagonal decay, and where there is a Xlarge distance between domains $E$ and $\overline{D}_j$, X\equ(4.19) Xresults in an expansion with superior convergence properties. XThe proof of the estimate of X$G_{\overline{B_{n+1}}\cup A}$ Xconsists in making choices of the sets X$E, \{D_j \}$ and $\overline{E}, \{ \overline{D}_j \}$, Xand estimating the formal sum X\equ(4.19). Cases are chosen as follows: X X\itemitem{Case 1:} XIf $\dist(x,S_{k}) > \half \ell_{n+1}$, Xfor all singular regions $S_{k} \subset A_{n+1}$, we set X$D_j = \{ S_j \}$, where the index runs over all singular Xregions in $A_{n+1}$. Then define X$E = \overline{B}_{n+1} \cup A - (\cup_j D_j)$. XAlso let $\overline{D}_j = C_{\ell_{n+1}}(S_j)$, and X$\overline{E} = \overline{B}_{n+1} X\cup A - (\cup_j \overline{D}_j)$. X X\itemitem{Case 2:} XIf $\dist(x,S_{k}) \le \half \ell_{n+1}$, Xfor some singular region $S_{k} \subset A_{n+1}$, chose X$D_j = C_{\ell_{n+1}}(S_j)$, for each of the singular Xregions $S_j \subset A_{n+1}$, and $E = \overline{B}_{n+1} X\cup A - (\cup_j D_j)$. Similarly, take X$\overline{D}_j = \{ S_j \}$, and X$\overline{E} = \overline{B}_{n+1} X\cup A - (\cup_j \overline{D}_j)$. X XAddress first the block diagonal term X$G_{E} \oplus_j G_{D_j}$ Xof \equ(4.19). In both cases, the domain X$E \subset \overline{B_n} \cup {A}$, for some set X${A}$ consisting entirely of non-singular sites. XThus, by the \clm(Greens), X $ \parallel G_{E} \parallel_{\gs_{n-1} - 2\ga_{n-1}} X \leq 2 C_G^n / (\gd_{n} \ga_{n-1}^{12})$. XOn the other hand, $G_{D_{\ell}}$ is either of the form X$ G_{ \{ S_{\ell} \} }(x,y;z)$ or X$ G_{C_{\ell_{n+1}}(S_{\ell})}(x,y;z)$, Xso \clm(L4.4) implies that X$\Vert G_{D_{\ell}} \Vert_{\sigma_{n-1} -4\gamma_{n-1}} X \le C_0 / (\gd_{n+1} \ga_{n-1}^{12})$ Xfor all singular regions of $A_{n+1}$. XThus, we have proved X X\CLAIM Lemma(blockdiag) If the hypotheses of X\clm(L4.4) hold then there exists a Xconstant $C>0$ such that X$$ X \parallel G_E \oplus_j G_{D_j} X \parallel_{\sigma_{n-1} - 4\gamma_{n-1}} \le X {{C C_G^n} \over { \delta_{n+1} \gamma_{n-1}^{12} }}~~. X\EQ(Gamma0) X$$ X XThe term in \equ(4.19) linear Xin the coupling $\Gamma$ Xinvolves the choices of decomposition more directly. In Case 1, X$$ X (G_E \oplus_j G_{D_j}) \Gamma_{ED} (G_{\overline{E}} X \oplus_j G_{\overline{D}_j})(x,y) = X \sum_{ \scriptstyle p \in E X \atop \scriptstyle q \in D_j } \, X G_E(x,p) \Gamma_{ED}(p,q) G_{\overline{D}_j}(q,y)~~. X\EQ(breakup) X$$ X XIn deriving this expression we used the fact that X$x \in E$. XBecause $G_E$ has block diagonal form, the only nonzero Xcontributions to the sum are for $ p \in E$. XThis implies that $\Gamma_{ED}(p,q) = 0$ unless X$q \in D_j $ for some index $j$. XThe estimates of exponential off diagonal decay of Xoperators are now used; X $|G_E(x,p)| \le X (2 C_G^n / (\delta_n \gamma_{n-1}^{12})) X e^{-(\sigma_{n-1}-2\gamma_{n-1}) |x-p|}$, Xby \clm(Greens), X $|\Gamma_{ED}(p,q)| \le ( C_W \|\pp \| /\gamma_{n-1}^3) X e^{-(\sigma_{n-1} - 2\gamma_{n-1}) |p-q|}$, Xand X$$ X|G_{\overline{D}_j}(q,y)| X \le (C / (\gd_{n+1} \gamma_{n-1}^{10})) X e^{-(\gs_{n-1} - 3\ga_{n-1})|q-y|}, X$$ Xfrom \clm(L4.4). Since X$\dist(x,S_{\ell}) > \ell_{n+1} / 2$ Xand in the sum the only nonzero coupling terms are for X$q \in D_j = \{ S_{\ell} \}$; $|x-p|+|p-q| > \ell_{n+1} /2$. XWe refer to this as the long step in the sum. XThe geometry of this path is illustrated in figure 1. X X\input fig1.tex X\centerline{\bf figure 1} X XThis is just the situation addressed by \clm (L4.6), thus Xthe sum \equ(breakup) is bounded by X$$\eqalign{ X |\sum_{ \scriptstyle p\in E X \atop \scriptstyle q \in D_j} & X G_E(x,p) \Gamma_{ED}(p,q) G_{\overline{D}_j}(q,y)| \cr X &\le \sum_{ \scriptstyle p \in E X \atop \scriptstyle q \in D_j} X { C C_G^n \| \pp \| \over X \gd_n \gd_{n+1} \ga_{n-1}^{25}} X e^{-(\gs_{n-1} - 3\ga_{n-1}) (|x-p| + |p-q| + |q-y|)} \cr X &\le { C C_G^n \| \pp \| \over X \gd_{n+1}^2 \ga_{n-1}^{27}} X e^{-\ga_{n-1} \ell_{n+1}/2} e^{-(\gs_{n-1} - 4\ga_{n-1})|x-y|} X \cr } X\EQ(step1) X$$ X X XIn case 2, the sums corresponding to \equ(breakup) are X$$\eqalign{ X\bigl| (G_E \oplus_j G_{D_j}) & \Gamma_{ED} (G_{\overline{E}} X \oplus_j G_{\overline{D}_j})(x,y) \bigr| \cr X = & \bigl| \sum_{ \scriptstyle p \in D_{\ell} \atop X \scriptstyle q \in E \cup_{k \not= \ell} D_k } X G_{D_{\ell}}(x,p) \Gamma_{ED}(p,q) X ( G_{\overline{E}}(q,y) \oplus_k X G_{\overline{D_k}}(q,y)) \bigr| \cr X \le& \sum_{ \scriptstyle p \in D_{\ell} \atop X \scriptstyle q \in E \cup_{k \not= \ell} D_k } X \left({{ C}\over{\delta_{n+1} \gamma_{n-1}^{10} }}\right) X \left({{C_W \| \pp \| }\over{ \gamma_{n-1}^3 }} \right) X \left( {{2}\over{\delta_{n+1} }} + {{C_G^n}\over{ X \delta_n \gamma_{n-1}^{12} }} \right) \times \cr X& \qquad \quad X \times e^{-(\gs_{n-1} - 3\gamma_{n-1})(|x-p| + |p-q| + |q-y|)} \cr X \le& {{ C C_G^n \| \pp \| X }\over{\delta_{n+1}^2 \gamma_{n-1}^{27} }} X e^{-\ga_{n-1} \ell_{n+1}/2} Xe^{-(\gs_{n-1}-4\ga_{n-1})|x-y|} X \cr } X\EQ(step2) X$$ X X\input fig2.tex X\centerline{\bf figure 2} X XIn this case the ``long step'' results from the Xfact that $q \in E \cup_{k \not= \ell} D_k$ while X$\dist (x,s_{\ell}) \leq \ell_{n+1} / 2$. Thus X$|x-p| + |p-q| > \ell_{n+1}/2$, and again \clm(L4.6) can be Xapplied to the exponential sum. The geometry of this case is Xillustrated in figure 2. Combining estimates \equ(step1) and X\equ(step2) we have the following decay result. X X\CLAIM Lemma(linear) There exists a constant $C>0$, independent of X$n$ such that if X$$ X{{r_0 C e^{-\gamma_{n-1} \ell_{n+1}/2} }\over{ X\delta_{n+1} \gamma_{n-1}^{15} }} < 1~~, X\EQ(linearest) X$$ Xthen X$$ X \parallel (G_E \oplus_j G_{D_j}) \Gamma_{ED} X (G_{\overline{E}} \oplus_j G_{\overline{D}_j}) X \parallel_{\gs_{n-1}-5\gamma_{n-1}} X \leq {{1}\over{4}} {{C_G^n}\over{\delta_{n+1} \gamma_{n-1}^{12} }}~~. X\EQ(Gamma1) X$$ X X\REMARK In the case where the nonlinearity contains no quadratic Xterms, \equ(linearest) can be replaced by X$$ X{{r_0^2 C e^{-\gamma_{n-1} \ell_{n+1}/2} }\over{ X\delta_{n+1} \gamma_{n-1}^{15} }} < 1~~. X\EQ(newlinearest) X$$ X X XFinally, we bound the infinite series in \equ(4.19). XThe estimates are much like those immediately preceeding. XWe will prove X X\CLAIM Lemma(series) There exists a constant $C>0$, independent Xof $n$, such that if X$$ X{{r_0^2 C G_G^n e^{-\gamma_{n-1} \ell_{n+1}/2} }\over{ X\delta_{n+1}^2 \gamma_{n-1}^{34} }} < {{1}\over{4}}~~, X\EQ(opest) X$$ Xthen X$$ X \parallel G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}} \parallel_{\gs_{n-1}-5\ga_{n-1}} X \leq {{1}\over{4}} ~~. X\EQ(4.26) X$$ X X X\PROOF Consider first case 1. Then, X$$ X |G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}}(x,y)| = X \bigl| \sum_{{p\in E \atop q \in D_j} \atop X r \in \overline{D}_j} X G_E(x,p) \Gamma_{ED}(p,q) G_{\overline{D}_j}(q,r) X \Gamma_{\overline{E} \overline{D} }(r,y) \bigr| X\EQ(step3) X$$ X XHere the restrictions on the location of $p$ and $q$ Xare the same as in \equ(breakup), Xwhile $r \in \overline{D_j}$ because X$q \in D_j \subseteq \overline{D}_j$. XIn fact the sum vanishes unless X$y \in \overline{E} \cup_{k \not= j} \overline{D}_j$, for Xonly then is X$\Gamma_{\overline{E}\overline{D}}(r,y) \not= 0$. XThese paths also have a long step between $x$ and $q$, Xwhence $|x-p|+|p-q|> \ell_{n+1}/2$. XUsing additionally the estimate X $|\Gamma_{\overline{E} \overline{D} }(r,y)| X \le ( \| \pp \| C_W / \gamma_{n-1}^3) Xe^{-(\gamma_{n-1} - X 2\gamma_{n-1} |r-y|)}$, X\equ(step3) is bounded by an exponential sum; X$$\eqalign{ X |G_{E} & \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}}(x,y)| \cr X \le & \sum_{ {{ {{p\in E }\atop{ q \in D_j}} }\atop{ X r \in \overline{D}_j} }} X \left( {{C C_G^n } \over X {\gd_n \gd_{n+1} \ga_{n-1}^{22} }} \right) X \left( {{C_W \| \pp \|} \over{\gamma_{n-1}^4 }} \right)^2 X e^{-(\gs_{n-1} - 3 \gamma_{n-1})(|x-p|+|p-q|+|q-r|+|r-y|)} \cr X \le& {{ C C_G^n \| \pp \|^2}\over{ X \delta_{n+1}^2 \gamma_{n-1}^{32} }} X e^{-\ga_{n-1} \ell_{n+1} /2} e^{-(\gs_{n-1} X - 4 \gamma_{n-1})|x-y|)}~~. \cr} X$$ X\clm(L4.6) is used above in estimating the Xexponential sums over paths with large steps. XIn Case 2, a very similar computation, using the estimates Xof \equ(step2) leads to an identical conclusion--we omit Xthe details. X XIf we now apply \clm(opnorm) we find X$$ X\parallel G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \Gamma_{\overline{ED}} X \parallel_{\gs_{n-1} - 5\gamma_{n-1}} X \le {{ C C_G^n \| \pp \|^2}\over{ X \delta_{n+1}^2 \gamma_{n-1}^{34} }} X e^{-\ga_{n-1} \ell_{n+1} /2} ~~. X$$ X X XThus, if X$$ X{{ C C_G^n \| \pp \|^2}\over{ X \delta_{n+1}^2 \gamma_{n-1}^{34} }} X e^{-\ga_{n-1} \ell_{n+1} /2} < 1/4, X$$ Xthe operator norm $\parallel G_{E} \oplus_j G_{D_j} X \Gamma_{ED} G_{\overline{E}} \oplus_j X G_{\overline{D}_j} \Gamma_{\overline{ED}} X \parallel_{\gs_{n-1}-5\gamma_{n-1}} \le 1/4$, Xand the \clm(series) follows. X X\endproof X X\REMARK If the nonlinearity in \equ(NLlattice) contains no quadratic Xterm, estimate \equ(opest) may be replaced by X$$ X{{r_0^4 C G_G^n e^{-\gamma_{n-1} \ell_{n+1}/2} }\over{ X\delta_{n+1}^2 \gamma_{n-1}^{34} }} < {{1}\over{4}}~~, X\EQ(newopest) X$$ X X X X XThe full resolvent Xexpansion \equ(4.19) is Xnow easily estimated in operator norm. XUsing estimates \equ(Gamma0), \equ(Gamma1) and X\equ(4.17), X$$\eqalign{ X \parallel G_{B} \parallel_{\gs_{n-1}-5\gamma_{n-1}} X \leq & \parallel G_{E} \oplus_j G_{D_j} X \parallel_{\gs_{n-1}-5\gamma_{n-1}} \cr X & \qquad \qquad \qquad + X\parallel G_{E} \oplus_j G_{D_j} \Gamma_{ED} X G_{\overline{E}} \oplus_j G_{\overline{D}_j} X \parallel_{\gs_{n-1}-5\gamma_{n-1}} \cr X & \quad + \sum^{\infty}_{\ell=1} X \parallel (G_{E} \oplus_j G_{D_j}) \Gamma_{ED} X (G_{\overline{E}} \oplus_j G_{\overline{D}_j}) X \Gamma_{\overline{ED}} X \parallel_{\gs_{n-1}- X 5\gamma_{n-1}}^{\ell} \times \cr X& \qquad \qquad \bigl( \parallel G_{E} \oplus_j G_{D_j} X \parallel_{\gs_{n-1}-5\gamma_{n-1}} \cr X & \qquad \qquad \quad X + \parallel (G_{E} \oplus_j G_{D_j}) \Gamma_{ED} X (G_{\overline{E}} \oplus_j X G_{\overline{D}_j}) X \parallel_{\gs_{n-1}-5\gamma_{n-1}} \bigr) \cr X \leq & \bigl( {{C C_G^n }\over{ \gd_{n+1} \ga_{n-1}^{12} }} X + {1 \over 4} {{ C C_G^n}\over{ X \gd_{n+1} \ga_{n-1}^{12}}} \bigr) X \sum_{\ell=0}^{+\infty} { 1 \over 4^{\ell} } X \cr X\leq & {{G_G^{n+1} }\over{ \delta_{n+1} \gamma_{n-1}^{12}}}~~. \cr X} X\EQ(N_series) X$$ XThe requirements on the choice of $\ga_{n-1}$ Xwill be easily satisfied in the induction, as the radii X$\ell_{n+1}$ of isolating neighborhoods of resonant sites Xgrow exponentially, while the tolerance $\delta_{n+1}$ for Xsmall divisors behaves similarly, but does not appear in the Xexponent of constants. This finishes the proof of X\clm(T4.5). X X\endproof X X\SUBSECTION The Newton step: X XThe second step in the Xinduction is to perform a correction to the approximation Xgiven by Newton's method. That is, the new approximate Xsolution is $u_{n+1}= u_{n}+v_{n}$, where $v_{n}$ is Xobtained as a solution of a linear equation on the domain X$\overline{B_{n+1}}$, using the Green's function X$G_{\overline{B_{n+1}}}(\pp,\Omega,u_{n})$. Linearizing Xthe nonlinear equation about this new approximation changes Xit; $H_{\overline{B_{n+1}}}(\pp,\Omega, u_{n+1})$ and its Xinverse must be estimated. Due to the rapid convergence of Xthe Newton scheme these corrections are small and the Xestimates are not difficult. From Xsection 3, we know that X$$ X\parallel v_{n}\parallel_{\gs_{n}-\ga_n} X \leq {{\| \pp \|^2 \gre_{n} C_G^{n+1} } \over {\delta_{n+1} X \gamma_n^{12} }}. X\EQ(4.30) X$$ X XThe following estimates are designed to correct the operator X$H_{B_{n+1}}(u_{n+1})$ and the Green's function X$G_{\overline{B_{n+1}}}$ due to these adjustments -- in fact Xthey are valid for any sequence $v(x)$ satisfying inequality X\equ(4.30). X X\CLAIM Lemma(L4.8) If one adds to the function X$u_{n}(x)$ any $v(x)$ satisfying the estimate X\equ(4.30), the modified Hamiltonian has the form X$$ XH_{E_{n+1}} (u_{n}+v)= H_{E_{n+1}}(u_{n}) X+R_{n+1}(u_{n},v)~~, X\EQ(4.31) X$$ Xfor any $E_{n+1} \subset \overline{B_{n+1} \cup A}$. XThen $R_{n+1}$ Xsatisfies the estimate X$$ X\parallel R_{n+1}(u_{n} ; v)\parallel_{\sigma_{n}-2\gamma_n} X\leq X{{ \| \pp \|^2 \epsilon_n C_W C_G^{n+1} X }\over{\delta_{n+1} \gamma_{n}^{15} X}} X\EQ(4.32) X$$ X X\PROOF The linear operator X$H_{B}(u)=V_{B}(\Omega)+ DW_{B}(\pp,u)$ consists of a Xlinear Xdiagonal part, which is independent of $u_n+v_n$, Xand an off diagonal piece $ D W( \phi(\pp) + u)$. X From estimate ${\bf H2}$ of Section 6, $D_u^2 W$ has operator Xnorm bounded by $C_W \|\pp\| \ga^{-3}$ on X$\HH_{\gs_n-\ga}$. XThe estimate on $R_{n+1}$ Xfollows from Taylor's theorem. X X\endproof X XWhen $(\pp,\Omega)\in \NN_{n+1}$ the spectrum of X$H_{E_{n+1}}(u_{n})$ Xis bounded by $\delta_{n+1}/2$ Xaway from zero. The induction provides for perturbations X$R_{n+1}$ satisfying \equ(4.32), which are substantially Xsmaller Xthan this. The Green's function $G_{E_{n+1}}(u_{n+1})$ X can be constructed from $G_{E_{n+1}}(u_{n})$ Xand $R_{n+1}$ via Neumann series for $z$ in the Xset $\{z; |z|\leq \delta_{n+1}/4\}$. This gives the Xfollowing result for the construction. X X\CLAIM Theorem(T4.9) Under perturbations of size X\equ(4.32), the Green's function remains a bounded operator Xon X${\cal{H}}_{\sigma_{n}-2\gamma_n}$ for all $|z|\leq X\delta_{n+1}/4$, as long as X$$ Xr_0^2 \epsilon_n \le {{\delta_{n+1}^2 \gamma_{n}^{27} }\over{ X2 C_W C_G^{2(n+1)} }}~~, X$$ Xand one Xhas the estimate Xthat X$$ X\parallel G_{E_{n+1}} X(u_{n+1})\parallel_{\sigma_{n}- 2\gamma_n} \leq X{{2 C_G^{(n+1)} }\over{\delta_{n+1} X \gamma_n^{12} }}. X\EQ(4.33) X$$ X X\PROOF Simple Neumann series gives that X$$ XG_{E_{n+1}}(u_{n}+v)=G_{E_{n+1}}(u_{n})\left(1 + R_{n+1} XG_{E_{n+1}}(u_{n})\right)^{-1}, X$$ Xso that it suffices to bound $\parallel R_{n+1} XG_{E_{n+1}}(u_{n})\parallel_{\gs_{n}- 2\ga_n} \leq 1/2$. XThe stated condition on the constants assures that this will Xbe the case. X X\endproof X X XNote that \clm(T4.5) and \clm(T4.9) imply \clm(Greens) Xwhich we used repeatedly in section 4. X X XThe same method allows us to carry through an estimate on the XGreen's function on nonsingular domains. One has the X\CLAIM Corollary(nonsingular) Let $A\subset \zsquared$ be a nonsingular Xdomain and $u_n = u_0 + \sum_{j=0}^{n-1} v_j$, for X$n\ge 1$. If X$$ X\| v_j \|_{\sigma_j - \gamma_j} \le X{{1}\over{4^{j+1} }} {{\gamma_{j+1}^6 \gamma_0^3 \delta_0}\over{ X 2C_W C_0 }} X$$ Xand if $|z| < \delta_{n+1}/2$, then X$$ X\| G_A(u_n;z)\|_{\sigma_{n-1}-2\gamma_{n-1} } \le {{2 C_0}\over{d_s X\gamma_0^3}}~~. X$$ X XThe proof is again a simple application of Neumann series--we Xdo not repeat it. X X X X END_OF_FILE if test 40593 -ne `wc -c <'sec5.tex'`; then echo shar: \"'sec5.tex'\" unpacked with wrong size! fi # end of 'sec5.tex' fi if test -f 'sec6.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sec6.tex'\" else echo shar: Extracting \"'sec6.tex'\" \(38791 characters\) sed "s/^X//" >'sec6.tex' <<'END_OF_FILE' X X X\SECTION Analysis of the nonlinear operator X XFor the nonlinear wave equation \equ(NLW) it is natural to Xassume that the nonlinear term $g(x,s)$ is periodic Xin $x$ with period $\pi$, and analytic in the region X$\{ (x,s) \in {\bf C}^2; X |{\rm Im} \ x| \leq \overline \sigma \}$. XThese conditions translate into conditions for the Xnonlinear operator $W$ applied to sequences Xdefined over the lattice $\ZZ^+ \times \ZZ$. XIn stating these Xestimates, we will fix an index $\sigma_* < X\overline \sigma$, and then give estimates Xthat will hold for all $\sigma < \sigma_*$. XThe constants will Xdepend upon $\overline \sigma$ and $\sigma_*$, but Xwill be uniform in $\sigma < \sigma_*$. X XThis section also addresses several Xof the properties of the nonlinear wave equation that Xhave been stated in Section 2, including the properties Xof genericity of $(d_0,L_0)$-nonresonant frequency Xsequences $\{ \omega_j \}_{j=1}^\infty$. This will Xfinish the proofs of the main theorems, X\clm(NLWperiodic) and \clm(NLWDir). Finally, there Xis a proof of \clm(density) on the accumulation of Xperiodic orbits at zero in the fully non-resonant Xcase. X X\SUBSECTION Analyticity properties of $W$ X XThe basic property of the nonlinear operator $W$ Xin the nonlinear lattice equation \equ(NLlattice) Xis that it is an analytic mapping $W: \HH_\gs X\rightarrow \HH_{\gs-\gamma}$, for any $\gs < \gs_*$ . XThis works well with the techniques of estimating Xexponentially decaying matrices in Section 5. XHowever we will use slightly stronger hypotheses Xon the nonlinearity. These hold in the case of Xthe nonlinear wave equation of the original problem, Xas we will prove below. Since we are interested in Xsolutions of the nonlinear problem which are small Xperturbations of the normal modes of the linear Xproblem, these properties are stated in terms Xof perturbations from a given linear mode $\phi(\pp)$. XThe additional assumptions are estimates on Xperturbations of $W(\phi(\pp))$ and its first and Xsecond derivatives. Let $0<\gs<\gs_*$. X X X X X X\item{{\bf H1}} If $u \in \HH_{\sigma}$, Xwith $Q u = 0$ and X$\|u\|_\gs \leq r_0$, then for any X$0 < \ga \leq \gs, \ X W(\phi(\pp) + u) \in \HH_{\gs-\ga}$, and X$$ X \|W(\phi(\pp) + u)\|_{\gs-\ga} \leq X {C_W \over \ga}(\|\pp\|^2 + X {1 \over \ga^2}\|u\|^2_\gs). X\EQ(s6.1) X$$ X X X X\item{{\bf H2}} If $u \in \HH_{\sigma}$, Xwith $ Q u = 0$ and X$\|u\|_\gs \leq r_0$, then for any $0 < \ga \leq \gs$, X$D_u W(\phi(\pp)+u)$ is a bounded operator on X$\HH_{\gs-\ga}$, Xwith operator norm given by the estimate X$$ X \| D_u W(\phi(\pp) + u)\|_{\gs-\ga} \leq X {C_W \over \ga^2} ( \|\pp\| + {1 \over \ga} X \|u\|_\gs) X\EQ(s6.2) X$$ X X X X\item{{\bf H3}} If $u \in \HH_\gs$, with $Qu = 0$ Xand $\|u\|_\gs \leq r_0$, then for any $0 < \ga \leq \gs$, XThe second derivative $D^2_u W(\phi(\pp) + u)$ Xsatisfies the estimates X$$ X \| D^2_u W (\phi(\pp) + u)[v,w] \|_{\gs-\ga} \leq X { C_W \over \ga^3} \| v \|_\gs \| w \|_{\gs-\ga}. X\EQ(s6.3) X$$ XFurthermore this bilinear operator can be written in the Xform $D_u^2 W = C + \Delta$, where $\Delta$ satisfies X$$ X \| \Delta [v,w]\|_{\gs-\ga} \leq X {C_W \over \ga^3} (\|\pp\| + {1 \over \ga}\| u \|_\gs) X \| v \|_\gs \| w \|_{\gs-\ga}. X\EQ(decomp1) X$$ XThe operator $C = D_u^2 W(0)$ satisfies the property X$$ X \langle \gd(z), C[\phi(\pp),\gd(z')] \rangle_0 X = 0 X\EQ(decomp2) X$$ Xfor any $z,z'$ with $(z-z') = (0,0)$ or $(\pm 1,0)$. X XThe hypotheses \equ(decomp1) and \equ(decomp2) are Xnatural, and always satisfied by the nonlinear wave Xequation, where $C$ depends only upon the coefficient X$g_2(x)$. In fact if $D^2_u W(0) = 0$, that is the Xnonlinear term is of cubic or higher order at zero, Xthen $C = 0$. The role of hypotheses \equ(decomp1) and X\equ(decomp2) is to avoid overly strong nonlinear Xcoupling between resonances associated with lattice Xsites within the same singular region. XThey are used in section 4 Xin the analysis of the linearized operators $D_u W$ and Xtheir dependence on parameters. In the above estimates, Xthe constant $C_W$ is of course independent of $\ga$. X X\SUBSECTION Circumstances in which $H1$-$H3$ are satisfied. X XWe consider two cases of boundary conditions for equation \equ(NLW) for Xwhich hypotheses $H1$-$H3$ are satisfied by the associated Xnonlinear operator on the lattice. XThe first case is for periodic boundary conditions on the interval X$[0,\pi]$, and the second is Dirichlet Xboundary conditions imposed on the endpoints $0,\pi$. In this second case Xthere is an additional restriction on the non-linearity. X X\noindent X{\bf Periodic Boundary Conditions:} Assign periodic Xboundary conditions to the Sturm-Liouville Xoperator $L_{g_1} = -{{d^2}\over{dx^2}} + g_1(x)$ Xon $[0,\pi]$. XThe potential $g_1$ is assumed to be periodic, of period $\pi$, Xand analytic in the strip X$\{ x; \ |{\rm Im} \ x| < \overline{\gs} \}$. XThe spectral asymptotics for Sturm-Liouville Xoperators guarantees that there Xare at most finitely many negative eigenvalues. XLabel the eigenvalues X$$ X \omega_0^2 < X \omega_{1}^2 \le \omega_2^2 < \omega_{3}^2 \le, \dots~~. X$$ Xand let $\{\psi_n(x)\}_{n=0}^{\infty}$ Xbe the corresponding Xsequence of eigenfunctions. XThe main technical result of this section is Xthat these eigenfunctions have Xuniform properties of analyticity in X$\{ x; \ |{\rm Im} \ x| < \overline{\gs} \}$. X X\CLAIM Proposition(expand) Expand $\psi_n(x) = X\sum_m \hat{\psi}_n(m) e^{i m x}$. XThen for $\sigma_* < \overline{\sigma}$, Xthere are constants $C_1(g_1,\sigma_* )$ Xand $C_2(g_1,\sigma_*)$, (independent of $n$) such that Xfor $m > 0$ one has X$$ X | \hat{\psi}_n(m) | \le C_1(g_1,\sigma_*) X e^{-\sigma_* |n - | m | |}~~. X\EQ(s6.4) X$$ XConversely, if one expands $e^{i n x} = X\sum_m \hat{e}_n(m) \psi_m(x)$ Xthen X$$ X | \hat{e}_n(m) | \le C_2(g_1,\sigma_* ) X e^{- \sigma_* | |n| - m | }~~. X\EQ(s6.5) X$$ X X\REMARK The eigenfunctions $\psi_n(x)$ Xfor the periodic problem are analytic Xin the strip $\{ x; |{\rm Im} \ x| < \overline \gs \}$, thus Xthis result ``almost'' follows from analyticity of the Xeigenfunctions. The one thing that must be checked is that the Xconstants $C_{1,2}(g_1,\sigma)$ can be chosen uniformly in $n$. XThe proof of this proposition will be given below, Xfollowing a discussion of some of its consequences. X X{}From this proposition we obtain the following useful Xcorollary. Let X$\DD_{\sigma} = X\{(x,\xi )\in {\bf C}^2 ;|{\rm Im} \ x| < \sigma , X{|\rm Im} \ \xi| < \sigma \}$. XIf $\tilde f (x,\xi )$ is analytic on $\DD_{\sigma }$, and periodic Xwith period $\pi$ in $x$ and $2\pi$ in $\xi$, Xdefine $||| \tilde f |||_{\sigma } = X\sup_{(x,\xi) \in \DD_{\sigma }}| \tilde f (x,\xi )|$. XLet $f$ be the coefficients in the eigenfunction Xexpansion of $\tilde f$. X X\CLAIM Corollary(analyticity) There exist constants X$C_1(g_1,\sigma_*)$ Xand $C_2(g_1,\sigma_*)$ such that for all X$0 < \ga \leq \gs < \gs_*$, X$||| \tilde f |||_{\sigma -\gamma } X < (C_1/\gamma) || f ||_{\sigma }$ Xand $||f||_{\sigma -\gamma } < X (C_2/\gamma) ||| \tilde f |||_{\sigma -\gamma }$. X X\PROOF One just writes out $\tilde f(x,\xi )$ Xin terms of its eigenfunction Xexpansion, substitutes for the factors $\psi_n(x)$ their expansion Xin terms of exponentials, and then uses the estimates of \clm(expand) Xto bound the resulting sums. To estimate $f$ one writes Xdown the definition of this quantity, and again bounds it using Xthe estimates of \clm(expand). X X\endproof X XWith the proposition in hand, verifying the hypotheses on $W$ is Xstraightforward. We begin by supposing that $u \in \BB_{\sigma }$ Xwith $\| u \|_{\sigma } \le C_u \|\pp\|^2$ and that $g(x,u)$ is analytic Xfor $(x,u) \in \{|{\rm Im} \ x| \leq \overline \sigma \}$, Xfor some $\overline \gs > 0$. Then X$\tilde W (\phi(\pp) + u) = (g-g_1)(x,\tilde \phi (\pp) + \tilde u)$. XBy \clm(analyticity), Xfor any $0 < \gamma \leq \gs $, $|||\tilde \phi (\pp) X + \tilde u |||_{\sigma -\gamma/2} X < (C/\gamma) (\|\pp\|\ + C_u\|\pp\|^2)) $. XThen, $(g-g_1)(x,\tilde \phi (\pp)(x,\xi )+\tilde u (x,\xi ))$ Xis analytic for X$(x,\xi) \in \DD_{\sigma -\gamma/2}$. Furthermore, X$$ X \| W(\phi + u)\|_{\sigma -\gamma } X \le {C_2 \over \gamma} X||| (g-g_1) (x,\tilde{\phi} + \tilde u (x,\xi) )|||_{\sigma -\gamma/2 } ~~. X$$ XSince $(g-g_1)$ is of at least quadratic order in $\tilde u$ for X$|||\tilde u||| \leq 1$, X$||| (g-g_1)(x,\tilde \phi + \tilde u)|||_{\sigma -\gamma/2} X\le C_3 |||\tilde \phi (\pp) + \tilde u |||^2_{\gs-\gamma/2}$ Xas long as $|||\tilde \phi (\pp) X + \tilde u |||_{\sigma -\gamma/2} \leq 1$. XThus, X$$ X\| W(\tilde \phi + \tilde u) \|_{\sigma -\gamma } X \le {C_W \over \gamma} (\|\pp\|^2 X + {1 \over \gamma^2}\|u\|^2_{\gs-\gamma}) X$$ XThis verifies {\bf H1}. X X\REMARK In the case when the nonlinearity contains no quadratic Xterm in its Taylor series, one can improve this estimate to X$$ X\| W(\phi(\pp) + u) \|_{\sigma - \gamma} X \le {C_W \over \gamma}(\|\pp\|^3 + X {1 \over \gamma^3}\|u\|^3_{\gs-\gamma}). X~~, X$$ Xwith similar improvements reflected in the estimates of Xthe first and second derivatives of $W$. X XWe now prove that the derivative of $W$ satisfies {\bf H2}. Let X$x = (j,k)$ and $x' = (j',k')$ be two points in X$\zsquared$. Then X$$\eqalign{ X D_u W(\phi(\pp) + u) (x , x') = & X \, \langle \gd(x), D_u W(\phi(\pp) + u) \gd(x') \rangle_0 \cr X = & \int \int \ \overline{\psi}_j(x) X \psi_{j'}(x) e^{i \xi (k' - k)} X D_u (g-g_1)(\tilde{\phi}(x, \xi ) + \tilde u (x,\xi )) dx d\xi \cr X} X$$ X XIf we write $\overline{\psi}_j(x) = X\sum_{n} \overline{\hat{\psi}}_j(n) e^{-inx}$, Xand a similar expression for $\psi_{j'}(x)$, we have X$$ X D_u W(x,x') = \sum_{n,n'} \overline{\hat{\psi}}_j(n) X {\hat{\psi}}_{j'}(n') \int \int e^{i x(n-n')} X e^{i \xi(k-k')} D_u (g-g_1)(\tilde \phi(\pp) X + \tilde u) \ dx d\xi X$$ XSince $D_u g(\tilde{\phi}(x,\xi ) + \tilde u (x,\xi ))$ Xis analytic for X$(x,\xi ) \in \DD_{\sigma -\gamma/2}$, the double integral is bounded Xin absolute value by X$$ Xe^{-(\sigma -\gamma/2)(|k-k'|+|n-n'|)} X |||D_u (g-g_1)(\tilde \phi (\pp) + \tilde u) |||_{\sigma - \gamma/2} ~~. X$$ XThus by \clm(expand) X$$\eqalign{ X |D_u W (x,x') | \le& X C ||| D_u (g-g_1)(\tilde \phi +\tilde u) |||_{\sigma - \gamma/2} \cr X &\times \sum_{n,n'} e^{- (\sigma - \gamma/2)(|k-k'|+|n-n'|) } X e^{-\sigma_* (|j-|n|| + |j'-|n'||)} \cr X \le& C ||| D_u (g-g_1)(x,\tilde \phi X + \tilde u) |||_{\sigma - \gamma/2} X e^{-(\sigma - \gamma/2 )(|k-k'|+|j-j'|)} \cr X} X$$ Xwith constant $C=C(\gs_*-\gs)$. XThen \clm(opnorm) implies X$$ X \| D_u W \|_{\sigma -\gamma } X \le {C \over \gamma^2} X |||D_u (g-g_1)(\tilde \phi + \tilde u)|||_{\sigma - \gamma/2} ~~. X$$ XNote that since $(g-g_1)(x,s)$ is at least quadratic in $s$, X$|D_u (g-g_1)(x,s)| \le C |s|$, so if X$\| u \|_{\sigma } \le C_u \|\pp\|^2$, then X$||| D_u (g-g_1)(\tilde{\phi} + \tilde u) |||_{ \sigma -\gamma/2} X\le C (\|\pp\| + |||\tilde u |||_{\gs-\gamma/2})$. XThis, when combined with the previous estimates of $|||\tilde u|||_{\gs/2}$ Xcompletes the verification of {\bf H2}. X X XFinally we turn to the verification of {\bf H3}. The boundedness of Xthe second derivative $D_u^2 W$ is almost the same as the Xestimate for {\bf H2}, so we will not include the details here. XThe decomposition $D_u^2 W = C + \Delta$ for the nonlinear wave Xequation arises from the Taylor expansion of the nonlinear term. X$$\eqalign{ X D^2_{\tilde u} g(x,\tilde \phi + \tilde u) =& \ X 2g_2(x) + 6g_3(x)(\tilde \phi + \tilde u) X + \cdots \cr X \equiv & \ \tilde C + \tilde \Delta \cr} X$$ XLet $z=(j,k)$, $z' = (j',k)$ with $(j-j') = 0, \pm 1$ Xand $p_1 + ip_2 = r e^{i\theta k}$. XThe behavior of the bilinear operator X$C$ claimed in \equ(decomp2) is verified by the integral X$$\eqalign{ X \langle \gd(z), &C[\phi(\pp),\gd(z')] \rangle_0 = \cr X \int_0^{2\pi} \int_0^\pi X {\overline \psi_j(x) e^{-i \xi k} } \ X &g_2(x) r \psi_1(x) \cos (\xi + \theta) e^{i\xi} X \psi_{j'}(x) e^{i \xi k} \ dxd\xi = 0. \cr} X$$ XThus all contributions to the second piece of the bilinear form X$\Delta$ are at least linear in $\phi(\pp) + u$. The hypothesis Xis that $u \in \HH_\gs$ so that the use of the Xestimates in \clm(analyticity) gives the statement X\equ(decomp1). X X X XCombining these estimates, we have shown that the Xnonlinear wave equation with periodic boundary conditions Xgives rise to a nonlinear lattice problem which satisfies Xthe hypotheses {\bf H1,H2} and {\bf H3}. X X\CLAIM Proposition(periodic) If we consider periodic Xboundary conditions in X\equ(NLlattice), and if the nonlinearity $g(x,u)$ Xis $\pi$ periodic in $x$ and Xanalytic in $(x,u)$ for X$x \in \{|{\rm Im} \ x| < \overline{\gs} \}$, for Xsome $\overline \gs > 0$, then Xfor $\gs_* < \overline \gs$, for all X$0 < \gs < \gs_*$ the requirements X{\bf H1} -- {\bf H3} are satisfied Xfor the operator $W$ and its derivatives. X X\noindent X{\bf Dirichlet Boundary Conditions:} XThe second problem that is considered Xis for Dirichlet boundary conditions Ximposed on the interval $[0,\pi]$. XWe assume that the nonlinear term $g(x,u)$ is odd Xwith respect to reflection through the origin -- {\it i.e.} X$g(-x,-u) = -g(x,u)$. XEigenfunction expansions are thus considered Xwith respect to the Sturm-Liouville operator X$L_{g_1} = -{{d^2}\over{dx^2}} + g_1(x)$, Xwith Dirichlet boundary conditions imposed Xon $[0,\pi]$. XThus in addition to assuming that $g_1(x)$ is analytic and Xperiodic in a strip of width X$\overline \sigma$ about the real axis, Xit is even with respect to $0$ -- {\it i.e.} $g_1(-x) = g_1(x)$. XWe label the eigenvalues of $L_{g_1}$ in order, X$$ X\omega_1^2 < \omega_2^2 < \omega_3^2 < , \dots~~ . X$$ XDenote $\{ \psi_n(x) \}_{n=1}^{\infty}$ Xthe eigenfunctions corresponding to X$\omega_n$. XSince $g_1$ is even, the eigenfunctions $\psi_n(x)$ can be Xextended as {\bf odd} $2\pi$ periodic Xanalytic functions, and hence their Fourier series Xexpansions are of the form X$$ X\psi_n(x) = \sum_{m=1}^{\infty} \hat{\psi }_n(m) \sin(mx)~~. X$$ XThe result analogous to \clm(expand) is X X\CLAIM Proposition(sinexpand) Expand $\psi_n(x) = X\sum_{m>0} \hat{\psi}_n(m) \sin(nx)$. XThen for $\sigma_* < \overline{\sigma}$, Xthere are constants $C_1(g_1,\sigma_* )$ Xand $C_2(g_1,\sigma_*)$, (independent of $n$) such that Xfor $m > 0$ one has X$$ X| \hat{\psi}_n(m) | \le C_1(g_1,\sigma_*) e^{-\sigma_*|n-m|}~~. X\EQ(s6.6) X$$ XConversely, if one expands X$\sin( m x) = \sum_n \hat{s}_n(m) \psi_m(x)$ Xthen X$$ X| \hat{s}_n(m) | \le C_2(g_1,\sigma_*) e^{-\sigma_*|n-m|}~~. X\EQ(s6.7) X$$ X XLet $\tilde f (x,\xi)$ be an analytic in $\DD_\gs$, X$2\pi$ periodic in $x,\xi$ and an odd function of $x$. XThe analog of \clm(analyticity) is the following; X X X X\CLAIM Corollary(sinanalyticity) There exist constants X$C_1(g_1,\gs_*)$ and $C_2(g_1,\gs_*)$ such that for $\gs < \gs_*$, X$$ X ||| \tilde f |||_{\sigma -\gamma } X < {C_1 \over \gamma} \| f \|_{\sigma-\gamma}, X$$ Xand X$$ X \| f \|_{\sigma -\gamma } X < {C_2 \over \gamma} X ||| \tilde f |||_{\sigma -\gamma }. X$$ X XUsing this corollary, the verification of X{\bf H1}-{\bf H3} for the nonlinear term $W$ Xproceeds along similar lines. XWe assume that $g(x,\tilde u)$ is odd under Xreflection through the origin so that the nonlinear term Xpreserves the class of $2\pi$ periodic functions which are odd in $x$. XThat is, $g(-x,\tilde u(-x,\xi )) = - g(x,\tilde u(x,\xi ))$ Xprovided $\tilde u(-x,\xi ) = -\tilde u(x,\xi )$. X X X\SUBSECTION Proof of Propositions 6.1 and 6.4: X XWe now turn to the proof of the technical results \clm(expand) Xand \clm(sinexpand). We will give the Xdetails for the case of periodic boundary conditions; of course Xthe case of Dirichlet boundary conditions is similar. XLet the function $g_1(x)$ be periodic with period $\pi$ Xand analytic in the strip X$\{ x \in {\bf C}; \ |{\rm Im} \ x| < \overline \gs \}$. XConsider the Sturm-Liouville problem X$$ X(-{{d^2}\over{dx^2}} + g_1(x) ) \psi_n(x) = \omega_n^2 \psi_n(x)~~, X\EQ(SL) X$$ Xwith periodic boundary conditions on $[0,\pi]$. XThe eigenfunctions $\{\psi_n \}_{n=0}^\infty$ Xare analytic in the same strip, thus their XFourier coefficients decay exponentially. XThe point of the proposition is to obtain Xa uniform bound on this decay. We present here a proof Xof this fact that is in the spirit of this paper. X X XTaking inner products of both Xsides of \equ(SL) with the exponential functions X$e_k(x) = e^{ikx}$, gives X$$ X k^2 \hat{\psi}_n(k) + (e_k, g_1 \psi_n) = X \omega_n^2 \hat{\psi}_n(k)~~, X\EQ(SLlattice) X$$ Xwhere $\hat{\psi}_n(k) = (e_k,\psi_n)$. XIf we now use the fact that $\{ e^{ikx}\}_{k\in \ZZ}$ Xis a basis for $L^2(0,2\pi)$, we can rewrite X\equ(SLlattice) as X$$ X \sum_j (e_k, g_1 e_j) \hat{\psi}_n(j) + X (k^2 - \omega_n^2) \hat{\psi}_n(k) = 0~~. X$$ XThis equation can be rewritten as X$$ X (V_n + D) \hat{\psi}_n = 0~~, X\EQ(operator) X$$ Xwhere $\hat{\psi}_n \in \ell^2(\ZZ)$ with components X$\hat{\psi}_n(k)$. $V_n$ is a diagonal operator Xon $\ell^2(\ZZ)$ with matrix elements X$V_n(k,k) = (k^2 - \omega_n^2)$, and $D$ is an operator Xwith matrix elements satisfying an estimate X$$\eqalign{ X |D(j,k)| \leq& \ \bigl| \int_0^{2\pi} e^{i x(j-k)} X g_1(x) dx \bigr| \cr X \leq& C_{g_1} e^{-\overline{\gs}|j-k|}. \cr} X\EQ(D) X$$ XAssume for convenience that $n$ is even. XThe case of $n$ odd follows from that Xbelow with only minor changes in the notation. X{}From the asymptotics of the eigenvalues of \equ(SL) we Xknow that $\omega_n \approx n$. Since X$k^2 - \omega_n^2 = (k-\omega_n)(k+\omega_n)$, we see that Xfor $k \ne \pm n$, and $n$ sufficiently large, Xthere exists a constant $c_{g_1} > 0$ such that X$$ X \left| {{1}\over{n}} V_n(k,k) \right| X = \left| {1 \over n}(k-\omega_n)(k+\omega_n) \right| X\ge c_{g_1} ~~, X$$ Xwhile $|V_n(\pm n,\pm n) /n| \le c_{g_1}/8$. XOn the other hand, the operator norm of $D/n$ Xon $\ell^2(\ZZ)$ is bounded by $C_{g_1}/n$, for X$C_{g_1}$ independent of $n$. Thus for $n$ sufficiently Xlarge so that $C_{g_1}/n < c_{g_1}/8$, the operator X$H_n \equiv (V_n + D)/n$ will have a pair of eigenvalues Xwhose distance from the origin is less than X$c_{g_1}/4$, and the remainder of the spectrum will Xbe a distance of at least $3 c_{g_1}/4$ from the origin. XWe decompose the lattice $\ZZ$ into two regions Xrelative to these two classes of spectra, X$S \equiv \{n,-n \}$, and $NS \equiv \ZZ - S$. XThe projection X$$ XP \equiv {{1}\over{2\pi i}} \oint_{\Delta} X{{d \zeta}\over{(H_n - \zeta)}}~~, X\EQ(P) X$$ Xwith $\Delta$ the circle of radius $c_{g_1}/2$ centered at the Xorigin, will project onto the two dimensional subspace Xcorresponding to the two `small' eigenvalues Xof $H_n$; this eigenspace is quite close to $\ell^2(S)$. XThe vector $\hat{\psi}_n \in \ell^2(\ZZ)$ Xis an element Xof this subspace, since $H_n \hat{\psi}_n = 0$. XWe estimate $\hat{\psi}_n(j)$ by X$$ X |\hat{\psi}_n(j)| = |(e_j,{\psi}_n)| \le X |(e_j, P e_j)|^{1/2}~~. X\EQ(s6.8) X$$ XThe right hand side of this inequality is bounded Xwith the aid of the following result. X X\CLAIM Lemma(projection) Let $\gs_* < \overline \gs$. XThere exists $N_0 > 0$ and a constant X$C_{N_0}>0$ such that for $n \ge N_0$, X$$ X |(e_j, P e_j)| \le C_{N_0} X e^{-2 \gs_*(||j| - n|)}~~. X$$ X XDefine a decomposition of the operator $H_n$ relative to the Xsubspaces $\ell^2(S) \oplus \ell^2(NS) = \ell^2(\ZZ)$, X$$\eqalign{ X H_0 =& {1 \over n} V_n |_{NS} \cr X H_S =& {1 \over n} (V_n + D)|_S \cr X H_{NS} =& {1 \over n} (V_n + D)|_{NS}. \cr} X\EQ(Haitch) X$$ XThe Green's functions associated with these reduced operators Xare X$$\eqalign{ X G_0(\zeta) =& (H_0 - \zeta)|_{NS}^{-1} \cr X G_S(\zeta) =& (H_S-\zeta)^{-1}_S \cr X G_{NS}(\zeta =& (H_{NS} - \zeta)|^{-1}_{NS} \cr} X\EQ(Gee) X$$ XThe coupling term in this decomposition is defined Xby the relation X$$ X H_n = G_{NS} \oplus G_S + \Gamma. X\EQ(Gam) X$$ XHilbert spaces analogous to those used in sections 3 Xand 5 are defined by the norm X$\|v\|_\gs^2 = \sum_{k \in \ZZ} |v(k)|^2 e^{2 \gs |k|}$, Xand $\|G\|_\gs$ refers to the operator norm. X X X X X X X X\CLAIM Lemma(G1) For $\zeta \in \Delta$, $|G(j,k;\zeta)| X\le {{4}\over{c_{g_1}}}$. X X\PROOF This follows immediately from our estimates on the Xlocation of the spectrum of $H_n$. X X\endproof X X\CLAIM Lemma(G2) Given $0 < \overline \ga \leq \overline \gs$ Xthere exists $N_0 > 0$ such for $n \ge N_0$, and for $\zeta \in \Delta$, X$$ X |G_{NS}(j,k;\zeta)| \le {8 \over c _{g_1}} X e^{-(\overline \gs - \overline \ga)|j-k|}~~. X$$ X X\PROOF This restricted Green's function is constructed via the Neumann Xseries, with $(H_{NS} - \zeta) = (H_0 - \zeta) + (D/n)$. XThis decomposition gives the series X$$ X G_{NS}(\zeta) = G_0(\zeta) \sum_{j=0}^\infty (-1)^j X ((D/N) G_0(\zeta))^j . X$$ XFor $\zeta \in \Delta$ and $n$ sufficiently large we Xwill show that the series converges. XThe first point is that $\|G_0(\zeta)\|_{\overline \gs} X\leq 4/c_{g_1}$, and, using \clm(opnorm), X$\|D/n\|_{\overline \gs - \overline \ga} X\leq |||g_1|||_{\overline \gs}/(\overline \ga n)$. XThus for $N_0$ chosen sufficiently large, and for $n \geq N_0$, Xwe have X$$ X {1 \over n}\|DG_0(\zeta)\|_{\overline \gs -\overline \ga} X \leq {C \over N_0} < {1 \over 2}. X$$ XHence $\|G_{NS}(\zeta)\|_{\overline \gs -\overline \ga} X\leq 8/c_{g_1}$. X X\endproof X X X X X XTo prove \clm(projection) we combine these lemmas Xwith the resolvent identity: X$$ X G(\zeta) = G_S(\zeta) \oplus G_{NS}(\zeta) X + \left( G_S(\zeta) \oplus G_{NS}(\zeta) \right) X \Gamma G(\zeta) X\EQ(res) X$$ XFor $k = \pm n$, the estimate of \clm(expand) is simply Xthe requirement of boundedness, Xso we assume that $k \ne \pm n$. X X XIterating the resolvent expansion, we find that (formally) X$$\eqalign{ XG(\zeta) = G_S(\zeta) &\oplus G_{NS}(\zeta) X + \left( G_S(\zeta) \oplus G_{NS}(\zeta) \right) X \Gamma \left(G_S(\zeta) \oplus G_{NS}(\zeta)\right) \cr X& + \sum_{\ell = 2}^{\infty} \left( X\left( G_S(\zeta) \oplus G_{NS}(\zeta) \right) \Gamma \right)^k X\left(G_S(\zeta) \oplus G_{NS}(\zeta)\right) \cr X} X\EQ(newresolvent) X$$ XBecause $G_S$ and $G_{NS}$ vanish unless both of their Xarguments lie in either the singular or non-singular region Xof the lattice respectively, and since $\Gamma$ vanishes Xunless one of its arguments lies in $S$ and the other in X$NS$, the second term in \equ(newresolvent) vanishes Xwhen evaluated at the arguments $(k,k)$, with $k \in NS$. XFurthermore, since $\dist. ( \spec H|_{NS},0) X > 3 c_{g_1} /4 $, we have X$$ X {{1}\over{2\pi i}} \oint_{\Delta} X G_{NS} (k,k;\zeta) d\zeta = 0 ~~, X$$ Xso the first term in \equ(newresolvent) also makes no contribution Xto the projection operator in\equ(P). X X XWe now turn to evaluate a typical term in the infinite sum. XWe will use X\CLAIM Lemma(decay) If $m \in S$, and $\zeta \in \Delta$, Xthen there exists a constant $C_{g_1} > 0$ (depending Xon $\overline{\sigma}$ and $\overline{\gamma}$), such that X$$\eqalign{ X|\left( \left(G_{S}(\zeta) \oplus G_{NS} X(\zeta) \right) \Gamma\right)(m,p)| & \le X{{C_{g_1}}\over{N}} e^{-(\overline{\sigma} -\overline{\gamma}) X| |p| - |n| |} \cr X|\left( \left(G_{S}(\zeta) \oplus G_{NS} X(\zeta) \right) \Gamma\right)(p,m)| & \le X{{C_{g_1}}\over{N}} e^{-(\overline{\sigma} -\overline{\gamma}) X| |p| - |n| |} ~~.\cr X} X$$ X X\PROOF We verify the first inequality--the second follows in Xlike fashion. Note that X$$ X\left( \left(G_{S}(\zeta) \oplus G_{NS} X(\zeta) \right) \Gamma\right)(p,m) = X\sum_{q \in NS} G_{NS}(p,q;\zeta) \Gamma(q,m)~~. X$$ XBy \equ(D), $|\Gamma(q,m)| \le (C_{g_1}/N)e^{\overline{\sigma} |q-m|}$. X{}From \clm(G2) we have X$|G_{NS}(p,q;\zeta)| \le {8 \over c _{g_1}} X e^{-(\overline \gs - \overline \ga)|p-q|}$. XSumming over $q$, and using the fact that $m = \pm n$, completes the Xproof. X X\endproof X XAs a consequence of this lemma we have X\CLAIM Corollary(iteratedbnd) If $m \in S$, $\ell$ Xis a non-negative integer and $\zeta \in \Delta$, then there exists Xa constant $C_{g_1} > 0$ (depending Xon $\overline{\sigma}$ and $\overline{\gamma}$), such that X$$ X| \left( \left( G_S(\zeta) \oplus G_{NS}(\zeta)\right) X\Gamma\right)^{\ell} |(m,p) \le {{C_{g_1} }\over{N}}^{\ell - 2}~~. X$$ X X\PROOF The proof is an easy induction argument, the details Xof which we omit. X XNow consider the term in the infinite sum, X$$\eqalign{ X\left\{ \left( \left( G_S(\zeta) \oplus G_{NS}(\zeta)\right) X\Gamma\right)^{\ell} \left( G_S(\zeta) X\oplus G_{NS}(\zeta)\right) \right\}(k,k) = & \cr X\sum_{ {{m,p,r \in S}\atop{ q,j \in NS }} } XG_{NS}(k,j,\zeta) \Gamma(j,m) \qquad \qquad\qquad \qquad & \cr X\left( \left( G_S(\zeta) \oplus G_{NS}(\zeta)\right) X\Gamma\right)^{\ell - 2} (m,p)& G_S(p,r;\zeta) \Gamma(r,q) XG_{NS}(q,k;\zeta)~~.\cr X} X$$ XThe various factors in this sum can be bounded by X\equ(D), \clm(G1), \clm(G2), and \clm(iteratedbnd). XInserting these bounds, we find that the right hand side of Xthis equality is bounded in magnitude by X$$ X\sum_{ {{m,p,r \in S}\atop{ q,j \in NS }} } X{{8}\over{c_{g_1}}} e^{-(\overline{\sigma} - \overline{\gamma}) X|k-j|} {{C_{g_1}}\over{N} } e^{-\overline{\sigma} |j-m|} X\left({{C_{g_1} }\over{N}}\right)^{\ell - 2} {{8}\over{c_{g_1}}} X{{C_{g_1}}\over{N}} e^{-\overline{\sigma} |r-q|} X{{8}\over{c_{g_1}}} e^{-(\overline{\sigma} - \overline{\gamma}) X|k-q|}~~. X$$ XIf we now sum over $q$ and $j$, and use the fact that $m$, X$p$, and $r$ are all either $\pm n$, we find Xthat this sum is bounded by X$$ XC \left({{C_{g_1} }\over{N}}\right)^{\ell - 2} Xe^{-2 \sigma_* | |k| - n|}~~, X$$ Xwhere we can choose $\sigma_*$ to be any constant smaller Xthan $\overline{\sigma} - \overline{\gamma}$. XIf $N$ is large, this quantity can be summed over $\ell\ge 2$, Xand the estimate of \clm(projection) follows. X X X\endproof X X X XThe proof of \clm(expand) follows from this result. XFor analytic potential functions $g_1$ the eigenfunctions X$\psi_n(x)$ are $\pi$ periodic, and analytic in the strip X$\{ x; |{\rm Im} \ x| < \overline \gs \}$, thus the XFourier coefficients decay exponentially, X$$ X |\hat \psi_n(m)| \leq C(n) e^{-\overline \gs |n-|m||}. X$$ XThe fact that the constant can be chosen independently of X$n$ follows from estimate \equ(s6.8) and \clm(projection). XStatement \equ(s6.5) and the results of \clm(sinexpand) Xfollow from similar arguments. X X X\SUBSECTION $(d_0,L_0)$ nonresonant spectra: X X XThis section is concerned with the result X\clm(dense-L-nonres), and the statements that appear in X\clm(NLWperiodic) and \clm(NLWDir) concerning the Xnature of the set of nonlinearities to which the Xexistence theorems apply. XWe will first make more precise sense of the statements Xin these theorems concerning the topology in which the Xadmissible coefficients are dense. XIndeed we will show that the Xcoefficients $g_1(x)$ which give rise to spectra X$\{ \omega_j \}$ which are X$(d_0,L_0)$-nonresonant with X$\omega_1$ form an open dense set Xin $L^2(0,\pi)$, and furthermore the intersection Xof this set with the class ${\cal A}_{\overline \gs}$ of coefficients X$g_1$ which are bounded and analytic in the strip X$\{ x ; |{\rm Im} \ x | \leq \overline \gs \}$ Xis open in all reasonable topologies. X XThe Dirichlet spectra of the Sturm-Liouville operators X$$ X L_{g_1} \psi = \bigl(-{d^2 \over dx^2} + g_1(x) \bigr) X$$ Xare characterized by the following representation Xfor their spectrum X$$ X \omega_n^2 = n^2 + \overline g_1 + d(n), X\EQ(Dir-spec-asym) X$$ Xwhere $\overline g_1 = \pi^{-1} \ \int_0^\pi g_1(x) \ dx$, Xand $d(n)$ are terms of a sequence in $\ell^2(\ZZ^+)$. XBy contrast, the set of potentials which are analytic Xin the strip $\{ x ; |{\rm Im} \ x | \leq \overline \gs \}$ Xare less easily classified by the properties of the Xsequence $\{ d(n) \}$. Without yet imposing conditions Xof analyticity for the potential $g_1$, Xwe will ask for all spectra which satisfy Xthe $(d_0,L_0)$-nonresonance conditions. That is, Xfor $(j,k)$ such that $|j|+|k| \leq L_0$, X$$\eqalign{ X |k^2 \omega_1^2 - \omega_j^2| >& d_0, X~~{\rm if}~~(j,k) \ne (1,\pm1)~~{\rm and}~~,\cr X|k \omega_1 - j| >& d_0(|j|+|k|)^{-\tau}~~{\rm for} X(j,k) \ne (0,0)~~. \cr X} X$$ XDenote $NR$ the set of $L^2(0,\pi)$ potentials Xwith Dirichlet spectrum $\{\omega_j\}_{j=1}^\infty$ Xgiving rise to a frequency sequence which is X$(d_0,L_0)$-nonresonant for some $L_0$, Xfor $d_0 = o(L_0^{-1/2})$. X X\CLAIM Proposition(Dir-spectra) XThe set $NR$ is open and dense in $L^2(0,\pi)$, and Xhas open intersection with ${\cal A}_{\overline \gs}$. X X\PROOF For $d_0,L_0$ fixed, denote by $NR(d_0,L_0)$ Xthe set of frequency sequences $\{ \omega_j \}_{j=1}^\infty$ Xwhich are $(d_0,L_0)$ nonresonant. We will use the fact that Xthere is an isomorphism between the potentials in $L^2(0,\pi)$ Xwith mean value zero, and the sequences X$\{d(j)\} \in \ell^2(\ZZ^+)$, [PT]. XThe sequences not in $NR(d_0,L_0)$ because Xthey violate the first condition of X\clm(L-nonresonance) are Xcharacterized in terms of their spectral Xrepresentation \equ(Dir-spec-asym), X$$ X |k^2 \omega_1^2 - \omega_j^2| X = |k^2 \omega_1^2 - (j^2+ \overline{g_1}) - d(j)| X \leq d_0, X\EQ(violate) X$$ Xfor some $(j,k), \ |j|+|k| \leq L_0$. XThe condition \equ(violate) involves only finitely many Xeigenvalues $\omega_j$, and leads clearly to a closed condition on Xboth the spaces $L^2(0,\pi)$ and ${\cal A}_{\overline \gs}$. XThus $NR(d_0,L_0)$ is open in both $L^2(0,\pi)$ Xand ${\cal A}_{\overline \gs}$. XIf $g_1$ is a potential, corresponding to the sequence X$\{d(j)\}_{j=1}^{\infty}$, and satisfying X\equ(violate), the maximal distance (in X$\ell^2 (\ZZ^+)$ ) to a sequence in $NR(d_0,L_0)$ is X$$ X \bigl( \sum_{j=1}^{L_0} \ d_0^2 \bigr)^{1/2} X = d_0 L_0^{1/2}. X$$ XThe second condition of \clm(L-nonresonance) X({\it i.e.} \equ(2.6))is Xa diophantine condition on the principal frequency X$\omega_1$. If $\{ \omega_j \}_{j=1}^\infty$ is not Xin $NR(d_0,L_0)$ because of violating this condition, Xthen there is an $\omega_1'$ such that X$|\omega_1 - \omega_1'| < C d_0$, and $\omega_1'$ is Xdiophantine. Then X$$\eqalign{ X {\rm dist}(\{ \omega_j \}_{j=1}^\infty, NR(d_0,L_0)) \ X &\leq d_0 L_0^{1/2} + {\rm dist}( \{ \omega_1' \} \cup X \{ \omega_j \}_{j=2}^\infty, NR(d_0,L_0)) \cr X & \leq d_0 L_0^{1/2} (1 + C L_0^{-1/2}) \cr} X$$ XThe set $NR = \cup_{L_0, d_0=o(L_0^{1/2})} \ NR(d_0,L_0)$ Xis open, and since $d_0 = o(L_0^{1/2})$, Xit is dense in $L^2(0,\pi)$. X X\endproof X X X X X XThe periodic eigenvalue problem has different spectral Xasymptotics than the Dirichlet case, in particular the Xeigenvalues come in pairs clustered about the even indexed XDirichlet eigenvalues. The analog of X\equ(Dir-spec-asym) is that X$$ X \omega_j^2 = j^2 + \overline{g_1} + p(j) X$$ Xif $j>0$ is even, and X$$ X \omega_j^2 = (j+1)^2 + \overline{g_1} + p(j) X$$ Xif $j$ is odd. Furthermore we know that $\sum p(j)^2 < \infty$, Xand $p(2j-1) \leq d(2j) \leq p(2j)$ for $j \geq 1$. The regularity Xof the potential is more clearly reflected in the sequence X$\{p(j)\}_{j+1}^\infty$; in fact for $g_1 \in L^2 $ X$$ X \sum_{j+1}^\infty |p(2j)-p(2j-1)|^2 < \infty X$$ Xand if $g_1(x) \in {\cal A}_{\overline \gs}$, then X$$ X |p(2j)-p(2j-1)| \leq C_{g_1} e^{-{\overline \rho}|j|}. X\EQ(lateexp) X$$ Xfor some ${\overline \rho}$. X XDenote $NR_{per}$ the set of $L^2(0,\pi)$ potentials for Xwhich the periodic spectrum gives a $(d_0,L_0)$-nonresonant Xfrequency sequence for some $L_0$, Xfor $d_0 = o(L_0^{-1/2})$. XUsing the Dirichlet sequences X$\{d (j) \}$ to approximate the Xlocation of the periodic spectrum, and the asymptotics X\equ(lateexp) to control the deviations of $p(2j-1),p(2j)$ Xfrom $d(2j)$, the excision procedure of \clm(Dir-spectra) Xleads to the following result. X X\CLAIM Proposition(per-spectra) XThe set $NR_{per}$ is open and dense in $L^2(0,\pi)$, and Xhas open intersection with ${\cal A}_{\overline \gs}$. X X X X X\SUBSECTION Proof of Theorem 2.8: X XWithout the hypothesis of full nonresonance it may Xcertainly happen that the point $(\pp,0) \in \Eta_0$ Xis eliminated from $\Eta$ at some stage of the Xinduction, and the Cantor set of Theorem 2.7 will not Xhave the origin as an accumulation point. In order to Xprove Theorem 2.8 with the hypothesis of full Xnonresonance we must inspect the process of excision Xmore carefully. The excisions are of two types; constructing Xsequentially $\Eta_{n+1}^{(1)}$ and $\Eta_{n+1}^{(2)}$. XThe former is simplest to analyze, so we address it first. XExcisions to enforce a diophantine condition on $\BGO$ Xare taken so that for all $(j,k) \in B_{n+1} \backslash XB_n$, X$$ X |j\BGO - k| > {d \over (|j|+|k|)^\tau}. X$$ XThus the values of $\BGO$ excised at the $n^{\rm th}$ step Xmust retain a certain distance from $\omega_1$. X$$\eqalign{ X |\omega_1 - \BGO| &> |\omega_1 - j/k| - |j/k - \BGO| \cr X &\geq {c_1 \over (|j|+|k|)^{({\overline \tau}+1)} } X - {d \over (|j|+|k|)^{(\tau+1)} } . \cr} X$$ XAs long as $d/(|j|+|k|)^{(\tau+1)} < X c_1/(2(|j|+|k|)^{({\overline \tau}+1)})$, for all X$n \geq N_0, \ (j,k) \in B_{n+1} \backslash B_n$, we have X$$ X |\BGO - \omega_1| > {c_1 \over X 2 L_{n+1}^{({\overline \tau}+1)} }, X$$ Xand thus $(\pp,0)$ is not excised. This requires the Xchoice ${\overline \tau} \leq \tau$. XAt the ${n}^{\rm th}$ step, ${\rm meas} \{ \BGO \in X (\omega_1 - r_0^2, \omega_1+r_0^2) X ; d/(|j|+|k|)^{\tau+1} > |\BGO - j/k| \} X \leq d L_n^{-\tau} (1 + r_0^2L_{n+1})$. XGiven a nondegenerate surface ${\cal C}$ described by X$\BGO(\pp) = \omega_1 + \Kappa\|\pp\|^2(1 + C(\|\pp\|))$, Xconsider the measure of the set X${\cal S}_{n+1}^{(1)}(r_1) = \{ 0 < r < r_1 \leq r_0; X \|\pp\| = r, \ (\pp,\BGO(\pp)) \in \Eta_{n+1}^{(1)} \}$. XIf $|\Kappa| r_1^2 < X c_1 /L_{n+1}^{({\overline \tau}+1)}$, then no Xexcisions have yet occurred in $(0,r_1)$, and X${\rm meas} {\cal S}_{n+1}^{(1)}(r_1) = r_1$. XFor $|\Kappa| r_1^2 \geq c_1 /L_{n+1}^{\overline \tau}$, Xthe construction of $\Eta_{n+1}^{(1)}$ could excise a Xsubset from ${\cal S}_{n+1}^{(1)}(r_1)$, of measure Xbounded by X$$ X C {d \over L_n^\tau }(1 + r_0^2L_{n+1}) X \times \sqrt{ {L_n^{({\overline \tau}+1)} \over |\Kappa|}}. X$$ XDenote $N_1 = N_1(r_1)$ the least step such that X$|\Kappa| r_1^2 > c/(L_{N_1}^{({\overline \tau}+1)})$, then X$$\eqalign{ X {\rm meas} \bigl(& (0,r_1) \backslash \cap_{n \geq N_1} X {\cal S}_{n+1}^{(1)}(r_1) \bigr) \cr X & \leq \sum_{n \geq N_1} {d \over L_n^\tau} X (1 + r_0^2L_{n+1}) X \times \sqrt{ {2 \over c_1 |\Kappa|}} X \times L_{n+1}^{({\overline \tau}+1)/2}. \cr X & \leq C{d \over \sqrt{|\Kappa|} } X \bigl( L_{N_1}^{-(\tau -({\overline \tau}+1)/2)} X + r_0^2 L_{N_1}^{-((\tau-1) X - ({\overline \tau}+1)/2)} \bigr). \cr} X$$ XIf we recall that $c /L_{N_1}^{({\overline \tau}+1)} X< |\Kappa| r_1^2 \leq c /L_{N_1 -1}^{({\overline \tau}+1)}$, Xthe result is that X$$ X {\rm meas} \ {\cal S}_{n+1}^{(1)}(r_1) X \geq \bigl( r_1 - C d X |\Kappa|^{(\tau - 1)/({\overline \tau}+1) - 1} X r_1^{2(\tau - 1)/({\overline \tau}+1) - 1} \bigr) ~~. X$$ XThus we ask that X $0 < \mu < 2(\tau - 1)/({\overline \tau}+1) - 1$. XIn order to make the choice of constants uniform in $L_0$, Xalso ask that X$d|\Kappa|^{(\tau - 1)/({\overline \tau}+1) - 1} < 1$. XThis is achieved with the present choices of Xinduction parameters X$d = L_0^{-\eta}, \ |\Kappa| \geq L_0^{-\nu}$ as long as X$\eta + \nu((\tau - 1)/({\overline \tau}+1) - 1) > 0$. X X XThe construction of $\Eta_{n+1}^{(2)}$ involves the Xexcision of $\gd_{n+1}/L_n^2$ neighborhoods of the Xzero sets $Z_i$ discussed in section 4. These Xintersect the $\Omega$ axis at points X$\omega_\ell/m, \ (\ell,m) \in B_{n+1} \backslash B_n$. XUsing the hypothesis of full nonresonance, if X$$\eqalign{ X |\omega_1 - \omega_\ell / m| X \geq & {c_2 \over X (|\ell|+|m|)^{({\overline \alpha}+2)} } \cr X \geq & {c_2 \over L_{n+1}^{({\overline \alpha}+2)} } X \geq {\gd_{n+1} \over L_n^2 }. \cr} X$$ Xthen we will retain the point $(\pp,0) \in \Eta_{n+1}^{(2)}$ Xthroughout the induction. This requires the choice X${\overline \alpha} \leq \alpha$. We also ask that X${\overline \alpha} + 1 \geq {\overline \tau}$, so that Xthe critical excisions of this construction will always Xfall within $\Eta_{n+1}^{(1)}$, thus the parabolic Xestimates \equ(parab-est) will hold in Xthe relevant neighborhood of $\omega_1$. X XThe zero set $Z_i$ may intersect the surface ${\cal C}$ Xcloser to $\omega_1$ than X$\omega_\ell / m$, however these must be at points where X$\omega_1 +|\Kappa/2| \|\pp\|^2 X \geq |\omega_\ell /m| - C_1 \|\pp\|^2 /L_n^2 $. XThat is, the points of intersection must satisfy X$$ X \|\pp\|^2 > \bigl| X \bigl|{\omega_\ell \over m}\bigr| - \omega_1 \bigr| X {1 \over |\Kappa| + C_1/L_n^2 }. X$$ XWe ask that $C_1/L_n^2 < 1$. Define the set X$$ X {\cal S}_{n+1}^{(2)}(r_1) = \{ 0 < r < r_1 \leq r_0; \ X \|\pp\| = r, \ {\rm and}~~ X (\pp,\BGO(\pp)) \in \Eta_{n+1}^{(2)} \}~~. X$$ XIf $r_1^2 < (|\Kappa|+1)^{-1} \bigl|c_2 / XL_{n+1}^{({\overline \alpha} + 2)} X \bigr| X - \gd_{n+1}/L_n^2$, Xthen in constructing $\Eta_{n+1}^{(2)}$ there is no Xexcision within ${\cal S}_{n+1}^{(2)}$. Thus we will Xselect $\overline \alpha$ so that X$\gd_{n+1}/L_n^2 \leq c_2 / X (2L_{n+1}^{({\overline \alpha} + 2)} )$. In particular Xwe require that ${\overline \alpha} \leq \alpha$. XLet $N_2 = N_2(r_1)$ be the first induction step at Xwhich excisions may occur within $(0,r_1)$. We now Xestimate the measure of the set X$(0,r_1) \backslash \cap_{n \geq N_2} {\cal S}_{n+1}^{(2)}$ Xas in the first case. X$$\eqalign{ X {\rm meas} \ \bigl(& (0,r_1) \backslash X \cap_{n \geq N_2} {\cal S}_{n+1}^{(2)}(r_1) \bigr) \cr X &\leq \sum_{n \geq N_2} L_n(1 + r_0^2L_{n+1}) X \times 2 \ell_{n+1}^2 X \times {\gd_{n+1} \over L_n^2} X \times \sqrt{ L_n^{{\overline \alpha} + 2} X \over |\Kappa| c_2 } \cr X &\leq {C \over \sqrt{ |\Kappa|}} X \bigl( L_{N_2}^{-(\alpha-(2\beta +{\overline \alpha}))} X + r_0^2 L_{N_2}^ X {-(\alpha-(2\beta + {\overline \alpha}+1))} \bigr) X \cr} X$$ XUsing that $C/L_{N_2+1}^{({\overline \alpha} + 2)} X \leq r_1^2 \leq C/L_{N_2}^{({\overline \alpha} + 2)}$, X the result is that X$$ X {\rm meas} \ {\cal S}_{n+1}^{(2)}(r_1) X \geq \bigl( r_1 - {C \over \sqrt{| \Kappa|}} X r_1^{2((\alpha - 2\beta -1)/ X ({\overline \alpha}+2) - 1)} \bigr). X$$ XThus we also require that X$$ X 0 < \mu X < 2((\alpha - 2\beta -1)/({\overline \alpha} + 2) -1). X$$ XThis finishes the proof of the theorem. X X X\endproof X X X\noindent{\bf Acknowledgements:} The authors would Xlike to thank the D\'epartement de Physique XTh\`eorique, Universit\'e de Gen\`eve, the XIHES--Bures sur Yvette, MSRI--Berkeley, and Oxford XUniversity for their hospitality, and the National XScience Foundation (grants \#DMS-8858218 and X\#DMS-9002059) for their support of our research. XAdditionally the first author is supported by a Xfellowship from the Alfred P. Sloan Foundation. X X X X X END_OF_FILE if test 38791 -ne `wc -c <'sec6.tex'`; then echo shar: \"'sec6.tex'\" unpacked with wrong size! fi # end of 'sec6.tex' fi if test -f 'title.tex' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'title.tex'\" else echo shar: Extracting \"'title.tex'\" \(904 characters\) sed "s/^X//" >'title.tex' <<'END_OF_FILE' X X\TITLE NEWTON'S METHOD AND PERIODIC SOLUTIONS XOF NONLINEAR WAVE EQUATIONS X\footnote*{\small Submitted to Communications in Pure and Applied Mathematics} X\AUTHOR Walter Craig X\FROM XDepartment of Mathematics XBrown University XProvidence, RI~ 02912 X\AUTHOR C. E. Wayne X\FROM XDepartment of Mathematics XPennsylvania State University XUniversity Park, PA~ 16802 X\ENDTITLE X\ABSTRACT XWe prove the existence of periodic solutions Xof the nonlinear wave equation X$$ X\partial_t^2 u = \partial_x^2 u - g(x,u)~, X$$ Xsatisfying either Dirichlet or periodic boundary conditions Xon the interval $[0,\pi]$. The coefficients of the eigenfunction Xexpansion of this equation satisfy a nonlinear functional Xequation. Using a version of Newton's method, Xwe show that this equation has Xsolutions provided the nonlinearity $g(x,u)$ Xsatisfies certain generic conditions of Xnonresonance and genuine nonlinearity. X X X\ENDABSTRACT X END_OF_FILE if test 904 -ne `wc -c <'title.tex'`; then echo shar: \"'title.tex'\" unpacked with wrong size! fi # end of 'title.tex' fi echo shar: End of shell archive. exit 0