\magnification=1200 \baselineskip=20pt \def\epf{\square} \def\ret{r^{\eta/4}} \def\rme{r^{-\eta/4}} \def\lto{L^1(a,\infty)} \def\re{{\rm Re}\,} \def\lor{L^1(\real)} \def\pmin{\psi_{-}} \def\ppl{\psi_{+}} \def\hvp{ H_{Q,p}} \def\lp{L^p(\real)} \def\lo{L^1(\real)} \def\mpg{M_p(\alpha,\gamma)} \def\tgf{t^{\gamma/4}} \def\tgmf{t^{-\gamma/4}} \def\lto{L^1(a,\infty)} \def\re{{\rm Re}\,} \def\lor{L^1(\real)} \def\pmin{\psi_{-}} \def\ppl{\psi_{+}} \def\real{{\bf R}} \def\epf{{\bf //}} \def\hvg{H_{Q,p}} \def\mag{M(\alpha,\gamma)} \def\plt{|\gamma|\leq 2\alpha/p} \def\pgt{|\gamma|> 2\alpha/p} \def\rmz{\real\setminus\{0\}} \def\cont{{\rm C }} \def\psiv{\Psi_+(x,V)} \def\ppv{\psi_+(r,V)} \def\pmv{\psi_-(r,V)} \def\ppp{\psi_+'(r,V)} \def\pmp{\psi_-'(r,V)} \def\fhk{f(x+h, y+k)} \def\fh{f(x+h,y)} \def\fk {f(x,y+k)} \def\fx{f(x,y)} \def\ino{\int_0^1} \def\ffi{\varphi} \def\imi{\int_{-\infty}^\infty} \def\test{C_0^\infty(\real)} \def\uvt{ U_Q(t)} \def\uzv{U_0(t)} \def\upq{U_{Q,p}(t)} \centerline{\bf PERTURBATIONS OF THE WIGNER-VON NEUMANN POTENTIAL} \medskip \centerline{\bf LEAVING THE EMBEDDED EIGENVALUE FIXED} \medskip \vskip.5cm \centerline{ \bf J. Cruz-Sampedro\footnote*{ Research supported in part by CONACYT, 32146-E, Mexico.}} \medskip \centerline{Instituto de Ciencias B\'asicas e Ingenier\'\i a, UAEH} \centerline{sampedro@uaeh.reduaeh.mx} \medskip \centerline{ \bf I. Herbst\footnote{**}{ Research partially supported by NSF grant DMS-96000056.}} \medskip \centerline{Mathematics Department, University of Virginia} \centerline{iwh@weyl.math.virginia.edu} \medskip \centerline{\bf and} \medskip \centerline{ \bf R. Mart\'\i nez-Avenda\~no} \centerline{ Department of Mathematics, Michigan State University} \centerline{ruben@math.msu.edu} \vskip1cm {\bf Abstract.} We investigate the Schr\"odinger operator $H=-d^2/dx^2+(\gamma/x)\sin \alpha x+V$, acting in $ L^p(\real)$, $1\leq p<\infty$, where $\gamma \in \real \setminus \{ 0 \} $, $\alpha >0$, and $V \in L^1(\real)$. For $\plt $ we show that $H$ does not have positive eigenvalues. For $\pgt$ we show that the set of functions $V\in L^1(\real)$, such that $H$ has a positive eigenvalue embedded in the essential spectrum $\sigma_{\rm ess}(H)=[0,\infty)$, is a smooth unbounded sub-manifold of $L^1(\real)$ of codimension one. \vfill\eject \centerline{\bf Perturbations du Potentiel Wigner-Von Neumann} \medskip \centerline{\bf Qui Fixent la Valeur Caract\'eristique Immerg\'ee} \vskip1cm {\bf R\'esum\'e.} On examine l'op\'erateur de Schr\"odinger $H=-d^2/dx^2+(\gamma/x)\sin \alpha x+V$ d\'efini dans $ L^p(\real)$, $1\leq p<\infty$, o\`u $\gamma \in \real \setminus \{ 0 \} $, $\alpha >0$, et $V \in L^1(\real)$. Si $|\gamma|\leq 2\alpha/p$, on montre que $H$ n'a aucune valeur caract\'eristique positive. Si $|\gamma|> 2\alpha/p$, on montre que l'ensemble des fonctions $V\in L^1(\real)$, telles que $H$ a une valeur caract\'eristique positive immerg\'ee dans le spectre essentiel $\sigma_{\rm ess}(H)=[0,\infty)$, est une sous-vari\'et\'e lisse non-born\'ee de $L^1(\real)$ de codimension \'egale \`a un. \vfill\eject \noindent{\bf 1. Introduction} \medskip In this paper we consider Schr\"odinger operators of the form $$\hvp=-{d^2\over dx^2}+Q, \eqno(1.1)$$ acting in $\lp$, $1\leq p<\infty$, where $Q=W+V$, $W(x)=(\gamma/x)\sin\alpha x$, $\alpha>0$ and $\gamma\in \real\setminus\{0\}$ are constants, and $V$ is a real-valued function in $\lo$. To give a precise definition of the operator $\hvp$ we use the {\it Feynman-Kac formula}. For $\displaystyle{f\in\cup_{p\geq 1}\lp}$ and $t\geq 0$ we define $$ U_Q(t)f(x)=E_x\left(\exp\left\{-\int_0^tQ(b(s))ds\right\}f(b(t)) \right),\eqno(1.2) $$ where $E_x$ denotes the expectation with respect to Brownian motion starting at $x$ with Brownian transition function given by $$p_t(x,y)={\exp\left({-(x-y)^2/ 4t}\right)\over\sqrt{4\pi t}}, \qquad\qquad x,y\in\real,\quad t\geq 0. \eqno(1.3) $$ We define $\hvp$ to be the negative of the infinitesimal generator of the C$_0$-semigroup $(\upq; t\geq 0) $, $ 1\leq p<\infty$, defined for $f\in\lp$ and $t\geq 0$ by $\upq f=U_Q(t)f$. Various classes of operators which contain the ones defined above have been investigated in [6, 12, 16, 17, 18, 26, 28, 30], and it is well known that the spectrum $\sigma(\hvp)$ is $p$-independent and that $\sigma_{\rm ess}(\hvp)=[0,\infty)$ for all $p\geq 1$. \medskip Schr\"odinger operators of the form (1.1) were introduced by {\it Wigner} and {\it Von-Neumann} [29] in order to construct an example of a Schr\"odinger operator, acting in $L^2(\real^3)$, with a spherically symmetric potential which vanishes at infinity and possesses a positive eigenvalue embedded in the continuum. The significance of the Wigner-Von Neumann example lies in the fact that at the time it contradicted physical intuition, which predicted that bound states of positive energy could not occur if the potential tended to zero at infinity. \medskip In this paper we study the structure of the set of functions $V\in\lo$ for which the operator $\hvp$ has a positive eigenvalue. \medskip Our main result is: \proclaim Theorem 1.1. Let $\hvp$ be as in (1.1). If $\plt$, then $\hvp$ does not have positive eigenvalues. If $\pgt$, then the set of functions $V\in \lo$ such that $\hvp$ has a positive eigenvalue embedded in the essential spectrum $\sigma_{\rm ess}(\hvp)=[0,\infty)$ is a smooth unbounded sub-manifold of $\lo$ of codimension one. In addition, if $V$ belongs to this sub-manifold then $\alpha^2/4$ is the unique positive eigenvalue of $\hvp$. It is well known that the eigenvalues in the discrete spectrum are, in an appropriate setting, stable under perturbations. On the other hand, it is also known that embedded eigenvalues in the continuum are rather unstable [1, 2, 9, 10, 20]. In [2] {\it Agmon}, {\it Herbst}, and {\it Skibsted} prove that generically, in a Baire category sense, arbitrarily small perturbations of a generalized $N$-body Hamiltonian remove all non-threshold eigenvalues embedded in the continuum, and conjecture that the set of perturbations that preserve a non-threshold embedded eigenvalue is something like a differentiable manifold. The result presented in this paper shows that the above conjecture is true for the simplest Schr\"odinger operators which possess an eigenvalue embedded in the continuum. A similar result for $p=2$ was announced in [11] without proof. For $\alpha>0$ and $\gamma \in \rmz$, let $\mag$ be the set of functions $V\in\lo$ such that, for some $k>0$, the differential equation $$-\psi''+\gamma{\sin\alpha r\over r}\psi+V\psi=k^2\psi, \qquad r\in \real,\eqno(1.4)$$ has a nonzero solution that goes to zero as $|r|$ goes to infinity. We say that a function $\psi$ is a solution of this differential equation if it is continuously differentiable, $\psi'$ is absolutely continuous, and (1.4) holds almost everywhere. Local existence of solutions to (1.4) is well known. We also prove \proclaim Theorem 1.2. Let $\mag$ be as defined above. Then $\mag$ is a smooth unbounded sub-manifold of $\lo$ of codimension one. In addition, if $V\in\mag$ then $k=\alpha/2$. \medskip Using the terminology of [19], $\mag$ is the set of functions $V$ in $\lo$ such that $H_{Q,2}$ has a half-bound state of positive energy. \medskip To prove the results stated above we determine, following {\it Cassell} [7], the exact asymptotic behavior at infinity of the solutions to (1.4) and then identify the set of functions $V$ in $\lo$ that produce positive eigenvalues of $\hvp$ with the zero set of a smooth function on $\lo$ for which zero is a regular value. \medskip Schr\"odinger operators with eigenvalues in the continuous spectrum have also been investigated in [3, 14, 23], and the asymptotic behavior of the solutions of (1.4) for various classes of potentials has also been studied in [4, 7, 13, 15, 22]. For perturbations of embedded eigenvalues in situations which are relevant to the automorphic Laplacian and $N$-body Schr\"odinger operators see [2, 5, 9, 10, 20, 25]. In a different context, results of the type presented here have been obtained in [21]. \medskip This paper is organized as follows. In Section 2 we investigate the asymptotic behavior at infinity of solutions to (1.4) and establish the existence of solutions that vanish at infinity. In Section 3 we prove the main results. In the Appendix we establish the connection between the eigenfuctions of $\hvp$ and the solutions of (1.4) that belong to $\lp$. \medskip {\bf Acknowledgments.} We thank S. Sontz for useful comments. \medskip \noindent{\bf 2. Existence of Solutions that Vanish at Infinity} \vskip.3cm In this section we follow {\it Cassell} [7] to determine the asymptotic behavior as $r$ goes to infinity of the solutions to (1.4). We will prove \medskip {\bf Theorem 2.1} {\sl For $\alpha>0$, $\gamma\in\rmz$, $k>0$, and $V\in\lo$ we have: \vskip.3cm i) If $k\not =\alpha/2$, then (1.4) has solutions $\phi$ and $\psi$ such that, as $r$ goes to $+\infty$, $$\phi(r)=\cos kr+o(1)\qquad \hbox{and}\qquad \psi(r)=\sin kr+o(1),$$ with $$\phi'(r)=-k\sin kr+o(1)\qquad\hbox{and}\qquad\psi'(r)=k\cos kr+o(1).$$ ii) If $k=\alpha/2$, then (1.4) has solutions $\phi$ and $\psi$ such that, as $r$ goes to $ +\infty$, $$\phi(r)=r^{-\gamma/{2\alpha}}(\cos kr+o(1))\qquad \hbox{and}\qquad \psi(r)=r^{\gamma/{2\alpha}}(\sin kr+o(1)),$$ with $$\phi'(r)=-kr^{-\gamma/2\alpha}(\sin kr+o(1))\qquad \hbox{and}\qquad \psi'(r) =kr^{\gamma/2\alpha}(\cos kr+o(1)).$$} \medskip {\it Proof.} Setting $\xi(r)=\psi(r/k)$, $\sigma=\alpha/k>0$, and $\eta=\gamma/k\in\rmz$ we find that $\xi$ satisfies $$ -\xi'' + \eta {\sin\sigma r \over r} \xi + W \xi = \xi,\eqno(2.1)$$ where $W(r)=V(r/k)/k^2\in L^1(\real)$. Using the transformation $$x=\pmatrix{\cos r&-\sin r\cr\sin r&\cos r\cr} \pmatrix{\xi\cr \xi'},$$ we see that (2.1) is equivalent to $$ x'= A(r) x,\eqno(2.2)$$ where $$ A(r)=a(r) \pmatrix{\sin r\cos r&\sin^2 r\cr-\cos^2 r&-\sin r\cos r\cr},$$ and $\displaystyle{a(r)= -\eta {\sin\sigma r \over r} - W(r)}$. \medskip Next we write $A(r)=(\eta/r)G(r) + R(r)$, where $R(r)$ is the $L^1$-matrix given by $$R(r)=-W(r)\pmatrix{\sin r\cos r&\sin^2 r\cr-\cos^2 r&-\sin r\cos r\cr},\eqno(2.3)$$ and $$G(r)\equiv \pmatrix{g_1(r)&g_2(r)\cr g_3(r)&-g_1(r)\cr}, \eqno(2.4)$$ with $$g_1(r)=-{1\over4}(\cos(\sigma-2)r-\cos(\sigma+2)r),$$ $$g_2(r)=-{1\over4}(2\sin\sigma r-\sin(\sigma+2)r-\sin(\sigma-2)r),$$ and $$g_3(r)={1\over4}(2\sin\sigma r+\sin(\sigma+2)r+\sin(\sigma-2)r).$$ Now we decompose $G$ as $G=G_1+G_2$, where $G_1=0$ and $G_2=G$ for $\sigma\not=2$, and $$G_1=\pmatrix{-{1\over4}&0\cr0&{1\over 4}\cr}$$ and $$G_2(r)={1\over 4}\pmatrix{\cos 4r&-2\sin 2r+\sin 4r\cr 2\sin 2r+\sin 4r &-\cos 4r\cr}$$ for $\sigma=2$. \medskip Setting $ S(r) = I + (\eta/r) G_2^* $, a crude approximation to a solution of $S'=(\eta/r)G_2S$, where $$ G_2^*(r) \equiv \int_0^r G_2(u)du,$$ we find that if $a$ is large then $S(r)^{-1}$ exists for $r\geq a$ and $\sup\{\|S(r)^{-1}\|:r>a\}<\infty$. Hence setting $$\tilde R=S^{-1}((\eta/r)^2(G G_2^* - G_2^* G_1) + R S +(\eta/r^2) G_2^*)$$ we have that $\tilde R\in\lto$ and defining $B=(\eta/r)G_1+\tilde R$ we have that $$S B = A S - S' .\eqno(2.5)$$ Therefore setting $ x=S(r) y $ and using (2.5) we find that (2.2) is equivalent to $$ y' =B(r) y. \eqno (2.6) $$ \medskip To finish the proof we proceed as follows: \medskip i) If $\sigma\not=2$ then $B=\tilde R\in\lto$. Hence, proceeding as in the proof of Theorem XI.65 of [22] we find that, as $r$ goes to $ +\infty$, (2.6) has a fundamental matrix $X=I+o(1)$, where $I$ denotes the $2\times 2$ identity matrix. Thus (2.1) has solutions $\psi_1$, $\psi_2$ such that $$\psi_1(r)=\cos r+o(1),\qquad \psi_2(r)=\sin r+o(1),$$ $$\psi_1'(r)=-\sin r+o(1),\qquad\hbox{and}\qquad \psi_2'(r)=\cos r+o(1),$$ from which i) of Theorem 2.1 follows. \medskip ii) If $\sigma=2$ and $\gamma >0$, then the change of variables $\tau = \eta \log r$ transforms (2.6) into $$ {d \ffi \over {d \tau}} = \left(G_1+ L \right) \ffi,\eqno(2.7)$$ where $L$ is in $L^1(\tau_0,\infty)$ for some $\tau_0$ independent of $V$. It is easily verified that this last system of O.D.E.s satisfies the conditions of a theorem due to Levinson. See Theorem 8.1 in Ch. 3 of [8]. Thus, as $\tau$ goes to $ +\infty$, (2.7) has solutions $\ffi_1$ and $\ffi_2$ such that $$\lim_{\tau\to\infty}\exp(\tau/4)\ffi_1(\tau)= \pmatrix{1\cr 0}, \qquad\hbox{and}\qquad \lim_{\tau\to\infty}\exp(-\tau/4)\ffi_2(\tau)=\pmatrix{0\cr 1}. $$ Hence (2.6) has solutions of the form $$y_1=\rme\left(\pmatrix{1\cr 0}+o(1)\right),\qquad y_2= \ret\left(\pmatrix{0\cr 1}+o(1)\right), \eqno(2.8)$$ and therefore (2.1) has solutions $\psi_1$, $\psi_2$ such that $$\psi_1(r)=\rme(\cos r+o(1)),\qquad \psi_2(r)=\ret(\sin r+o(1)),$$ $$\psi_1'(r)=-\rme(\sin r+o(1)),\qquad\hbox{and}\qquad \psi_2'(r) =\ret(\cos r+o(1)).$$ Changing the signs that need to be changed, we see that the same result is true when $\gamma < 0$, and thus ii) of Theorem 2.1 follows. $\epf$ \medskip {\it Remark.} Clearly an analogous result holds in a neighborhood of $-\infty$. \vskip 1cm \noindent{\bf 3. Proof of the main result} \vskip.3cm First we prove Theorem 1.2 and then Theorem 1.1. \medskip {\it Proof of Theorem 1.2.} Clearly we may assume $\gamma>0$. Let $\mag$ be as in the statement of Theorem 1.2. First we show that $\mag$ is nonempty. By Theorem 2.1, for any given $V\in\lo$ we may choose nonzero solutions $\psi_-$ and $\psi_+$ of (1.4) with $k=\alpha/2$, such that $\psi_{-}(r)$ goes to zero as $r$ goes to $-\infty$, and $\psi_{+}(r)$ goes to zero as $r$ goes to $+\infty$. By the same theorem we can also choose $a>0$ such that $\psi_{-} (-a) \psi_{+}(a)>0$. Now, if we define $\psi(r)=\psi_{-}(r)$ for $r\leq-a$, $\psi(r)=\psi_{+}(r)$ for $r\geq a$, and $\psi(r)=\varphi(r)$ for $|r|\leq a$, where $\varphi$ is any $C^2$ function of constant sign that smoothly joins $\psi_-$ and $\psi_+$ on $[-a, a]$, and set $\tilde V(r)=V(r)$ for $|r|\geq a$ and $\tilde V(r)=(\alpha^2/4)- (\gamma/r)\sin \alpha r+\varphi''/\varphi$ for $|r|\leq a$, then $\tilde V\in\lo$ and $\psi$ is a nonzero continuously differentiable function which goes to zero as $|r|$ goes to infinity and satisfies $$-\psi'' + \gamma {\sin \alpha r\over r} \psi + \tilde V \psi =(\alpha^2/4) \psi,\qquad\quad\hbox{a.e in $\real$.}$$ Hence $\tilde V\in\mag$. In addition, it follows from the construction of $\tilde V$ that $\mag$ is unbounded in $\lor$. Furthermore, if $V$ belongs to $\mag$, then in view of Theorem 2.1 we must have $k=\alpha/2$. \medskip To complete the proof of Theorem 1.2 we only need to show that $\mag$ is a smooth sub-manifold of $\lo$ of codimension one. This is proved in the following lemma. \medskip {\bf Lemma 3.1.} {\sl Let $\mag$ be as in Theorem 1.2. Then there exists a $C^\infty$ function $F : \lor \longrightarrow \real$ such that zero is a regular value of $F$ and $\mag = F^{-1}( \{ 0\}) $. } \medskip {\it Proof.} For every $V\in\lo$ let $\psi_+$ be the solution of $$-\psi'' + \gamma {\sin \alpha r \over r}\psi + V \psi =(\alpha^2/4) \psi, \qquad r\in \real, \eqno(3.1)$$ which coincides for large positive $r$ with the function $\phi$ given in (ii) of Theorem 2.1. First we will show that $\psi_+$ and $\psi_+'$ depend smoothly on $V$. For $r_0\in\real$, let $X_{r_0}$ be the Banach space of continuous functions $\varphi$ from $[r_0,\infty)$ into $\real^2$ with the norm $$ \| \varphi \|_{r_0} \equiv \sup_{\tau\geq r_0} \|\varphi(\tau) \exp\left(\tau/ 4\right) \| < \infty. $$ It is easy to verify that the smoothness of $\psi_+$ and $\psi_+'$ with respect to $V$ follows from the fact that for all $r_0\in\real$, the solution $\varphi_1$ of (2.7) that satisfies $$ \lim_{\tau\to +\infty} \exp(\tau/ 4)\ffi_1 =\pmatrix{1\cr 0}, $$ is a smooth a function of $L \in \lor$ with values in $X_{r_0}$. To prove this last define $\Phi: L^1(\real)\times X_{\tau_0} \to X_{\tau_0}$ as $$\Phi(L, \varphi)(\tau) = \varphi(\tau) - \psi_1(\tau) + \int_\tau^\infty \Psi(\tau) \Psi^{-1}(s) L(s) \varphi(s) ds, $$ where, $\tau_0$ is as in (2.7), $\displaystyle{\psi_1(\tau)=\exp(-\tau/ 4)\pmatrix{1\cr 0}}$, and $$ \Psi(\tau)=\pmatrix{ \exp (-\tau/4) & 0 \cr 0 & \exp (\tau/4)}.$$ Note that $\Phi(L,\varphi)=0$ if and only if $\varphi=\ffi_1$. Next we fix $L_0 \in L^1(\real)$ and let $\varphi_0 \in X_{\tau_0}$ be so that $\Phi(L_0, \varphi_0)=0$. We prove that if $\tau_0$ is sufficiently large, then $\Phi(L,\varphi)=0$ implicitly defines $\varphi_1$ as a smooth function of $L$, with values in $X_{\tau_0}$, on a neighborhood of $L_0$. Since $\Phi (\cdot,\cdot)$ is jointly smooth, by the implicit function theorem it suffices to show that for $\tau_0$ sufficiently large the operator $d_2 \Phi(L_0, \varphi_0)$ is invertible from $X_{\tau_0}$ onto $X_{\tau_0}$. Clearly $$ (d_2 \Phi(L_0, \varphi_0))( H )= H + P(H), \qquad \hbox{ for all $H \in X_{\tau_0}$,} $$ where $$ P(H)(\tau) \equiv \int_\tau^\infty \Psi(\tau) \Psi^{-1}(s) L_0(s) H(s) ds.$$ Using the definition of $\Psi$ it is easily verified that $$ \| P(H) \|_{\tau_0} \leq \| H \|_{\tau_0} \int_{\tau_0}^\infty \| L_0(s) \| ds, \qquad \hbox{ for all $H\in X_{\tau_0}$,} $$ from which the invertibility of $d_2 \Phi(L_0, \varphi_0)$ for large $\tau_0$ follows. Thus $\ffi_1$ is smooth in $L$ in a neighborhood ${\cal O}$ of $L_0$ in the Banach space $X_{\tau_0}$. Since $\ffi_1(\tau_0)$ is smooth in $L$, the smoothness of $\ffi_1$ as a function from ${\cal O}$ to $X_{r_0}$ follows from the fact that the solutions to the initial value problem $${d\ffi\over d\tau}=(G_1+L)\ffi,\qquad\qquad\ffi(\tau_0)=\ffi_1(\tau_0), $$ are smooth in $L$ and the initial value $\ffi_1(\tau_0)$. This last can be proved in the usual way using the integral equation. \medskip Analogous arguments show that the solution $\psi_-$ of (3.1) that satisfies $$\psi_-(r)=|r|^{-\gamma/2\alpha} ( \cos kr + o(1) ) \qquad \hbox{ and } \qquad \psi_-'(r)=- k|r|^{-\gamma/2\alpha} ( \sin k r + o(1) ), $$ as $r \to -\infty$, is a smooth function of $V$ and so is $\psi'_-$. \medskip Now we define $F:L^1(\real) \to \real $ as $$F(V)=\left| \matrix{ \ppv& \pmv \cr \ppp&\pmp } \right|.$$ This function is well defined since the Wronskian of any pair of solutions of (3.1) is constant as a function of $r$. Moreover $V \in \mag$ if and only if $F(V)=0$; or equivalently, if and only if $\psi_-= \lambda \psi_+$, where $\lambda \not= 0$ is a function of $V$, constant as a function of $r$. Thus $\mag=F^{-1}(\{0\})$. Since $F$ is a smooth function of $V$, to finish the proof it remains to show that zero is a regular value of $F$, that is to say that for every $V\in\mag$ we have $dF(V)\not= 0$. Differentiating $F$ with respect to $V$ we find that for every $V$ and $H$ in $L^1(\real)$ we have $$dF(V) (H) = \left| \matrix{ d\ppv (H) & \pmv \cr d\ppp (H) & \pmp \cr} \right| + \left| \matrix{ \ppv & d\pmv (H) \cr \ppp & d\pmp (H) \cr} \right|, \eqno(3.2) $$ where $d$ indicates differentiation with respect to $V$ and the prime differentiation with respect to $r$, with $V$ fixed. In order to prove that $dF(V)$ is not zero we note first that, for any fixed $a\in\real$, $$\psi_+'(r,V)=\psi_+'(a,V)+\int_a^r\Lambda(t,V)\psi_+(t,V)dt, $$ where $\Lambda(r,V) \equiv (\gamma / r) \sin \alpha r + V-(\alpha^2/4)$. Using the fact that for any interval $[c,d]$ the map $ V\mapsto \psi_+(\cdot, V) $, from $\lo$ to the space $C[c,d]$ is smooth we have $$ d\psi_+'(r,V)(H)=d\psi_+'(a,V)(H)+\int_a^r H(t)\psi_+(t,V)dt +\int_a^r \Lambda(t,V)d\psi_+(t,V)(H)dt. $$ Thus $d\psi_+'(r,V)(H)$ is absolutely continuous with derivative $$(d\psi_+'(r,V)(H))'=H(r)\psi_+(r,V)+\Lambda(r,V)d\psi_+(r,V)(H)\qquad\quad \hbox{a.e.} $$ Now for any fixed $b\in\real$ and $\beta\geq b$ we consider $$\eqalign{\int_b^\beta\psi_+(t,V)(d\psi_+'(t,V)&(H))'dt\cr =&\int_b^\beta(\psi_+(t,V))^2H(t)+ \Lambda(t,V)\psi_+(t,V)d\psi_+(t,V)(H)dt. }$$ Integrating by parts we also have $$ \eqalign{\int_b^\beta\psi_+(t,V)(d\psi_+'(t,V)&(H))'dt\cr =&d\psi_+'(t,V)(H)\psi_+(t,V)\bigg|_b^\beta -\int_b^\beta\psi_+'(t,V)d\psi_+'(t,V)(H)dt. } $$ Since $\psi_+(t,V)$ is not a jointly $C^2$-function of $t$ and $V$, in order to perform another integration by parts we show first that for any $t\in\real$ and $V$ and $H$ in $\lo$ we have $$ d\psi_+'(t,V)(H)=(d\psi_+(t,V)(H))'. \eqno(3.3)$$ To prove (3.3) just note that $\displaystyle{\psi_+(t,V)=\psi_+(c,V)+\int_c^t\psi'_+(\tau,V)d\tau}$. Since $\psi'_+(\cdot,V)$ is smooth in $V$ as a function in $C[c,d]$ for any $d>c$, we can differentiate with respect to $V$ under the integral sign and obtain $$ d\psi_+(t,V)(H)=d\psi_+(c,V)(H)+\int_c^t d\psi_+'(\tau,V)(H)d\tau, $$ where for fixed $V$ and $H$, $d\psi_+'(\tau,V)(H) $ is continuous in $\tau$. Thus (3.3) follows immediately. So another integration by parts yields $$ \eqalign{ \int_b^\beta(\psi_+(t,V))^2H(t)+& \Lambda(t,V)\psi_+(t,V)d\psi_+(t,V)(H))dt\cr = & \psi_+(t,V)d\psi_+'(t,V)(H) \bigg|_b^\beta -\psi_+'(t,V)d\psi_+(t,V)(H)\bigg|_b^\beta\cr &\qquad +\int_b^\beta\Lambda(t,V) \psi_+(t,V)d\psi_+(t,V)(H)dt,} $$ which gives $$\int_b^\beta\psi_+(t,V)^2H(t)dt=\left(\psi_+(t,V)d\psi_+'(t,V)(H)- \psi_+'(t,V)d\psi_+(t,V)(H) \right)\bigg|_b^\beta. $$ Taking $\beta$ to infinity and utilizing the fact that for fixed $V$ and $H$, the functions $\psi_+$, $\psi_+'$, $d\psi_+$, and $d\psi_+'$ all approach zero at infinity we obtain $$\int_b^\infty\psi_+(t,V)^2H(t)dt=\psi_+'(b,V)d\psi_+(b,V)(H)- \psi_+(b,V)d\psi_+'(b,V)(H). $$ \medskip Analogously, $$ \int_{-\infty}^b \psi_-(t,V)^2 H(t) dt = \psi_-(b,V) d\psi_-'(b,V)(H) - \psi_-'(b,V) d\psi_-(b,V)(H).$$ Finally, combining (3.2) with these last two identities and using the fact that for $V \in \mag$ we have $\psi_-=\lambda \psi_+ $, $\lambda\not =0$, we find that $$ \eqalign{ dF(V)(H) & = \lambda \int_b^\infty \psi_+(t,V)^2 H(t) dt + {1 \over \lambda} \int_{-\infty}^b \psi_-(t,V)^2 H(t) dt \cr &= \int_{-\infty}^\infty \psi_-(t,V)\psi_+(t,V) H(t) dt, } $$ for all $H\in\lo$. Therefore, if $V\in\mag$ then $dF(V)$ is the linear functional on $\lo$ defined by the function $\psi_-\psi_+\in L^\infty(\real)\setminus\{0\}$. $\epf$ \medskip {\it Proof of Theorem 1.1.} For $p\geq 1$, $\alpha>0$, and $\gamma\in\real\setminus \{0\}$, let $\mpg$ be the set of functions $V\in\lo$ for which the operator $\hvp$ has a positive eigenvalue. It follows from Theorem 2.1 and Proposition A.1 that $$ \mpg=\cases{ \emptyset, & if $ \plt$, \cr \mag, &if $\pgt$.} \qquad\qquad \epf $$ \vskip.5cm \noindent{\bf 4. Appendix } \vskip.3cm Here we establish the connection between the eigenfunctions of the operator $\hvp$ and the decaying solutions of (1.4). Below we use {\it Duhamel's formula} [27] in the form $$(\ffi, \uvt f)=(\ffi, \uzv f) -\int_0^t(U_Q(t-u)\ffi, Q U_0(u) f) du, \eqno({\rm A.1}) $$ for $\ffi\in\test$ and $f\in L^\infty(\real)\cap \lp$, where $Q$ is as in (1.1) and $\uvt$ is as introduced in (1.2). Formula (A.1) is readily established by an approximation argument starting with bounded $Q$. Here $$ (\phi,\psi)=\imi\overline{\phi(x)}\psi(x) dx.$$ The main result of this section is \proclaim Proposition A.1. Let $p$, $Q$, and $\hvp$ be as in (1.1). Then $f\in\lp$ is an eigenfunction of $\hvp$ corresponding to the eigenvalue $\lambda\in\real$ if and only if $f$ is a differentiable function that vanishes at infinity, such that $f'$ is absolutely continuous on every finite interval of $\real$ and $$ -f''+Qf=\lambda f\qquad\qquad\hbox{a.e.} \eqno({\rm A.2})$$ {\it Proof.} Suppose $f\in\lp$ is a differentiable function that vanishes at infinity, such that $f'$ is absolutely continuous on every finite interval of $\real$ and that (A.2) is satisfied. We will show that $$ \upq f=\exp(-\lambda t) f,\qquad\quad t\geq 0.\eqno({\rm A.3})$$ For any given $\ffi\in C_0^\infty$ we define $$ g(s)=(\ffi,U_Q(s) f),\qquad\qquad s\geq 0. $$ We show first that for any $s\geq 0$ we have $$ \lim_{t\to 0^+}{g(s+t)-g(s)\over t}=-\lambda g(s). $$ Set $\psi=U_Q(s)\ffi$. Using Duhamel's formula and the fact [6] that $(\phi,U_Q(s)\xi)=(U_Q(s)\phi, \xi)$, for $\phi\in\lp$ and $\xi\in L^{p'}(\real)$, we find that $$\eqalign{ {g(s+t)-g(s)\over t}=& \left( \psi, {\uvt -1\over t}f \right)\cr =& \left( \psi, {\uzv -1\over t}f \right)-{1\over t}\int_0^t(U_Q(t-u)\psi, Q U_0(u)f)du. }\eqno ({\rm A.4})$$ We show next that as $t\to 0^+$ the right side of (A.4) goes to $(\psi,f'') -(\psi, Qf) =-\lambda g(s)$. In fact, using the kernel $p_t(x,y)$ of $\uzv$ introduced in (1.3) we have $$\eqalign{ \left( \psi, {\uzv-1\over t}f \right)=& \left( \psi, {1\over t}\imi p_t(x,y)(f(y)-f(x))dy \right) \cr =&\left( \psi, {1 \over t}\imi p_t(0,y)(f(x+y)-f(x))dy\right)\cr =&\left(\psi,{1\over t} \imi p_t(0,y)\int_0^y(y-u)f''(x+u)du dy \right)\cr = & \left( \psi, f''+\imi p_t(0,y) \int_0^y (y-u){ (f''(x+u)-f''(x))\over t}du dy\right), }$$ where in the third equality we have used Taylor's formula $$f(x+u)-f(x)=yf'(x)+\int_0^y(y-u)f''(x+u)du. $$ Setting $z=y/\sqrt t$ and then $u=\sqrt t w$ we find that $$\eqalign{ \imi p_t(0,y) \int_0^y (y-u) &{ (f''(x+u)-f''(x))\over t}du dy\cr =& \imi p_1(0,z)\int_0^z(z-w)(f''(x+\tau w) -f''(x))dw\,dz,} $$ where $\tau\equiv \sqrt t$. Hence $$\eqalign{ \bigg|\bigg( \psi, \imi & p_t(0,y) \int_0^y (y-u){ (f''(x+u)-f''(x))\over t}du dy\bigg) \bigg|\cr \leq & {1\over\sqrt{4\pi}}\imi |\psi(x)|\imi \exp(-z^2/4) \int_{-|z|}^{|z|} 2|z| |f''(x+\tau w)-f''(x)|dw\,dz\,dx\cr \leq & {1\over\sqrt{4\pi}}\imi|\psi(x)|\imi\int_{|w|}^\infty 2z \exp(-z^2/4)|f''(x+\tau w)-f''(x) |dz\,dw\,dx \cr =& {2\over\sqrt \pi}\imi|\psi(x)|\imi \exp(-w^2/4) |f''(x+\tau w)-f''(x)|dw\,dx } $$ Thus, using (A.2), the fact that $f\in \lp\cap L^\infty(\real)$, and the dominated convergence theorem we see that the right side of the last inequality goes to zero as $t\to 0^+$ and therefore $$ \lim_{t\to 0^+}\left( \psi, {\uzv-1\over t}f \right)=(\psi,f''). $$ By the continuity of the function $(U_Q(u)\psi, Qf)$ with respect to $u$, the second term on the right side of (A.4) approaches $-(\psi,Qf)$ since $${1\over t}\int_0^t\left(U_Q(t-u)\psi, Q(U_0(u)-1)f\right)du $$ goes to zero as $t\to 0^+$. To see this last we use the fact that $U_Q(t)$ maps $L^\infty(\real)$ into $L^\infty(\real)$ and that $\displaystyle{\|U_Q(t)\|_{L^\infty\rightarrow L^\infty }\leq C}$, for small $t$, that $Q=W+V$, with $V\in\lo$ and $W\in \lp$, for $p>1$, and that $(\uzv -1)f$ converges uniformly to zero as $t\to 0^+$ since $f$ vanishes at infinity and hence is uniformly continuous on $\real$. Thus we have proved that the right derivative $D_+g $ of the function $g(s)$ satisfies $ D_+g(s)=-\lambda g(s)$, for all $s\geq 0$. It follows that $ D_+(\exp(\lambda s)g(s))=0$ for $s\geq 0$ and therefore [24] that $g(s)=\exp(-\lambda s)g(0)$, which proves (A.3). Suppose now that $f\in\lp$ satisfies (A.3). Then [6, 26] $f\in L^\infty(\real)$, $f$ is continuous, vanishes at infinity, and $f'$ exists and belongs to $ L^2_{\rm loc}(\real)$. 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