Content-Type: multipart/mixed; boundary="-------------0705201019931" This is a multi-part message in MIME format. ---------------0705201019931 Content-Type: text/plain; name="07-123.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-123.keywords" magnetic fields, translation invariance, spectral theory, dispersion curves, group velocities, long-time evolution ---------------0705201019931 Content-Type: application/x-tex; name="Magn-Transl.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Magn-Transl.tex" \documentclass[reqno,12pt]{amsart} %\documentclass[reqno]{amsart} %\documentclass[12pt]{article} %\documentclass{article} \usepackage{amsfonts,amssymb,amsbsy,amsmath,amsthm} %\usepackage{ showkeys} \topmargin -1cm \textheight21.4cm \textwidth15.7cm \oddsidemargin 0.5cm \evensidemargin 0.5cm %\newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}{Lemma}[section] %\newtheorem{follow}{Corollary}[section] %\newtheorem{pr}{Proposition}[section] %\theoremstyle{definition} %\newtheorem{remark}[theorem]{Remark} %\newtheorem{remarks}[theorem]{Remarks} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}[lemma]{Theorem} \newtheorem{follow}[lemma]{Corollary} \newtheorem{pr}[lemma]{Proposition} \theoremstyle{definition} \newtheorem{definition}[lemma]{Definition} \newtheorem{example}[lemma]{Example} \newtheorem{counterexample}[lemma]{Counterexample} \newtheorem{condition}[lemma]{Condition} \newtheorem{assumption}[lemma]{Assumption} \newtheorem{problem}[lemma]{Problem} \newtheorem{remark}[lemma]{Remark} %\theoremstyle{remark} \newcommand{\bel}{\begin{equation} \label} \newcommand{\ee}{\end{equation}} \newcommand{\co}{(a^*)} \newcommand{\tgb}{\tilde{G}_{\mu}^{(\beta)}} \newcommand{\adtgb}{{\cal G}_{q, \mu}^{(\beta)}} \newcommand{\lm}{l_m} \newcommand{\eodp}{\frac{1}{2\pi}} \newcommand{\ldz}{E \downarrow 0} \newcommand{\Homu}{{\mathbb H}_0(\mu)} \newcommand{\Hmu}{{\mathbb H}(\mu)} \newcommand{\Pomu}{{\mathbb P}_0(\mu)} \newcommand{\Pomum}{{\mathbb P}_0^-(\mu)} \newcommand{\Pomup}{{\mathbb P}_0^+(\mu)} \newcommand{\Tomum}{\tilde{{\mathbb P}}_0^-(\mu)} \newcommand{\Tomup}{\tilde{{\mathbb P}}_0^+(\mu)} \newcommand{\Pmu}{{\mathbb P}(\mu)} \newcommand{\Somu}{{\mathbb H}_0(\mu)} \newcommand{\Smu}{{\mathbb H}(\mu)} \newcommand{\Domu}{{\mathbb D}_0(\mu)} \newcommand{\Dmu}{{\mathbb D}(\mu)} \newcommand{\Dl}{{\cal D}(E)} \newcommand{\pim}{p(\mu)} \newcommand{\qm}{q(\mu)} \newcommand{\Pim}{P(\mu)} \newcommand{\Qm}{Q(\mu)} \newcommand{\xp}{X_\perp} \newcommand{\mper}{m_\perp} \newcommand{\bx}{{\bf x}} \newcommand{\rt}{{\mathbb R}^{3}} \newcommand{\rd}{{\mathbb R}^{2}} \newcommand{\re}{{\mathbb R}} \newcommand{\qn}{q \in {\mathbb Z}_+} \newcommand{\kn}{k \in {\mathbb Z}_+} \newcommand{\gqkmpd}{\gamma_{q,k}^{(\beta)}(\mu + \delta)} \newcommand{\gqkmmd}{\gamma_{q,k}^{(\beta)}(\mu - \delta)} \newcommand{\gqkmpmd}{\gamma_{q,k}^{(\beta)}(\mu \pm \delta)} \newcommand{\gqkm}{\gamma_{q,k}^{(\beta)}(\mu)} \newcommand{\gzkm}{\gamma_{0,k}^{(\beta)}(\mu)} \newcommand{\gzkmpd}{\gamma_{0,k}^{(\beta)}(\mu + \delta)} \newcommand{\gzkmmd}{\gamma_{0,k}^{(\beta)}(\mu - \delta)} \newcommand{\gzkmpmd}{\gamma_{0,k}^{(\beta)}(\mu \pm \delta)} \newcommand{\nzkrd}{\nu_{0,k}(r_{\delta})} \newcommand{\nzkrp}{\nu_{0,k}(r_{+})} \newcommand{\nzkrm}{\nu_{0,k}(r_{-})} \newcommand{\bp}{{\cal P}} \newcommand{\bq}{{\cal Q}} \newcommand{\bz}{{\cal Z}} \newcommand{\pika}{$\spadesuit$ } \newcommand{\kupa}{$\heartsuit$ } \newcommand{\karo}{$\diamondsuit$ } \newcommand{\trefa}{$\clubsuit$ } \newcommand{\bpika}{\spadesuit } \newcommand{\bkupa}{\heartsuit } \newcommand{\bkaro}{\diamondsuit } \newcommand{\btrefa}{\clubsuit } \newcommand{\q}{\quad} \newcommand{\curl}{\operatorname{curl}} \newcommand{\ran}{\operatorname{Ran}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\dive}{\operatorname{div}} \newcommand{\grad}{\operatorname{grad}} \newcommand{\slim}{\operatorname{s-lim}} \newcommand{\supp}{\operatorname{supp}} {\renewcommand{\theequation}{\thesection .\arabic{equation}} \let\goth\mathfrak \let\Bbb\mathbb \let\cal\mathcal %\renewcommand\Im{\operatorname{Im}} %\renewcommand\Re{\operatorname{Re}} \begin{document} \title[Translationally Invariant Magnetic Operators]{On Spectral Properties of Translationally Invariant Magnetic Schr\"odinger Operators} %\today \author{ D. Yafaev} \address{ IRMAR, Universit\'{e} de Rennes I\\ Campus de Beaulieu, 35042 Rennes Cedex, FRANCE} \email{yafaev@univ-rennes1.fr} \subjclass[2000]{47A40, 81U05} \keywords{magnetic fields, translation invariance, spectral theory, dispersion curves, group velocities, long-time evolution} %{\Large \bf On the Spectrum of a Class of Cylindrically % Symmetric Magnetic Quantum Hamiltonians} \begin{abstract} We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr\"odinger operator ${\bf H}$ with such a potential. In particular, we show that the spectrum of ${\bf H}$ is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions $\exp(-i {\bf H} t)f$ of the time dependent Schr\"odinger equation. It turnes out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to a evolution of a one-dimensional free particle but ``exits" to $+\infty$ and $-\infty$ might be essentially different. \end{abstract} \maketitle \section{Introduction} \setcounter{equation}{0} {\bf 1.1.} Traslationally invariant magnetic fields $B(x)=(b_{1}(x), b_{2}(x), b_{3}(x))$, $x=(x_{1}, x_{2}, x_{3})$, $\dive B(x)=0$, give important examples where a non-trivial information can be obtained about spectral properties of the corresponding Schr\"odinger operators ${\bf H} $. We suppose for definiteness that $B(x)$ does not depend on the $x_{3}$-variable so that ${\bf H} $ commute with translations along the $x_{3}$-axis. There are two essentially different (and in some sense extreme) classes of traslationally invariant magnetic fields. The first class consists of fields $B(x)=(0,0,b_{3}(x_{1}, x_{2}))$ of constant direction. For such fields, the momentum $p$ of a classical particle in the $x_{3}$-direction is conserved, and in the Schr\"odinger equation the variable $x_{3}$ can be separated. Thus, we arrive to a two-dimensional problem in the $(x_{1}, x_{2})$-plane. Furthermore, if $b_{3}$ is a function of $r=(x_{1}^2+ x_{2}^2)^{1/2}$ only, then we get a set of problems on the half-line $r>0$ labelled by the magnetic quantum number $m$. The most important example of this type is a constant magnetic field $b_{3}(r)=const$ (see \cite{LL}). Some class of functions $b_{3}(r)$ decaying as $r\to\infty$ was discussed in \cite{MS} (see also \cite{CFKS}) where new interesting effects were found. Another famous case is $b_{3}(x_{1}, x_{2})=\delta(x_{1}, x_{2})$ ($\delta(\cdot)$ is the Dirac delta-function) studied in \cite{AB}. Scattering by an arbitrary short-range (decaying faster than $|x|^{-2-\varepsilon}$, $\varepsilon>0$, as $|x|\to \infty$) magnetic field $b_{3}(x_{1}, x_{2})$ turns out to be rather similar to this particular case (see \cite{Yaf}). The second class consists of fields $B(x)=(b_{1}(x_{1}, x_{2}),b_{2}(x_{1}, x_{2}), 0)$ orthogonal to the $x_{3}$-axis. In this case the corresponding magnetic potential $A(x)$, defined (up to gauge transformations) by the equation $\curl A(x)=B(x)$, can be chosen as \bel{w3} A(x)=(0,0,-a(x_{1}, x_{2})) \ee so that it has the constant direction. In contrast to fields of the first class, now the variable $x_{3}$ cannot be separated in the Schr\"odinger equation. Nevertheless due to the invariance with respect to translations along the $x_{3}$-axis the operator (we always suppose that the charge of a particle is equal to $1$) \bel{MH} {\bf H} = (i\nabla + A (x))^2 \ee can be realized, after the Fourier transform in the variable $x_{3}$, in the space $L^2(\re;L^2(\re^2))$ as the operator of multiplication by the operator-valued function \bel{u1ax} H (p) = -\Delta + (a(x_{1}, x_{2})+p)^2: L^2(\re^2)\to L^2(\re^2),\q p\in \re. \ee Moreover, if $a(x_{1}, x_{2})=a(r)$, then the subspaces with fixed magnetic quantum number $m \in {\mathbb Z}$ are invariant subspaces of $H (p)$ so that the operator $H (p)$ reduces to the orthogonal sum over $m \in {\mathbb Z}$ of the operators \bel{u1a} H_m(p) = -\frac{1}{r} \frac{d}{dr}r \frac{d}{dr} + \frac{m^2}{r^2} + (a(r)+p)^2 \ee acting in the space ${\cal H}=L^2(\re_+;rdr)$. In this case the field is given by the equation \bel{magnfi} B(x)=b(r)(-\sin \theta, \cos \theta,0) \ee where $b(r)=a'(r)$ and $\theta$ is the polar angle. Thus, vectors $B(x)$ are tangent to circles centered at the origin. An important example of such type is a field created by a current along an infinite straight wire (coinciding with the $x_{3}$-axis). In this case $b(r)=b_{0} r^{-1}$ so that $a(r)=b_{0} \ln r$. The Schr\"odinger operator with such magnetic potential was studied in \cite{Y}. \medskip {\bf 1.2.} In this article we consider magnetic fields \eqref{magnfi} with a sufficiently arbitrary function $b(r)$. Our goal is to study basic spectral properties of the corresponding Schr\"odinger operator ${\bf H} $ such as the absolute continuity, location and multiplicity of the spectrum, as well as the long-time behaviour of the unitary group $\exp{(-i{\bf H}t)}$. We emphasize that for magnetic fields considered here, the problem is genuinely three-dimensional, and actually the motion of a particle in the $x_{3}$-direction is of a particular interest. % Clearly, the sum of magnetic fields of the first and second types is also translationally invariant, but the problem in such a general framework does not probably make much sense. Anyway it is out of the scope of the present paper. Using the cylindrical invariance of field \eqref{magnfi}, we can start either from translational or from rotational (around the $x_{3}$-axis) symmetries. The rotational invariance implies that the operator ${\bf H}$ is the orthogonal sum of its restrictions ${\bf H}_m$ on the subspaces of functions with magnetic quantum number $m \in {\mathbb Z}=\{0,\pm 1, \pm 2,\ldots\}$. It can be identified with the operator (we keep the same notation for this operator) \bel{Hm1} {\bf H}_m = -\frac{1}{r} \frac{d}{dr}r \frac{d}{dr} + \frac{m^2}{r^2} + (i\frac{d}{dx_{3}} +a(r) )^2 \ee acting in the space ${\goth H}=L^2(\re_+\times \re ;rdr dx_{3})$. In view of the translation invariance, every operator ${\bf H}_m$ can be realized (again after the Fourier transform in the variable $x_{3}$) in the space $L^2(\re;L^2(\re_{+};rdr))$ as the operator of multiplication by the operator-valued function $ H_m(p) $ defined by \eqref{u1a}. Suppose now that $b(r)$ does not tend to zero too fast so that $a(r) \to \infty$ as $r \to \infty$. Then the spectrum of each operator $H_{m}(p)$ is discrete. Let $ \lambda_{n,m}(p)$, $n \in {\mathbb N}=\{1,2,\ldots\}$, be the increasing sequence of its eigenvalues (they are simple and positive), and let $ \psi_{n, m}(r,p)$ be the corresponding sequence of its eigenfunctions. The functions $\lambda_{n,m}(p)$ are known as dispersion curves of the problem. They determine the spectral properties of the operator ${\bf H}_{m}$. Note that if $a(r)$ is replaced by $-a(r)$, then $\lambda_{n,m}(p)$ is replaced by $-\lambda_{n,m}(p)$, so that the case $a(r) \to -\infty$ as $r \to \infty$ is automatically included in our considerations. Recall that, for a magnetic field $B(x)$, the magnetic potential $A(x)$ such that $\curl A(x) =B(x)$ is defined up to a gauge term $\grad \varphi(x)$. In particular for magnetic fields \eqref{magnfi} in the class of potentials $A(x)=(0,0, - a(r))$ one can always add to $a(r)$ an arbitrary constant $c$. This leads to the transformations $\lambda_{n,m}(p)\mapsto \lambda_{n,m}(p-c)$ and $\psi_{n,m}(r,p)\mapsto \psi_{n,m}(r,p-c)$. \medskip {\bf 1.3.} The precise definitions of the operators ${\bf H}_{m}$ and ${\bf H} $ and their decompositions into the direct integrals over the operators $H_{m}(p)$ and $H (p)$ are given in Section 2. To put it differently, we construct a complete set of eigenfunctions of the operator ${\bf H}$. They are parametrized by the magnetic quantum number $m$, the momentum $p$ in the direction of the $x_{3}$-axis and the number $n$ of an eigenvalue $\lambda_{m,n}(p)$ of the operator $H_m(p)$. Thus, if we set \bel{Psi1} {\pmb \psi}_{n,m, p}(r,\theta, x_{3})=e^{ip x_{3}} e^{im\theta } \psi_{n, m}(r,p), \ee then \bel{Psi2} {\bf H}{\pmb \psi}_{n,m, p}=\lambda_{n, m}(p) {\pmb \psi}_{n,m, p}. \ee In Section 3, we show that for all $n \in {\mathbb N}$ and $m \in {\mathbb Z}$ : \begin{itemize} \item Under very general assumptions $\lambda_{n,m}(p) \to \infty$ as $p \to \infty$ (Proposition \ref{t31}). \item If $b (r) \to 0$ as $r \to \infty$, then $\lambda_{n,m}(p) \to 0$ as $p \to -\infty$ (Proposition \ref{t32}). \item If $b (r)$ admits a finite positive limit $b_{0}$ as $r \to \infty$, then $\lambda_{n,m}(p) \to (2n-1)b_{0}$ for all $m$ as $p \to -\infty$ (Proposition \ref{t33}). \item If $b (r) \to \infty$ as $r \to \infty$, then $\lambda_{n,m}(p) \to \infty$ as $p \to -\infty$ (Proposition \ref{t33}). \end{itemize} Related results concerning the dispersion curves for Schr\"odinger operator with constant magnetic fields defined on unbounded domains $\Omega \subset \rd$ have been obtained in \cite{GS} (the case where $\Omega$ is a strip) and in \cite{dBP}, \cite[Section 4.3]{H} (the case where $\Omega$ is a half-plane). In Theorem \ref{f31} we formulate the main spectral results which follow from the asymptotic properties of the dispersion curves $\lambda_{n,m}(p)$, $p \in \re$. First, the analyticity and the asymptotics as $p \to \infty$ of $\lambda_{n,m}(p)$ imply immediately that the spectra $ \sigma({\bf H}_m)$ and $\sigma({\bf H})$ of the operators ${\bf H}_m$, $m \in {\mathbb Z}$, and ${\bf H}$ are purely absolutely continuous. Moreover, \bel{kr37} \sigma({\bf H}_m) = [{\mathcal E}_m,\infty), \quad m \in {\mathbb Z},\q \sigma({\bf H}) = [{\mathcal E}_0,\infty), \ee where \bel{kr34} {\mathcal E_m} = \inf_{p \in \re} \lambda_{1,m}(p)\geq 0. \ee Next, in the case where the magnetic field tends to $ 0$ as $r \to \infty$, the spectra of ${\bf H}_m$, $m \in {\mathbb Z}$, coincide with $[0,\infty)$ and have infinite multiplicity. On the other hand, in the case where the magnetic field tends as $r \to \infty$ to a positive finite limit, or to infinity, we have that ${\mathcal E}_m > 0$ for all $m \in {\mathbb Z}$ and each of the spectra $\sigma({\bf H}_m)$ contains infinitely many thresholds. Further, in Section 4, we obtain a convenient formula for the derivatives $\lambda_{n,m}'(p)$ which play the role of asymptotic group velocities. Our formula for $\lambda_{n,m}'(p)$ yields sufficient conditions (see Theorem \ref{t41}) for positivity of these functions. The leading example when these conditions are met, is \bel{bb} b(r)=b_{0}r^{-\delta} , \quad b_{0} >0, \quad \delta\in [0,1], \ee and $m \neq 0$. If $\delta=1$, this result remains true for all $m$ (cf. \cite{Y}). On the contrary, if $\delta =0$ and $m=0$, then $\lambda_{n,0}'(p) <0$ for all $n$ on some interval of $p$ (lying on the negative half-axis). Similar results concerning for the dispersion curves for the Schr\"odinger operator with constant magnetic field, defined on the half-plane with Dirichlet (resp., Neumann) boundary conditions, can be found in \cite{dBP} (resp., \cite{DH} and \cite[Section 4.3]{H}). Finally, in Section 5 we discuss the long-time behaviour of a quantum particle. The time evolution of a quantum system is determined by the unitary groups $\exp{(-i{\bf H}_m t)}$, $m \in {\mathbb Z}$, so that an analysis of its asymptotics as $t\to\pm\infty$ relies on spectral properties of the operators ${\bf H}_m$. Since these operators have discrete spectra, a quantum particle remains localized in the $(x_{1}, x_{2})$-plane. Its propagation in the $x_{3}$-direction is governed by the group velocities $\lambda_{n,m}'(p)$. In particular, the condition $\lambda_{n,m}'(p)>0$ for all $n \in {\mathbb N}$ and $p \in {\mathbb R}$ implies that a quantum particle with the magnetic quantum number $m$ propagates as $t\to+\infty$ in the positive direction of the $x_{3}$-axis. Let us compare these results with the long-time behaviour of a classical particle in magnetic field \eqref{magnfi}. As shown in \cite{Y}, the function $x_{3}'(t)$ is periodic with period $T$ determined by initial conditions. Its drift $x_{3} (T) - x_{3} (0)$ over the period is nonnegative if $b(r)\geq 0$ and $b'(r)\geq 0$. Morever, it is strictly positive if $b'(r)> 0$ for all $r$. In the case $b(r) = {\rm const}$ it is still strictly positive if the angular momentum $m$ of a particle is not zero. Thus, our results for functions \eqref{bb} correspond completely to the classical picture if $\delta=1 $ or $\delta\in [0,1)$ and $m\neq 0$. In the case $\delta =0$ and $m =0$ the behaviour of quantum and classical particles turn out to be qualitatively different. %%%%%%%%%%%%%%%%%%%%%% \section{ Hamiltonians and their diagonalizations} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%% Here we give precise definitions of the Hamiltonians and discuss their reductions due to the cylindrical symmetry. \medskip {\bf 2.1.} For an arbitrary magnetic potential $A : \re^3 \to \re^3$ such that $A \in L^2_{\rm loc}(\re^3)^3$, the self-adjoint Schr\"odinger operator \eqref{MH} can be defined via its quadratic form \bel{w1} {\bf h}[{\bf u}]= \int_{\re^3} \left| i (\nabla {\bf u})(x)+ A(x){\bf u}(x)\right|^2dx, \q x=(x_{1}, x_{2}, x_{3}) . \ee It is easy to see that this form is closed on the set of functions $u \in L^2 (\re^3) $ such that $\nabla {\bf u} \in L^1_{\rm loc} (\re^3)^3$ and $i\nabla {\bf u} + A {\bf u} \in L^2 (\re^3)^3$. Similarly, if $a \in L^2_{\rm loc}(\re^2)$, then the self-adjoint Schr\"odinger operator \eqref{u1ax} can be defined via its quadratic form \bel{w1a} h[u;p]= \int_{\re^2} \left(\left| (\nabla u)(x)\right|^2 + (a(x)+p)^2 |u(x)|^2\right)dx, \q x=(x_{1}, x_{2}), \quad p \in \re. \ee This form is closed on the set of functions $u \in L^2 (\re^2) $ such that integral \eqref{w1a} is finite. Clearly, this set does not depend on the parameter $p\in \re$. Let ${\mathcal F}: L^2(\re^3 )\to L^2(\re; L^2(\re^2 ) )$ be the Fourier transform with respect to $x_3$, i.e. $$ ({\mathcal F}{\bf u})(x_1,x_2,p) = \frac{1}{\sqrt{2\pi}} \int_{\re} e^{-ix_3p} {\bf u} (x_1,x_2,x_3) dx_3. $$ If $A(x)$ is given by formula \eqref{w3}, then \[ {\bf h}[{\bf u}]=\int_{-\infty}^\infty h[ ({\mathcal F} {\bf u})(p);p] dp \] which implies the equation \bel{w5} ( {\mathcal F} {\bf H} {\bf u})(x_{1}, x_{2}; p)= ( H(p) {\mathcal F} {\bf u})(x_{1}, x_{2}; p). \ee This equation can be regarded as a ``working" definition of the operator ${\bf H}$. \medskip {\bf 2.2.} Assume now that the function $a$ in \eqref{w3} depends only on $r$, and \bel{kr8} a \in L^2_{\rm loc}([0,\infty);rdr). \ee If we separate variables in the cylindrical coordinates $(r,\theta, x_{3} )$ and denote by ${\goth H}_m \subset L_2({\re}^3)$ the subspace of functions ${\bf f}(r,x_{3})e^{im\theta }$ where ${\bf f} \in L_2({\re}_+ \times {\re};rdr dx_{3})$ and $m \in {\mathbb Z} $ is the magnetic quantum number, then %\begin{equation}\label{eq:HHm3} \[ L_2({\re}^3)=\bigoplus_{m \in {\mathbb Z}} {\goth H}_m. \] The subspaces ${\goth H}_m$ are invariant with respect to $\bf H$ so that restrictions ${\bf H}_m$ of ${\bf H}$ on ${\goth H}_m$ are related with $\bf H$ by formula \bel{Hm} {\bf H}=\bigoplus_{m \in {\mathbb Z}}{\bf H}_m. \ee Every ${\goth H}_m$ can obviously be identified with the space $L^2(\re_+\times \re;rdr dx_{3})=:{\goth H}=$; then ${\bf H}_m$ is identified with operator \eqref{Hm1}. Quite similarly, if ${\cal H}_m \subset L_2({\re}^2)$ is the subspace of functions $f(r) e^{im \theta}$ where $f \in L_2({\re}_+;rdr)$, then %\begin{equation}\label{eq:HHm} \[ L_2({\re}^2)=\bigoplus_{m \in {\mathbb Z}} {\cal H}_m. \] The subspaces ${\cal H}_m$ are invariant with respect to $H(p)$ so that restrictions $H_m(p)$ of $H (p)$ on ${\cal H}_m$ are related with $H (p)$ by formula \begin{equation}\label{eq:HHm1} H (p)=\bigoplus_{m \in {\mathbb Z}} H_m(p). \end{equation} Every ${\cal H}_m$ can obviously be identified with the space $L^2(\re_+ ;rdr )=: {\cal H}$; then $H_m(p)$ is identified with operator \eqref{u1a}. Let ${\mathcal F}_{m} : {\goth H}_m \to L^2(\re; L^2(\re_+ ;rdr ) )$ be the restriction of ${\mathcal F}$ on the subspace ${\goth H}_m$. Then we have (cf. \eqref{w5}) \bel{w5m} ( {\mathcal F}_{m} {\bf H}_{m} {\bf f}) (r; p) = ( H_{m}(p){\mathcal F}_{m} {\bf f})(r; p). \ee Sometimes it is more convenient to consider instead of $H_{m}(p)$ the operator \bel{Lm} L_m(p) = r^{1/2} H_m(p) r^{-1/2} = - \frac{d^2}{dr^2} + \frac{m^2-1/4}{r^2} + (a(r)+p)^2 \ee acting in the space $L^2(\re_+)$ and unitarily equivalent to the operator $H_m(p)$. It is easy to see that the operator $L_m(p)$ corresponds to the quadratic form \bel{w1ay} l_{m}[g; p] = \int_0^{\infty} \left( |g'(r)|^2 + (m^2-1/4) r^{-2} |g (r)|^2 + (a(r)+p)^2 |g (r) |^2\right) dr, \ee defined originally on $C_0^{\infty}(\re_+)$, and then closed in $L^2(\re_+)$. \medskip {\bf 2.3.} If \bel{j1} a(r) \to \infty \quad {\rm as} \quad r \to \infty, \ee then the spectrum of the operator $H_m(p)$, $p \in \re$, $m \in {\mathbb Z}$, is discrete. Thus, it consists of the increasing sequence $\lambda_{n,m}(p)$ of simple eigenvalues. Since $H_m(p)$, $p \in \re $, is a Kato analytic family of type (B) (see \cite[Chapter VII, Section 4]{K}), all the eigenvalues $\lambda_{n,m}(p)$ are real analytic functions of $p \in \re$. Moreover, $\lambda_{n,m}(p)>0$ because form \eqref{w1a} is strictly positive. In view of formula \eqref{w5m} spectral analysis of the operators $ {\bf H}_m$ reduces to a study of a family of functions $\lambda_{n,m}(p) $, $n\in {\mathbb N}$. Indeed, let $\Lambda_{n,m} $ be the operator of multiplication by the function $\lambda_{n,m}(p) $ in the space $L^2 (\re)$. We denote by $\psi_{n,m}(r;p)$ real normalized eigenfunctions (defined up to signs) of the operators $H_{m}(p)$ and introduce an isometric mapping \[ \Psi_{n, m} : L^2(\re ) \rightarrow L^2({\re}_+\times {\re}; rdr d p) \] by the formula %\[ (\Psi_{n,m} u) (p)=\int_0^\infty u (r,p) \psi_{n,m}(r,p) r^{1/2} dr\] \bel{Psi} (\Psi_{n,m} w) (p)= \psi_{n,m}(r,p) w(p). \ee % Let us consider the direct sum of infinite number of copies of the space $L^2(\re )$ labelled by $n\in {\Bbb N}$. Then \[ L^2({\re}_+\times {\re}; rdr d p)=\bigoplus_{n\in {\Bbb N}} {\ran }\Psi_{n,m} \] and \bel{UI} {\bf H}_{m} =\bigoplus_{n\in {\Bbb N}} {\cal F}_{m}^* \Psi _{n,m} \Lambda_{n,m} \Psi_{n,m}^* {\cal F}_{m}. \ee Together with \eqref{Hm}, formulas \eqref{Psi} and \eqref{UI} justify equations \eqref{Psi2} for functions \eqref{Psi1}. %%%%%%%%%%%%%%%%%%%%% \section{Dispersion curves and spectral analysis } \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%% {\bf 3.1.} In this subsection we consider the operators $H(p)$ acting in the space $ L^{2} (\rd)$ by formula \eqref{u1ax}. Under the assumption $ a \in L^{2}_{\rm loc}(\rd)$ they are correctly defined by their quadratic forms \eqref{w1a}. If \bel{Di} a(x )\to\infty\q {\rm as} \q |x|\to\infty,\q x=(x_{1}, x_{2}) , \ee then the spectrum of $H(p)$ consists of eigenvalues $\lambda_{n }(p)$, $n \in {\mathbb N}$. We enumerate them in the increasing order with multiplicity taken into account. Our goal is to investigate the asymptotic behaviour of the eigenvalues $\lambda_{n}(p)$ as $p \to \infty$. Below we denote by $C$ and $c$ different positive constants whose precise values are of no importance. We use the following elementary \begin{lemma} \label{l31} Let $v(x)\geq 0$. For an arbitrary $\varepsilon>0$, we have the inequality \bel{2y} \int_{\rd} v(x) | u(x)|^2 dx \leq C \sup_{x\in {\rd}}\left( \int_{|x-y|\leq \varepsilon} v^2(y)dy\right)^{1/2} \int_{\rd} \left( \varepsilon |\nabla u(x)|^2 +\varepsilon^{-1} | u(x)|^2\right) dx \ee provided the supremum in the right-hand side is finite. \end{lemma} \begin{proof} Let $\Pi_{\varepsilon}\subset \rd$ be a square of length $\varepsilon$. We proceed from the estimate \[ \left( \int_{\Pi_{\varepsilon}} | u(x)|^4 dx \right)^{1/2}\leq C \left( \varepsilon \int_{\Pi_{\varepsilon}} |\nabla u(x)|^2 dx + \varepsilon^{-1} \int_{\Pi_{\varepsilon}} | u(x)|^2 dx\right) \] which follows from the Sobolev embedding theorem by a scaling transformation. Using the Schwarz inequality, we deduce from this estimate that \bel{2x} \int_{\Pi_{\varepsilon}} v(x) | u(x)|^2 dx \leq C \left( \int_{\Pi_{\varepsilon}} v^2(x) dx \right)^{1/2} \left( \varepsilon \int_{\Pi_{\varepsilon}} |\nabla u(x)|^2 dx + \varepsilon^{-1} \int_{\Pi_{\varepsilon}} | u(x)|^2 dx\right). \ee Let us split the space $\rd$ in the lattice of squares $\Pi_{\varepsilon}^{(n)}$ of length $\varepsilon$. Applying \eqref{2x} to every $\Pi_{\varepsilon}^{(n)}$ and summing over all $n$, we arrive at \eqref{2y}. \end{proof} In the following assertion we do not assume \eqref{Di}. \begin{pr} \label{l31xx} Let $ a \in L^{2}_{\rm loc}(\rd)$. Set $a_{-}(x)=\max\{-a(x),0\}$, \[ \alpha(\varepsilon) = \sup_{x\in {\rd}} \int_{|x-y|\leq \varepsilon} a_-^2(y)dy \] and suppose that $\alpha(\varepsilon)\to 0$ as $\varepsilon\to 0$. Then we have \bel{2} \liminf_{p \to \infty} p^{-2} \inf \sigma(H(p)) \geq 1. \ee \end{pr} \begin{proof} Applying estimate \eqref{2y} with $\varepsilon=p^{-1}$ to the function $v=a_{-}$, we find that \begin{eqnarray*} \int_{\rd} \left(|\nabla u|^2 + (p + a)^2 \right) |u|^2 dx \geq \int_{\rd} \left( |\nabla u|^2 + (- 2p a_- +p^2) |u|^2 \right) dx %\nonumber \\ \geq \int_{\rd} \left(|\nabla u|^2 + p ^2 \right) |u|^2 dx -C \sqrt{\alpha(p^{-1}) } \int_{\rd} \left(|\nabla u|^2 + p^2 \right) |u|^2 dx. \end{eqnarray*} Since $\alpha(p^{-1}) \to 0$ as $p\to\infty$, this implies \eqref{2}. \end{proof} \begin{pr} \label{t31} Let $ a \in L^{2}_{\rm loc}(\rd)$ and let condition \eqref{Di} be satisfied. Then, for all $n \in {\mathbb N}$, we have \bel{kr4} \lambda_{n}(p) = p^2(1 + o(1)), \quad p \to \infty. \ee \end{pr} \begin{proof} Under condition \eqref{Di} the function $a_{-}$ has compact support so that we can use Proposition~\ref{l31xx} and estimate \eqref{2} implies \bel{kr4a} \liminf_{p \to \infty} p^{-2} \lambda_{n }(p) \geq 1. \ee Set $G (\varepsilon)=-\Delta + (1 + \varepsilon^{-1}) a^2(x) $, $\varepsilon > 0$. The spectrum of $G (\varepsilon)$ is discrete; let $ \nu_{n }$, $n \in {\mathbb N}$, be the increasing sequence of its eigenvalues. By the elementary inequality $$(a+p)^2 \leq (1 + \varepsilon^{-1}) a^2 + (1 + \varepsilon) p^2, \q \varepsilon > 0, $$ we have $H (p) \leq G (\varepsilon) + (1 + \varepsilon) p^2$ so that by the minimax principle $$\lambda_{n }(p) \leq \nu_{n }(\varepsilon) + (1 + \varepsilon) p^2. $$ Therefore, for all $\varepsilon>0$, $$\limsup_{p \to \infty} p^{-2} \lambda_{n }(p) \leq 1 + \varepsilon, $$ which combined with \eqref{kr4a} yields \eqref{kr4}. \end{proof} \begin{follow} \label{t31f} Suppose that the function $a$ depends on $r$ only. Let conditions \eqref{kr8} and \eqref{j1} be satisfied. Then, for all $n \in {\mathbb N}$, $m \in {\mathbb Z}$, we have % \bel{kr4X} \[ \lambda_{n,m}(p) = p^2(1 + o(1)), \quad p \to \infty. \] \end{follow} \medskip {\bf 3.2.} {\it From now on we always assume that the function $a$ depends on $r$ only and that conditions \eqref{kr8} and \eqref{j1} are satisfied}. In this subsection we investigate the asymptotics as $p \to -\infty$ of the eigenvalues $\lambda_{n,m}(p)$ of the operators $H_{m}(p)$. Actually, it is more convenient to work with the operators $L_{m}(p)$ acting in the space $ L^{2} (\re_{+})$ by formula \eqref{Lm}. % These operators are correctly defined by their quadratic forms \eqref{w1ay}. The operators $H_{m}(p)$ and $L_{m}(p)$ are unitarily equivalent. We suppose that the function $a$ is differentiable at least for suffiently big $r$ and formulate the results in terms of the function $b(r)=a'(r)$ related to the magnetic field by formula \eqref{magnfi}. Remark first that if $k= -p>0$ is big enough, then the equation \bel{e5} a(r) = k \ee has at least one solution. We denote by $\rho_k$ the greatest solution of (\ref{e5}). Clearly, $\rho_k\to\infty$ as $k\to\infty$. \begin{pr} \label{t32} % Let \bel{kr1} a \in L^{\infty}_{\rm loc}([0,\infty)). \ee Suppose that \bel{e3} \lim_{r\to\infty} b(r) = 0. \ee Then for each $n \in {\mathbb N}$ and $m \in {\mathbb Z}$ we have \bel{e4} \lim_{k \to\infty} \lambda_{n,m}(-k) = 0. \ee \end{pr} \begin{proof} Set \bel{E} {\bf b}(r) = \sup_{x \geq r} |b (x)| \q {\rm and}\q \gamma_k= {\bf b} (\rho_k)^{-1/2}. \ee % and choose a function %$\gamma_k$ satisfying the conditions % \bel{e5a} \lim_{k\to\infty} \gamma_k =\infty \ee %and %\bel{E1} \lim_{k \to \infty} \gamma_k\Phi(\rho_k)=0.\ee %These two conditions are compatible because in view of %(\ref{e3}) $\Phi(r) \to 0$ as $ r \to \infty$. Let us fix $n \in {\mathbb N}$. We pick a function $\phi_1 \in C_0^{\infty}(\re)$ such that ${\rm supp}\,\phi_1 = \left[0, \frac{1}{2n}\right]$ and, for $n>1$, set $$ \phi_j(x) = \phi_1(x - (j-1)/n), \quad x \in \re, \quad j=2,\ldots, n. $$ For $k>0$ large enough, we put \bel{e5xx} \varphi_{j}(r;k) = \gamma_k^{-1/2} \phi_j \left(\frac{r-\rho_k}{\gamma_k}\right), \quad r \geq 0, \quad j=1,\ldots, n. \ee %Let the quadratic form $ l_{-k,m} $ be defined by equation \eqref{w1ay}. We will prove now that for quadratic form \eqref{w1ay} \bel{e8} \lim_{k \to \infty} l_{m} [\varphi_{j}(k); -k] = 0. \ee It follows from \eqref{e5xx} that \bel{e9} \int_0^{\infty} |\varphi'_{j} (r;k)|^2 dr \leq C \gamma_k^{-2} \ee with $C$ independent of $k$. Further, since $\supp \varphi_{j}(k) \subset [\rho_k, \rho_k + \gamma_k ]$, we have \bel{e10} \int_0^{\infty} r^{-2}|\varphi_{j} (r;k)|^2 dr \leq C \rho_k ^{-2}. \ee Similarly, \bel{e11} \int_0^{\infty} (a(r)-k)^2 |\varphi_{j} (r;k)|^2 dr \leq C \sup_{r \in (\rho_k, \rho_k + \gamma_k )} (a(r)-k)^2. \ee Using the condition $a(\rho_{k})=k$, we obtain, for $r\geq \rho_{k}$, the bound %\bel{e11x} \[ (a(r)-k)^2 = (a(r)- a(\rho_{k}))^2 = \left( \int_{\rho_{k}}^r b (s)ds\right)^2\leq (r-\rho_{k})^2 {\bf b}^2(\rho_k) \] where ${\bf b} $ is function (\ref{E}). Thus, the right-hand side in (\ref{e11}) is bounded by $C \gamma_k^2 {\bf b}^2(\rho_k)$. Putting together this result with inequalities (\ref{e9}), (\ref{e10}) and taking into account (\ref{E}), we get $$ l_{m}[\varphi_{j}(k); -k] \leq C \left({\bf b} (\rho_k) + \rho_k^{-2} \right). $$ This yields (\ref{e8}). Let us use now that the supports of the functions $\varphi_{j}(k)$, $j=1,\ldots, n $, are disjoint and set \bel{j9a} {\mathcal L}_{n}(k) = {\rm span}\, \left\{\varphi_{1}(k), \ldots, \varphi_{n}(k)\right\}. \ee Then $ {\rm dim}\; {\mathcal L}_{n}(k) = n$ and according to (\ref{e8}) $ l_{ m} [\varphi (k); -k] \to 0$ as $k \to \infty$ for all $\varphi (k)\in {\mathcal L}_{n}(k) $ with $\|\varphi (k)\|=1$. By the mini-max principle this implies (\ref{e4}). \end{proof} The proof of Proposition~\ref{t33} relies on a comparison of the operator $L_{m}(-k)$ with the ``model" operator \bel{j2c} T(k) = -\frac{d^2}{dx^2} + b^2 (\varrho_k) (x-\varrho_k)^2, \quad x \in \re, \ee acting in the space $L^2(\re)$. Let $f_j$ be the normalized in $L^2(\re)$ real-valued eigenfunctions (defined up to sign) of the harmonic oscillator, i.e. \bel{HO} -f_j''(x) + x^2 f_j(x) = (2j-1) f_j(x), \quad x \in \re, \quad j \in {\mathbb N}. \ee Then \bel{j3} \psi_{j}(x;k) = b (\varrho_k)^{1/4} f_j (b (\varrho_k)^{1/2}(x-\varrho_k)) \ee are normalized eigenfunctions of the operator $T (k)$, that is \bel{w30} T (k) \psi_{j} (k) = b (\varrho_k) (2j-1) \psi_{j} (k), \quad j \in {\mathbb N}. \ee The proof of the following result follows the general lines of the proof of \cite[Theorem 11.1]{CFKS}. \begin{pr} \label{t33} Suppose that $a(r)$ is locally semibounded from above. For $r>0$ large enough, we assume that the function $b(r)$ is differentiable and that conditions \bel{j2} b (r) > 0, \ee \bel{j2bX} \lim_{r \to\infty} r^{2} b (r) =\infty, \ee as well as \bel{j3a} \lim_{r \to \infty} b (r)^{-3} {\bf b}_{1}^2 (r) = 0, \q {\rm where} \q {\bf b}_{1}(r) = \sup_{r/2\leq x \leq 3r/2} |b'(x)|, \ee are satisfied. Let also \bel{j2b} \lim_{k \to\infty} k^{-2} b (\rho_{k}) =0. \ee Then, for all $n \in {\mathbb N}$, $m \in {\mathbb Z}$, we have \bel{j4} \lambda_{n,m}(-k) = b (\varrho_k) (2n-1 + o(1)), \quad k \to \infty. \ee \end{pr} \begin{proof} Due to the minimax principle, it suffices to show that:\\ (i) For each $n \in {\mathbb N}$ and sufficiently large $k$ there exists a subspace ${\mathcal L}_{n} (k)$ of $L^2(\re_+)$ such that ${\rm dim}\,{\mathcal L}_{n} (k) = n$, ${\mathcal L}_{n} (k) \subset D(L_m(-k))$, and for each $\varphi (k) \in {\mathcal L}_{n} (k)$ we have \bel{j5} \langle L_{m}(-k) \varphi (k), \varphi (k)\rangle \leq b (\varrho_k) (2n-1 + o(1))\|\varphi (k) \|^2, \quad k \to \infty. \ee (ii) For each $n \in {\mathbb N}$ there exists a bounded operator $R_n (k)$ such that ${\rm rank}\,R_n (k)\leq n-1$ (hence, $R_1 (k) = 0$), and \bel{j6} L_m(-k) \geq b (\varrho_k) (2n-1 + o(1))I + R_n (k), \quad k \to \infty. \ee We pick $\gamma_k > 0$ such that \bel{j7} \gamma_k \to 0, \ee \bel{j8} \gamma_k \, \varrho_k b(\varrho_k)^{1/2} \to \infty, \ee \bel{j8a} \gamma_k^{-3} \, b(\varrho_k)^{-3/2}{\bf b}_{1} (\varrho_k)\to 0 \ee as $k\to\infty$. Note that \eqref{j8} is compatible with \eqref{j7} due to \eqref{j2bX}, and \eqref{j8a} is compatible with \eqref{j7} due to \eqref{j3a}. \medskip {\em Proof} of (i). Let $\zeta \in C_0^{\infty}(\re)$ be such that $0\leq \zeta(x) \leq 1$ , $\zeta(x) = 1$ for $|x|\leq 1/2$ and $\supp \zeta = [-1,1]$. For $k$ large enough, set \bel{j9b} \zeta (r;k) = \zeta(\gamma_k b (\varrho_k)^{1/2}(r-\varrho_k)), \quad r \in \re_{+}, \ee and \bel{j9} \varphi_{j}(r;k) = \psi_{j}(r;k) \zeta (r;k), \quad r \in \re_{+}, \quad j \in {\mathbb N}, \ee the functions $\psi_{j} (r;k)$ being defined in \eqref{j3}. It follows from \eqref{j8} that $$ \supp {\varphi_{j}} (k) = [ \varrho_k - \gamma_k^{-1} b (\varrho_k)^{-1/2}, \varrho_k + \gamma_k^{-1}b (\varrho_k)^{-1/2} ] \subset [\varrho_k/2, 3\varrho_k/2] $$ and, in particular, ${\varphi_{j}}(k) \in D(L_m(-k))$. Note that \bel{j19a} \langle \varphi_{j}(k), \varphi_{l}(k)\rangle_{L^2(\re_+)} = \delta_{jl} - \int_{\re} \psi_{j}(x;k)\psi_{l}(x ; k) (1 - \zeta^2(x; k))dx = \delta_{jl} + o(1) \ee as $k \to \infty$. Indeed, the integral here can be estimated by $$ \int_{\re} |f_{j}(x)f_{l}(x)| (1 - \zeta^2(\gamma_k x) )dx \leq \int_{|x| \geq (2\gamma_k)^{-1} } |f_{j}(x)f_{l}(x)| dx $$ which tends to zero according to \eqref{j7}. In particular, \eqref{j19a} implies that for all $n \in {\mathbb N}$ the functions $\varphi_{1}(k), \ldots, \varphi_{n}(k)$ are linearly independent if $k$ is large enough. Thus, the space $ {\mathcal L}_{n}(k)$ defined by \eqref{j9a} has dimension $n$. Let us set \bel{psi} \psi (x;k ) = \sum_{j=1}^n c_j \psi_j(x;k ), \q c_j \in {\mathbb C}, \ee $ \varphi (r;k) = \psi (r;k) \zeta (r;k)$ and consider $ \langle L_{m}(-k) \varphi (k), \varphi (k)\rangle$. Integrating by parts, we find that \[ -2{\rm Re} \, \langle \psi'(k) \zeta'(k) , \psi (k)\zeta(k)\rangle-\langle \psi(k) \zeta''(k) , \psi (k)\zeta(k)\rangle = \| \psi(k) \zeta'(k)\|^2 \] so that \begin{eqnarray} \langle L_{m}(-k) \varphi (k), \varphi (k)\rangle &=& {\rm Re} \, \langle -\psi''(k) + (a(r)-k)^2 \psi (k), \psi (k)\zeta^2(k)\rangle \nonumber\\ &+& \| \psi(k) \zeta'(k)\|^2 + (m^2-1/4) \| r^{-1} \varphi (k) \|^2. \label{j5x1} \end{eqnarray} We assume that $\| \varphi (k)\|=1$ and hence according to \eqref{j19a} $\| \psi(k)\|=1+ o(1)$. The second and third terms in the right-hand side of \eqref{j5x1} are negligible. Indeed, differentiating \eqref{j9b} and using condition \eqref{j7}, we find that \bel{j12} \|\psi (k)\zeta' (k)\|^2 = O(b (\varrho_k) \gamma_k^2) = o(b (\varrho_k)). \ee Since $ r^{-1} \leq 2 \varrho_k^{-1}$ on the support of $\varphi (k)$, relation \eqref{j2bX} implies \bel{j11} \|r^{-1} \varphi(k) \|^2= O (\varrho_k^{-2}) = o( b (\varrho_k)), \quad k \to \infty. \ee Further we consider the first term in the right-hand side of \eqref{j5x1}. It follows from equation \eqref{w30} that \bel{j14} -\psi_{j}''(k) + (a(r)-k)^2 \psi _{j}(k) = b (\varrho_k)(2j-1) \psi_{j} (k) + \alpha(k) \psi_{j}(k) \ee where the function \bel{alpha} \alpha(r;k)= (a(r)-k)^2 - b ^2 (\varrho_k) (r-\varrho_k)^2. \ee Let us estimate the right-hand side. In view of the equation $a(\rho_{k})=k$, a second-order Taylor expansion of $a$ at $\varrho_k$ yields $$ a(r) = k + b (\varrho_k) (r-\varrho_k) + \int_{\varrho_k}^r b '(s) (r-s) ds. $$ Therefore, $$ \alpha(r;k) = 2 b(\varrho_k) (r-\varrho_k)\int_{\varrho_k}^r b'(s) (r-s) ds + \left(\int_{\varrho_k}^r b '(s) (r-s) ds\right)^2, $$ and hence $$ |\alpha(r;k)| \leq b (\varrho_k) {\bf b}_{1}(\varrho_k) | r-\varrho_k |^3 + 4^{-1}{\bf b}_{1}^2 (\varrho_k) (r-\varrho_k)^4 $$ % \bel{j17} \[ \leq \gamma_k^{-3} b (\varrho_k)^{-1/2} {\bf b}_{1}(\varrho_k) + 4^{-1} \gamma_k^{-4} b (\varrho_k)^{-2} {\bf b}_{1}^2(\varrho_k), \] provided that $|r-\varrho_k| \leq \gamma_k^{-1} b (\varrho_k)^{-1/2}$. In view of conditions \eqref{j7} and \eqref{j8a}, this gives us the estimate \bel{j17n} \sup_{|r-\varrho_k| \leq \gamma_k^{-1} b (\varrho_k)^{-1/2}} |\alpha(r;k) | =o(b (\rho_{k})) \ee so that % \bel{j14v} \[ \left( -\psi_{j}''(k) + (a(r)-k)^2 \psi _{j}(k) \right) \zeta(k)= b (\varrho_k)(2j-1) \varphi_{j} (k)+ o(b (\rho_{k})). \] Thus, using also \eqref{j19a} we obtain that \begin{eqnarray} {\rm Re} \, \langle -\psi''(k) + (a(r)-k)^2 \psi (k), \psi (k)\zeta^2(k)\rangle &=& b (\varrho_k) \sum_{j,l=1}^n (2j-1) c_{j}\bar{c}_{l} \langle \varphi _{j}(k), \varphi_{l} (k)\rangle + o(b (\rho_{k})) \nonumber\\ &\leq& b (\varrho_k) (2n-1) + o(b (\rho_{k})). \label{j5x2} \end{eqnarray} Together with \eqref{j12} and \eqref{j11}, this implies estimate \eqref{j5} for each $\varphi (k) \in {\mathcal L}_{n}(k)$. \medskip {\em Proof} of (ii). Let functions $\zeta \in C_{0}^{\infty}(\re)$ and $\eta \in C^{\infty}(\re)$ satisfy $ \zeta^2(x) + \eta^2(x) = 1$, $ x \in \re$; moreover, as before, we require that $0\leq \zeta(x) \leq 1$ , $\zeta(x) = 1$ for $|x|\leq 1/2$ and $ \supp \zeta = [-1,1]$. By analogy with \eqref{j9b} set \bel{j21q} \eta(r;k) = \eta(\gamma_k b (\varrho_k)^{1/2} (r-\varrho_k)), \q r \in \re_{+}. \ee Then we have % \bel{j20} \[ \zeta^2(r;k) + \eta^2(r;k) = 1, \quad r \in \re_{+}. \] We proceed from the localization formula (known as the IMS formula -- see e.g. \cite[Section 3.1]{CFKS}) % \bel{j21} \[ L_m(-k) = \zeta(k) L_m(-k)\zeta (k) + \eta (k) L_m(-k)\eta (k) - \zeta'(k)^2 - \eta'(k)^2, \] where $\zeta(k)$, $\eta (k)$, $\zeta'(k)$ and $\eta'(k)$ are understood as operators of multiplication by the functions $\zeta(r,k)$, $\eta (r,k)$, $\zeta'(r,k)$ and $\eta'(r,k)$, respectively. According to \eqref{j7} it follows from definitions \eqref{j9b} and \eqref{j21q} that \bel{j22} \max_{r\in\re_{+}}\,(\zeta' (r,k) ^2 + \eta'(r,k) ^2 )= O\left(\gamma_{k}^2 b (\varrho_k)\right) = o\left(b (\varrho_k)\right), \quad k \to \infty. \ee Next, we check that \bel{j23w} \eta (k) L_{m}(-k) \eta (k)\geq \nu_k b (\varrho_k) \eta^2 (k) \ee with $\nu_k \to \infty$ as $k \to \infty$. By virtue of the Hardy inequality % \bel{j23} \[ \eta (k) \left( -\frac{d^2}{dr^2} + \frac{m^2-1/4}{r^2}\right) \eta (k) \geq 0, \] it suffices to check that \bel{j24} (a(r) - k)^2 \geq \nu_k b(\varrho_k) \ee for \bel{j24c} r \geq \varrho_k+ \left(2\gamma_k b (\varrho_k)^{1/2}\right)^{-1}=:\varrho_k^{(+)}\q {\rm and} \q r \leq \varrho_k-\left(2\gamma_k b (\varrho_k)^{1/2}\right)^{-1}=:\varrho_k^{(-)}. \ee According to \eqref{j2} there exists $r_{0}$ such that the function $a(r)$ is increasing for $r\geq r_{0}$. Let first $r\geq r_{0}$. Then \bel{j24d} | a(r) - k | = | a(r) - a(\varrho_k)| \geq \pm (a(\varrho_k^{(\pm)}) - a(\varrho_k)) \ee if $\pm (r- \varrho_k^{(\pm)})\geq 0$ and $r\geq r_{0}$. It follows from definition \eqref{j24c} of the numbers $\varrho_k^{(\pm)} $ that \bel{j24e} a(\varrho_k^{(\pm)}) - a(\varrho_k) = \int_{\varrho_k}^{\varrho_k^{(\pm)}}b (s)ds= \pm (2\gamma_k)^{-1} b (\varrho_k)^{1/2}+\int_{\varrho_k}^{\varrho_k^{(\pm)}}(b (s)-b (\varrho_k))ds. \ee The absolute value of the integral in the right-hand side can be estimated by % \bel{j24f} \[ \left| \int_{\varrho_k}^{\varrho_k^{(\pm)}}ds \int_{\varrho_k}^s | b '(\sigma)|d\sigma\right| \leq 2^{-1} {\bf b}_{1} (\rho_{k}) (\varrho_k^{(\pm)}-\varrho_k)^2 = 8^{-1} {\bf b}_{1} (\rho_{k}) \gamma_k^{-2}b (\varrho_k)^{-1} \] where the function ${\bf b}_{1} $ is defined in \eqref{j3a}. By virtue of conditions \eqref{j7} and \eqref{j8a} this expression is $o(\gamma_k^{-1} b (\varrho_k)^{1/2})$ as $k\to\infty$. Therefore the absolute value of expression \eqref{j24e} is bounded from below by $ (3\gamma_{k})^{-1} b(\varrho_k)^{1/2}$. Thus, for $r\geq r_{0}$, estimate \eqref{j24} with $\nu_{k}=(3\gamma_{k})^{-2}\to\infty$ is a consequence of \eqref{j24d}. If $r\leq r_{0}$, we take into account that $a(r)$ is semibounded from above so that $(a(r) - k)^2\geq 2^{-1} k^2$. Hence estimate \eqref{j24d} with $\nu_{k}= 2^{-1} k ^{2}b (\varrho_k)^{-1} \to\infty$ is satisfied according to condition \eqref{j2b}. Putting together definitions \eqref{Lm} and \eqref{j2c} of the operators $L_m(-k)$ and $T(k)$, we see that \bel{j27} \zeta (k) L_m(-k) \zeta (k)= \zeta (k) T (k) \zeta (k) + \alpha(k) \zeta^2 (k), \ee where $\alpha(k)$ is the operator of multiplication by function \eqref{alpha}. The first term in the right-hand side is bounded from below by $ b (\varrho_k) \zeta^2 (k)$ because $b (\varrho_k)$ is the first eigenvalue of the operator $T(k)$. By virtue of \eqref{j17n} the second term satisfies the estimate \bel{j27Y} \| \alpha(k) \zeta^2 (k)\|=o(b (\varrho_k)) . \ee It follows that operator \eqref{j27} is bounded from below by $ b (\varrho_k) \zeta^2 (k) -o(b (\varrho_k))I$. Combining this result with \eqref{j22} and \eqref{j23w}, we get estimate \eqref{j6} in the case $n=1$. If $n \geq 2$, we denote by $ P_{n}(k)$ the orthogonal projection onto the span of the first $n-1$ eigenfunctions of the operator $T (k)$. Then $T(k)(I-P_{n}(k))\geq (2n-1)(I-P_{n}(k))$ and hence $$ \zeta (k) T (k) \zeta (k) = \zeta (k) T (k)(I-P_n(k)) \zeta (k) + \zeta(k)T (k) P_n (k) \zeta (k) $$ \bel{j28} \geq b (\varrho_k) (2n-1) \zeta (k) (I-P_n (k)) \zeta (k) + \zeta (k) T (k) P_n (k) \zeta (k) = b (\varrho_k) (2n-1) \zeta ^2 (k) + R_n (k) \ee where $$ R_n (k) = \zeta (k)( T (k) - b (\varrho_k) (2n-1) I) P_n(k) \zeta(k). $$ Clearly, ${\rm rank}\,R_n (k)\leq n-1$. Putting together \eqref{j22}, \eqref{j23w} and \eqref{j27} -- \eqref{j28}, we obtain \eqref{j6} in the case $n \geq 2$. \end{proof} \begin{example} \label{ex} Let $b(r)=b_{0}r^{-\delta}$, $b_{0}>0$, $\delta \leq 1$, for sufficiently large $r$. Then ${\bf b}_{1}(r)=b_{0}\delta r^{-\delta-1} $ and conditions \eqref{j2} -- \eqref{j3a} are satisfied. Moreover, $\rho_{k}= c_{1} k^\nu$ and $k^2 b (\rho_{k}) = c_2 k^{ -1-\nu}$ where $\nu= (1-\delta)^{-1}$ and $c_{1}, c_{2}>0$ if $\delta < 1$. If $\delta = 1$, then $\rho_{k}= \exp (b_{0}^{-1} k)$ and $k^2 b (\rho_{k}) = k^2 b_0 \exp (- b_{0}^{-1} k)$. In both cases condition \eqref{j2b} is also satisfied. Thus, Proposition~\ref{t33} implies the following results. If $\delta> 0$, then $\lambda_{n,m}(p)\to 0$ as $p\to -\infty$ (this result follows also from Proposition~\ref{t32}). If $\delta= 0$, then the functions $\lambda_{n,m}(p)$ have finite limits $b_{0}(2n-1)$ as $p\to -\infty$. If $\delta < 0$, then these functions tend to $+\infty$ as $p\to -\infty$. \end{example} %Let $a(r)=a_{0}r^\gamma$, $a_{0}>0$, for sufficiently large $r$. Then $\rho_{k}= (a_{0}^{-1} k)^{1/\gamma}$, $\Psi(r)=c_{1}r^{\gamma-2}$, $a'(r)^{-3}\Psi^2(r)= c_{2} r^{-\gamma-1}$, $k^2 a' (\rho_{k}) = c_{3} k^{ -1-1/\gamma}$ so that the conditions of Proposition~\ref{t33} are satisfied for all $\gamma>0$. %If $\gamma<1$, then $\lambda_{n,m}(p)\to 0$ as $p\to -\infty$ (this result follows also from Proposition~\ref{t32}). %If $\gamma=1$, then the functions $\lambda_{n,m}(p)$ have finite limits $a_{0}(2n-1)$ as $p\to -\infty$. If $\gamma>1$, then these functions tend to $+\infty$ as $p\to -\infty$. \medskip {\bf 3.3.} Let us return to the Hamiltonians ${\bf H}_{m} $ and ${\bf H} $ defined in Section~2. \begin{theorem} \label{f31} Assume \eqref{kr8} and \eqref{j1}. % and that, for sufficiently large $r$, the derivative $b(r)=a'(r)$ exists. $(i)$ Then all operators ${\bf H}_m$, $m \in {\mathbb Z}$, and hence ${\bf H} $ are absolutely continuous and their spectra coincide with the half-axes defined by equations \eqref{kr37} and \eqref{kr34}. $(ii)$ If the hypotheses of Proposition~$\ref{t32}$ hold true, then ${\mathcal E}_m=0$ for all $m \in {\mathbb Z}$. Moreover, the multiplicities of all spectra $\sigma({\bf H}_m)$ and hence of $\sigma({\bf H})$ are infinite. $(iii)$ Let the hypotheses of Proposition~$\ref{t33}$ hold true. If $b (r) \to \infty$, then the infimum in \eqref{kr34} is attained $($at a finite point$)$ so that for all $m \in {\mathbb Z}$ %\bel{z5} \[ {\mathcal E_m} =\min_{p\in \re}\lambda_{1,m}(p) > 0. \] $(iv)$ Let the hypotheses of Proposition~$\ref{t33}$ hold true. If $b (r)$ admits a finite positive limit $b_{0}$ as $r \to \infty$, then ${\mathcal E}_m \in (0,b_{0}]$ for all $m \in {\mathbb Z}$. \end{theorem} \begin{proof} It suffices to prove only the assertions concerning the operators ${\bf H}_m$. In view of decomposition \eqref{UI} they reduce to corresponding statements about the operators $\Lambda_{n,m}$. These operators are absolutely continuous because the eigenvalues $\lambda_{n,m}(p)$ are real analytic functions of $p \in \re$ which are non constants since according to Corollary~\ref{t31f} $\lambda_{n,m}(p)\to\infty$ as $p\to\infty$. Moreover, we have that \bel{kr34x} \sigma(\Lambda_{n,m})=[ {\mathcal E}_{n,m},\infty)\q {\rm where} \q {\mathcal E}_{n,m} = \inf_{p \in \re} \lambda_{n,m}(p)\geq 0 \ee because $\lambda_{n,m}(p)>0$ for all $p\in \re$. This implies relations \eqref{kr37} with ${\mathcal E_m} $ defined by \eqref{kr34}. In case (ii) it suffices to use that according to \eqref{e4} ${\mathcal E}_{n,m} =0$ and hence $\sigma(\Lambda_{n,m})=[0,\infty)$ for all $m$ and $n$. In case (iii) Proposition~\ref{t33} implies that $ \lambda_{n,m}(p)\to\infty$ as $p\to-\infty$ for all $n$ and $m$ so that \bel{kr34xy} {\mathcal E}_{n,m} = \min_{p \in \re} \lambda_{n,m}(p) > 0 \ee and hence infimum in \eqref{kr34} can be replaced by minimum. In case (iv) we use that according to \eqref{j4} ${\mathcal E}_{n,m} \leq (2n-1)b_{0}$. Moreover, ${\mathcal E}_{n,m} >0$ because $\lambda_{n,m}(p)> 0$ for all $p \in \re$. For $n=1$, this gives the desired result. \end{proof} \begin{remark} \label{multi} According to \eqref{UI} and \eqref{kr34x} the spectrum of the operator ${\bf H}_{m}$ consists of the ``branches" $[ {\mathcal E}_{n,m},\infty)$ where the points ${\mathcal E}_{n,m}$ are called thresholds. In cases (iii) and (iv) \bel{kr34yy} {\mathcal E}_{n,m} < {\mathcal E}_{n+1,m} \ee for all $n\in{\Bbb N}$. Indeed, in case iii) \eqref{kr34yy} is a consequence of the estimate $\lambda_{n,m}(p)< \lambda_{n+1,m}(p)$ valid for all $p\in\re$ and of formula \eqref{kr34xy}. In case (iv) one has to take additionally into account that the limit of $\lambda_{n,m}(p)$ as $p\to-\infty$ is strictly smaller than that of $\lambda_{n+1,m}(p)$. Inequality \eqref{kr34yy} means that there are infinitely many distinct thresholds in each of the spectra $\sigma({\bf H}_m)$, $m \in {\mathbb Z}$, and hence in $\sigma({\bf H})$. \end{remark} \begin{remark} \label{multi1} In case( iii) the multiplicity of the spectrum of all operators $\Lambda_{n,m}$ equals at least to $2$ whereas in cases (ii) and (iv) it might be equal to $1$. \end{remark} %%%%%%%%%%%%%%%%%% \section{Group velocities} \setcounter{equation}{0} %%%%%%%%%%% {\bf 4.1.} In this subsection we obtain a formula for the derivative $\lambda'_{n,m}(p)$, $n \in {\mathbb N}$, $m \in {\mathbb Z}$, which yields sufficient conditions for the monotonicity of $\lambda_{n,m}(p)$ as a function of $p$. Recall that the operators $H_m(p)$, $m \in {\mathbb N}$, $p \in \re$, were defined in the space ${\cal H}$ by formula \eqref{u1a}. The proof of Theorem~\ref{t41} relies on integration by parts. To prove that non-integral terms disappear at $r=0$, we use standard bounds on $\psi_{n,m}(r;p)$. Unfortunately, we were unable to find necessary results in the literature and therefore give their brief proofs. Let us consider the differential equation of Bessel type \bel{Be1} - r^{-1} ( r y ')' + m^2 r^{-2} y + q(r)y=0, \quad m=0, 1,2,\ldots, \ee in a neighborhood $(0,r_{0})$ of the point $r=0$. If $q(r)=0$, then it has the regular $y_{0}^{(reg)}(r)=r^m$ and singular $y_{0}^{(sing)}(r)=r^{-m}$ solutions for $m\neq 0$ and $y_{0}^{(reg)}(r)=1 $ and $y_{0}^{(sing)}(r)=\ln r$ for $m = 0$. %As far as their behaviour at $r=0$ is concerned, we use the following standard facts from the theory of ODE of Bessel type (see e.g. \cite[Section 21.5]{N}). \begin{lemma} \label{l40} Let $m\neq 0$, and let the function $r q(r)$ belong to the class $L^{1}(0,r_{0})$. Then equation \eqref{Be1} has a solution $y^{(reg)}(r)$ satisfying the relation \bel{Be2} y^{(reg)}(r)=r^m + o(r^m), \quad r\to 0. \ee For its derivative, we have the bound \bel{Be3} dy^{(reg)}(r)/dr=O(r^{m-1}). \ee Let $m=0$. Suppose that the function $r \ln r q(r)$ belongs to the class $L^{1}(0,r_{0})$. Then equation \eqref{Be1} has a solution $y^{(reg)}(r)$ satisfying relation \eqref{Be2} where $m=0$. For its derivative, we have the bound \bel{Be4} dy^{(reg)}(r)/dr=O\left(\int_{0}^r |q(s)| ds\right). \ee Moreover, if the function $r \ln^2 r q(r)$ belongs to the class $L^{1}(0,r_{0})$, then equation \eqref{Be1} has a solution $y^{(sing)}(r)$ satisfying the relation \bel{Be5} y^{(sing)}(r)= \ln r + o(1), \quad r\to 0. \ee In this case any bounded solution of equation \eqref{Be1} coincides $($up to a constant factor$)$ with the regular solution $y^{(reg)}(r)$. \end{lemma} \begin{proof} We construct the function $y^{(reg)}(r)$ as the solution of the Volterra integral equation \bel{Int1} y^{(reg)}(r)=y^{(reg)}_{0}(r)+\varkappa_{m}\int_{0}^r s (y^{(reg)}_{0}(r)y^{(sing)}_{0}(s)-y^{(reg)}_{0}(s)y^{(sing)}_{0}(r))q(s) y^{(reg)}(s) ds \ee where $\varkappa_{m}=(2m)^{-1}$ for $m\neq 0$ and $\varkappa_{0}= -1$. Differentiating it twicely, we see that $y^{(reg)}(r)$ satisfies equation \eqref{Be1}. Equation \eqref{Int1} can be solved by iterations, that is \bel{it} y^{(reg)}(r)=\sum_{n=0}^\infty y^{(reg)}_{n}(r). \ee Hereby the $n^{th}$-iteration obeys the bound % \bel{Int2} \[ | y_{n}^{(reg)}(r)|\leq \frac{C^n}{n!}r^m\Big(\int_{0}^r s |q(s) | ds\Big)^n \] if $m\neq 0$; if $m = 0$, then $s |q(s) |$ should be replaced by $s |\ln s| |q(s) |$. This ensures the convergence of series \eqref{it} as well as relation \eqref{Be2}. Differentiating equation \eqref{Int1} and using \eqref{Be2}, we get bounds \eqref{Be3} and \eqref{Be4} on the derivative of $y^{(reg)}(r)$. If $m=0$, we can construct the function $y^{(sing)}(r)$ as the solution of equation \eqref{Int1} where the first term, $y_{0}^{(reg)}(r)$, in the right-hand side is replaced by $ y_{0}^{(sing)}(r)$, that is \[ y^{(sing)}(r)=\ln r+\int_{0}^r s \ln (r/s) q(s) y^{(sing)}(s) ds. \] This equation can again be solved by iterations which, in particular, implies estimate \eqref{Be5}. \end{proof} This result can be supplemented by the following \begin{lemma} \label{l41} Let $m\neq 0$, and let the function $ r q^2 (r)$ belong to the class $L^{1}(0,r_{0})$. Assume additionally that $q =\bar{q }$. If $\psi$ is a solution of equation \eqref{Be1} from the class $L^{2}((0,r_{0}); rdr)$, then it coincides $($up to a constant factor$)$ with the regular solution $y^{(reg)}(r)$ and hence satisfies estimates \eqref{Be2} and \eqref{Be3}. \end{lemma} \begin{proof} Let us extend the function $q(r)$ to $(r_{0},\infty)$ by zero, and let us consider the differential operator $$ hy= - r^{-1} ( r y ')' + m^2 r^{-2} y + q(r)y $$ in the space $L^{2}({\Bbb R}_{+}; rdr)$ on domain $C_{0}^\infty({\Bbb R}_{+} )$. If $q=0$, we denote this operator by $h_{0}$. The operator $h_{0}$ is essentially self-adjoint. To prove the same for $h$, it suffices to check that \bel{ess} \int_{{\Bbb R}_{+}}q^2(r) |f(r)|^2 r dr\leq \varepsilon \| h_{0}f\|^2 +C \| f\|^2 , \q f\in C_{0}^\infty({\Bbb R}_{+} ), \q \varepsilon<1. \ee Let us use the estimate \begin{eqnarray*} \int_{|x|\leq r_{0}}q^2(|x|) |u(x)|^2 dx\leq \int_{|x| \leq r_{0}}q^2(|x|) dx \, \max_{x\in {\Bbb R}^2} |u(x)|^2 \nonumber\\ \leq \varepsilon \int_{{\Bbb R}^2} |(\Delta u)(x)|^2 dx +C\varepsilon^{-1} \int_{{\Bbb R}^2} | u(x)|^2 dx, \q \forall \varepsilon>0 . \end{eqnarray*} Restricting it on the subspace of functions $u(x)=f(r)e^{i m\theta}$, we obtain estimate \eqref{ess} which implies that $h$ is essentially self-adjoint as well as $h_{0}$. Thus, equation \eqref{Be1} has at most one solution from $L^{2}((0,r_{0}); rdr)$ which is necessarily proportional to $y^{(reg)}(r)$. \end{proof} Now we are in a position to obtain a formula for the derivative $\lambda'_{n,m}(p)$. In addition to our usual assumptions that $b(r)$ is not too singular at $r=0$, an integration-by-parts marchinery requires that $b(r)$ does not vanish too rapidly as $r\to 0$. The precise conditions are formulated rather differently in the cases $m\neq 0$ and $m=0$. We start with the first case. \begin{theorem} \label{t41} Let $m\neq 0$. Suppose that $b \in C^3(\re_+)$ and $b(r) > 0$, $r \in \re_+$. Assume \eqref{j1} and that $b(r)=O(e^{cr})$ for some $c>0$ as $r\to\infty$. At $r=0$ we suppose that $b(r)=O(r^{-\gamma})$ where $\gamma<3/2$. Moreover, we assume that for some $\beta< 2|m|- 1 $ \bel{beta} | (b(r)^{-1})^{(k)}| \leq C r^{-\beta-k}, \q k=0,1,2,3, \q r\to 0. \ee Put \[ v(r)=r (r^{-1}(rb(r)^{-1})')'. \] Then \begin{eqnarray} \lambda_{n,m}'(p) = - 2\int_0^{\infty} r b^{-2}(r)b'(r) \psi_{n,m}'(r;p)^2 dr \nonumber\\ -2^{-1}\int_0^{\infty} v' (r) \psi_{n,m}^2 (r; p) dr +2 m^2 \int_0^{\infty} r^{-2} b^{-1}(r) \psi_{n,m}^2(r; p) dr , \label{sa8}\end{eqnarray} where the eigenfunctions $\psi_{n,m}(r;p)$ of the operator $H_{m}(p)$ are real and normalized, that is $\|\psi_{n,m}\| = 1$. \end{theorem} \begin{proof} In view of the equation \bel{Bess} (a(r) + p)^2 \psi_{n,m} =r^{-1} ( r\psi_{n,m}' )' -m^2 r^{-2} \psi_{n,m} + \lambda_{n,m}\psi_{n,m} \ee we can apply to the function $\psi_{n,m} $ the results of Lemmas~\ref{l40} and \ref{l41} where $q(r)=(a(r) + p)^2 -\lambda_{n,m}$. Thus, Lemma~\ref{l41} implies that $\psi_{n,m}(r;p)= O(r^{|m|} )$ and $\psi_{n,m}'(r;p)= O(r^{|m|-1})$ as $r\to 0$ which ensures that non-integral terms disappear at $r=0$. To prove the same for non-integral terms corresponding to $r\to \infty$, we use super-exponential decay of eigenfunctions $\psi_{n,m}(r;p )$ of the operators $H_m(p)$. This result is valid \cite{Sh} (see also \cite{Gl}) for all one-dimensional Schr\"odinger operators with discrete spectra. In view of the condition $a(r)=O(e^{cr})$, it follows from equation \eqref{Bess} that the derivatives $\psi_{n,m}'(r ;p)$ also decay super-exponentially. Let us proceed from the formula of the first order perturbation theory (known as the Feynman-Hellman formula) \bel{F-H} \lambda_{n,m}'(p) = \int_0^{\infty} \frac{\partial (a(r) + p)^2}{\partial p} \psi_{n,m}^2 (r;p) rdr = \int_0^{\infty} \frac{\partial (a(r) + p)^2}{\partial r} \psi_{n,m}^2(r;p ) \tau(r) dr \ee where $\tau(r)=r b(r)^{-1}$. Using that $a(r)=O(r^{1-\gamma})$ and $\tau (r)=O(r^{1-\beta})$, we integrate by parts and get % \bel{sa4} \[ \lambda_{n,m}'(p) = - \int_0^{\infty} (a(r) + p)^2 \psi_{n,m}(r;p ) ( \tau'(r) \psi_{n,m} (r;p) + 2 \tau(r) \psi_{n,m}'(r;p) )dr. \] Now it follows from equation \eqref{Bess} that \begin{eqnarray} \lambda_{n,m}'(p) = -\lambda_{n,m}(p) \int_0^{\infty} ( \tau(r) \psi_{n,m}^2 (r;p) )' dr +m^2 \int_0^{\infty} r^{-2} ( \tau(r) \psi_{n,m}^2 (r;p) )'dr \nonumber\\ - \int_0^{\infty} r^{-1} ( r\psi_{n,m}' (r;p))' ( \tau'(r) \psi_{n,m} (r;p) + 2\tau (r) \psi_{n,m}'(r;p) )dr. \label{sa6}\end{eqnarray} By the condition $ \tau (r) \psi_{n,m}^2 (r;p) \to 0$ as $r\to 0$, the first term in the right-hand side equals zero. In the second term we integrate by parts which yields \[ \int_0^{\infty} r^{-2} ( \tau (r) \psi_{n,m}^2 (r;p) )'dr = 2 \int_0^{\infty} r^{-3} \tau (r) \psi_{n,m}^2 (r;p) dr \] because $r^{-2} \tau (r) \psi_{n,m}^2 (r;p) \to 0$. In the last integral in the right-hand side of \eqref{sa6}, we also integrate by parts using that $\psi_{n,m}'(r;p) \psi_{n,m}(r ;p) \tau'(r)\to 0$ as $r\to 0$. Thus, we have that \begin{eqnarray} - \int_0^{\infty} r^{-1} ( r\psi_{n,m}'(r;p) )' \tau'(r) \psi_{n,m} (r;p) dr &=& \int_0^{\infty} \tau'(r) \psi_{n,m}' (r;p)^2 dr \nonumber\\ &+ & \int_0^{\infty} v(r) \psi_{n,m}' (r;p) \psi_{n,m} (r;p) dr . \label{sa5}\end{eqnarray} The last integral in the right-hand side equals \[ - 2^{-1} \int_0^{\infty} v'(r) \psi_{n,m}^2 (r;p) dr \] because $ v(r) \psi_{n,m}^2(r;p) \to 0$ as $r\to 0$. % Note that, if $m\neq 0$, then the non-integral term $ r (r^{-1} t'(r))' \psi_{n,m}^2 (r) $ vanishes at $r=0$ according to Lemma~\ref{l41}. If $m= 0$, we have to use additionally condition \eqref{m=0}. Similarly, we get that \begin{eqnarray*} - 2 \int_0^{\infty} r^{-1} ( r\psi_{n,m}'(r;p) )' \tau(r) \psi_{n,m}'(r;p) dr &=&- \int_0^{\infty} r^{-2} \tau (r) d ( r \psi_{n,m}'(r;p)^2 ) \nonumber\\ &=& \int_0^{\infty} r^2 (r^{-2} \tau (r))' \psi_{n,m}'(r;p)^2 dr \end{eqnarray*} since $\tau (r) \psi_{n,m}'(r;p)^2\to 0$ as $r\to 0$. Putting the results obtained together, we arrive at representation \eqref{sa8}. \end{proof} \begin{follow} \label{t41cc} If $b'(r)\leq 0$ and $ r^{ 2} b(r) v'(r)\leq 4 m^2$ for all $r\geq 0$, then $ \lambda_{n,m}'(p)\geq 0$ for all $p\in {\Bbb R}$ and $ n$. If, moreover, one of these inequalities is strict on some interval, then $ \lambda_{n,m}'(p) >0$. \end{follow} \begin{follow} \label{t41c} If $b(r)=b_{0} r^{-\delta}$, $\delta\in [0, 1]$, then $\tau(r)= b_{0} ^{-1} r^{1+\delta}$, $v (r)= b_{0} ^{-1} (\delta^2-1) r^{\delta -1}$ and \begin{eqnarray*} \lambda_{n,m}'(p) = 2 b_{0}^{-1} \delta \int_0^{\infty} r^\delta \psi_{n,m}'(r;p)^2 dr %\nonumber \\ + b_{0}^{-1} (2 m^2- 2^{-1} (1-\delta)^2 (1+\delta))\int_0^{\infty} r^{-2+\delta} \psi_{n,m}^2(r;p) dr. % \label{sa8a} \end{eqnarray*} For $b_{0}> 0$, this expression is strictly positive $($so that the functions $ \lambda_{n,m}(p)$ are strictly increasing for all $p\in {\Bbb R})$ for $m\neq 0$ since $ (1-\delta)^2 (1+\delta) \leq 1$. Moreover, for $\delta=1$ this result is true for all $m\in {\Bbb Z}$. % If $a(r)=c \ln r $, $c>0$, then $t(r)= c ^{-1} r^{2 }$ and % \begin{eqnarray} \lambda_{n,m}'(p) = 2 c^{-1} \int_0^{\infty} r \psi_{n,m}'(r)^2 dr % + 2 m^2 c^{-1} \int_0^{\infty} r^{-1 } \psi_{n,m}^2(r) dr >0 \label{sa8b}\end{eqnarray} \end{follow} In the case $m=0$ we consider for simplicity only fields \eqref{bb}. \begin{pr} \label{t41cx} %Let $m=0$. If $b(r)=b_{0} r^{-\delta}$, $\delta\in [0, 1]$, then \begin{eqnarray*} \lambda_{n,0}'(p) = 2 b_{0}^{-1} \delta \int_0^{\infty} r^\delta \psi_{n,0}'(r;p)^2 dr % \nonumber \\ - b_{0}^{-1} 2^{-1} (1-\delta)^2 (1+\delta) \int_0^{\infty} r^{-2+\delta} (\psi_{n,0}^2(r;p) -\psi_{n,0}^2(0;p)) dr. % \label{sa8ax} \end{eqnarray*} If $b_{0}> 0$ and $\delta=1$, then $ \lambda_{n,0}'(p)>0$ for all $p\in {\Bbb R}$. % If $a(r)=c \ln r $, $c>0$, then $t(r)= c ^{-1} r^{2 }$ and % \begin{eqnarray} \lambda_{n,m}'(p) = 2 c^{-1} \int_0^{\infty} r \psi_{n,m}'(r)^2 dr % + 2 m^2 c^{-1} \int_0^{\infty} r^{-1 } \psi_{n,m}^2(r) dr >0 \label{sa8b}\end{eqnarray} \end{pr} \begin{proof} Let us proceed again from formula \eqref{F-H}. We use now that the function $\psi_{n,0}(|x|;p) $ of $x\in {\Bbb R}^2$ belongs to the Sobolev class $\mathsf{H}^2_{loc}({\Bbb R}^2)$, and therefore $\psi_{n,0}(r;p)$ has a finite limit as $r\to 0$. Thus, by Lemma~\ref{l40} $\psi_{n,0}'(r;p)= O(r^{1-\varepsilon} )$ for any $\varepsilon>0$ as $r\to 0$. These results allow us to intergrate by parts as in the case $m\neq 0$. The only difference is with the second integral in the right-hand side of \eqref{sa5}. Now $v (r)= b_{0} ^{-1} (\delta^2-1) r^{\delta -1}$ and this integral equals \begin{eqnarray*} \int_0^{\infty} v (r) \psi_{n,0}' (r;p) \psi_{n,0} (r;p) dr & = & 2^{-1} \int_0^{\infty} v(r) d (\psi_{n,0}^2 (r;p) - \psi_{n,0}^2 (0;p)) \\ & = & - 2^{-1} \int_0^{\infty} v'(r) (\psi_{n,0}^2 (r;p) - \psi_{n,0}^2 (0;p))dr \end{eqnarray*} because $v(r)(\psi_{n,0}^2 (r;p) - \psi_{n,0}^2 (0;p))$ as $r\to 0$. \end{proof} \medskip {\bf 4.2.} In this subsection we show that for linear potentials, that is for magnetic fields not depending on $r$, all eigenvalues $\lambda_{n,0}(p)$, $n \in {\mathbb N}$, of the operator $H_0(p)$ are not monotonous functions of $p\in\re$. We follow closely the proof of the first part of Proposition~\ref{t33}. However we now use that eigenfunctions of the harmonic oscillator decay faster than any power of $r^{-1}$ at infinity (actually, they decay super-exponentially). \begin{pr} \label{p41} Assume that for sufficiently large $r$ \bel{kr31} b (r) = b_{0}>0. \ee Then, for all $n \in {\mathbb N}$, some $\gamma_{n}>0$ and sufficiently large $k>0$, we have \bel{san30} \lambda_{n,0}(-k) \leq ( 2n-1)b_{0} -\gamma_{n}k^{-2}. \ee \end{pr} \begin{proof} Let $\zeta$ be the same function as in the proof of the first part of Proposition~\ref{t33}. We set $\rho_{k}=b_{0}^{-1} k$, $\gamma_{k}=2 b_{0}^{1/2} k^{-1}$ and define the functions $\zeta(r;k)$ and $\varphi_{j}(r;k)$ by formulas \eqref{j9b} and \eqref{j9}, respectively. It suffices to check that \bel{san35} \langle L_0(-k)\varphi(k),\varphi (k) \rangle \leq 2n-1 -\gamma_{n}k^{-2}. \ee for sufficiently large $k$ and all normalized functions from subspace \eqref{j9a}. Let us proceed from formula \eqref{j5x1}. Since the functions $\psi_{j}(x;k)$ decay faster than any power of $|x|^{-1}$ as $|x| \to\infty$, the term $o(1)$ in \eqref{j19a} is actually $O(k^{-\infty})$. Similarly, estimate \eqref{j12} can be formulated in a more precise form as \bel{psi1} \|\psi(k) \zeta'(k)\|^2= O(k^{-\infty}). \ee Since $r\leq 2^{-1} 3k$ on the support of $\varphi(k)$, we have that \bel{psi2} \| r^{-1} \psi(k) \|^2 \geq (2/3)^2 k^{-2}. \ee Now function \eqref{psi} is zero if $r$ and $k$ are large enough. Therefore equation \eqref{j14} yields the exact equality \begin{eqnarray*} {\rm Re} \, \langle -\psi''(k) + (b_{0}r -k)^2 \psi (k), \psi (k)\zeta^2(k)\rangle = b_{0} \sum_{j,l=1}^n (2j-1) c_{j}\bar{c}_{l} \langle \varphi _{j}(k), \varphi_{l} (k)\rangle \end{eqnarray*} (cf. \eqref{j5x2}). Up to terms $O(k^{-\infty})$, the right-hand side here is estimated by $b_{0}(2n-1)$. Together with \eqref{psi1} and \eqref{psi2}, this implies estimate \eqref{san35}. \end{proof} Combining relations \eqref{j4} and \eqref{san30}, we see that the eigenvalues $\lambda_{n,0}(p)$ tend as $p\to -\infty$ to their limits $(2n-1) b_{0}$ from below. On the other hand, according to \eqref{kr4} $\lambda_{n,0}(p)\to \infty$ as $p\to \infty$. Thus, all functions $\lambda_{n,0}(p)$ have necessarily local minima. We can obtain an additional information using the following elementary \begin{lemma} \label{f41h} Suppose that \eqref{kr31} is satisfied for all $r>0$ and that $a(r)=b_{0} r$. Then \bel{harm} \lambda_{n,m}(0)= 2 b_{0} ( 2n-1+ |m|) \ee for all $n \in {\mathbb N}$ and $m \in {\mathbb Z}$. \end{lemma} \begin{proof} Let us consider the two-dimensional harmonic oscillator ${\bf T}=-\Delta + b^2_{0} ( x_{1}^2+x_{2}^2)$. Separating the variables $x_{1}$, $x_{2}$, we see that its spectrum consists of the eigenvalues $2 b_{0} (l_{1}+l_{2}-1)$ where $l_{1}, l_{2} \in {\mathbb N}$. It follows that the operator ${\bf T}$ has the eigenvalues $2b_{0}j$, $j \in {\mathbb N}$, of multiplicity $j$. On the other hand, separating the variables in the polar coordinates, we see that the spectrum of ${\bf T}$ consists of the eigenvalues $\lambda_{n,m}(0)$ of the operators $H_{m}(0)$. For the proof of \eqref{harm} we take into account that all eigenvalues $\lambda_{n,m}(0)$ are simple and that $\lambda_{n,m+1}(0) > \lambda_{n,m}(0)$ for all $n$ and $m\geq 0$. Clearly, the operator $H_{0}(0)$ has an eigenvalue $2b_{0}j$ if and only if its multiplicity $j$ is odd. This gives formula \eqref{harm} for $m=0$. We shall show that for every $j\in {\mathbb N}$ \bel{harm1} \lambda_{1,j-1}(0)=\lambda_{2,j-3}(0)=\ldots=\lambda_{2,-j+3}(0)=\lambda_{1,-j+1}(0)=2b_{0}j \ee which is equivalent to formula \eqref{harm} for all $m$. % For $j=1$ equality \eqref{harm1} is true because the lowest eigenvalues of the operators $K$ and $H_{0}(0)$ coincide so that $ \lambda_{1,0}(0) = 2 b_{0}$. Let us choose some $j_{0}$ and suppose that \eqref{harm1} holds for all $j\leq j_{0}$. Then we check it for $j=j_{0}+1$. First we remark that if an operator $H_{m}(0)$ for some $m>0$ has $n$ eigenvalues in the interval $[2b_{0}, 2b_{0}(j_{0}+1)]$, then the operator $H_{m-1}(0)$ has at least $n$ eigenvalues in the interval $[2b_{0}, 2b_{0} j_{0}]$. Then using \eqref{harm1} for $j\leq j_{0}$, we see that if an operator $H_{m}(0)$ has the eigenvalue $2b_{0}(j_{0}+1)$, then necessarily the operator $H_{m-1}(0)$ has the eigenvalue $2b_{0}j_{0} $. Therefore according to \eqref{harm1} for $j=j_{0}$, only the operators $H_{m}(0)$ with $m=j_{0}, j_{0}-2,\ldots, -j_{0}+2, -j_{0}$ might have the eigenvalue $2b_{0}(j_{0}+1)$. There are $j_{0}+1$ of such operators and the multiplicity of this eigenvalue equals $j_{0}+1$. Thus, all the operators $H_{m}(0)$ for $m=j_{0}, j_{0}-2,\ldots, -j_{0}+2, -j_{0}$ and only for such $m$ have the eigenvalue $2b_{0}(j_{0}+1)$. This proves \eqref{harm1} for $j_{0}+1$. \end{proof} Comparing this result with \eqref{j6}, we see that, for potentials $a(r)=b_{0} r$, \[ \lim_{p\to-\infty} \lambda_{n,m}(p)=b_{0}(2n-1)< 2 b_{0}( 2n-1+ |m|) = \lambda_{n,m}(0). \] Together with \eqref{san30}, this implies that the functions $\lambda_{n,0}(p)$ have negative local minima. Thus, we get the following \begin{theorem} \label{f41} Under the hypotheses of Proposition~$\ref{p41}$ the eigenvalues $\lambda_{n,0}(p)$, $n \in {\mathbb N}$, of the operator $H_0(p)$ are not monotonous functions of $p \in \re$. Moreover, if \eqref{kr31} is satisfied for all $r>0$ and $a(r)=b_{0} r$, then the functions $\lambda_{n,0}(p)$ lose their monotonicity for $p<0$. \end{theorem} We do not know how many minima have the functions $\lambda_{n,0}(p)$. The problem of monotonicity of the eigenvalues $\lambda_{n,0}(p)$ for fields $b(r)=b_{0}r^{-\delta}$ where $\delta\in (0,1)$ remains also open. \medskip {\bf 4.3.} In a somewhat similar situation the break down of monotonicity of group velocities was exhibited in \cite{H}. In this paper one considers the Schr\"odinger operator ${\bf H}^{(N)} = -\frac{\partial^2}{\partial x^2}+ \left(i\frac{\partial}{\partial y} - bx\right)^2$ with constant magnetic field $b>0$, defined on the semi-plane $\left\{(x,y) \in \rd \, : \, x>0\right\}$ with the Neumann boundary condition at $x=0$. Let $H^{(N)}(p)=-d^2/dx^2 + (bx+p)^2$, $p \in \re$, be the self-adjoint operator in the space $L^2(\re_+)$ corresponding to the boundary condition $u'(0)=0$. Then the operator ${\bf H}^{(N)}$ is unitarily equivalent under the partial Fourier transform with respect to $y$, to the direct integral $\int_{\re}^{\oplus} H^{(N)}(p) dp$. It is shown in \cite[Section 4.3]{H} that the lowest eigenvalue $\mu_1(p)$ of $H^{(N)}(p)$ is not monotonous for $p<0$. This follows from the inequality $ \mu_1'(0) > 0$ proven\footnote{Note that in \cite{DH} and \cite{H} the parameter $p$ is chosen with the opposite sign.} in \cite{DH} and the relations \bel{san41} \lim_{p \to -\infty} \mu_1(p)= \mu_1(0) = b. \ee Our proof of non-monotonicity of the functions $\lambda_{n,0}(p)$ is essentially different since in contrast with \eqref{san41} we have $\lim_{p \to -\infty} \lambda_{n,0}(p) < \lambda_{n,0}(0)$. %%%%%%%%%%%%%%%%%% \section{Asymptotic time evolution } \setcounter{equation}{0} %%%%%%%%%%% {\bf 5.1.} Combined with the stationary phase method, the spectral analysis of the operators ${\bf H} ={\bf H} (a)$ allows us to find the asymptotics for large $t$ of solutions $u(t)=\exp(-i{\bf H} t)u_0$ of the time dependent Schr\"odinger equation. It follows from (\ref{MH}) that \[ \overline{\exp(-i{\bf H} (a) t) u_0}=\exp(i{\bf H} (-a) t) \overline{ u_0}. \] Therefore it suffices to consider the case $a(r)\to+\infty$. Moreover, on every subspace ${\goth H}_m$ with a fixed magnetic quantum number $m$, the problem reduces to the asymptotics of the function $u(t)=\exp(-i {\bf H}_m t)u_0$. Let us proceed from decomposition (\ref{UI}). Suppose that ${\cal F}_{m} u_{0}\in\ran \Psi_{n,m} $. Then (see (\ref{Psi})) \begin{equation}\label{eq:InV} ( {\cal F}_{m} u_{0} ) (r,p)=\psi_{n,m} (r,p)f(p) \end{equation} where $f= \Psi_{n,m}^* {\cal F}_{m} u_{0}$ and $u(t)= {\cal F}_{m}^*\Psi_{n,m} e^{-i\Lambda _{n,m}t} f$, that is \begin{equation}\label{eq:TE} u_{n,m}(r,x_{3},t)=(2\pi)^{-1/2}\int_{-\infty}^\infty e^{ip x_{3} -i\lambda_{n,m}(p) t}\psi_{n,m}(r,p)f(p)dp. \end{equation} The analytic function $\lambda_{n,m}''(p)$ might have only a countable set of zeros $p_{n,m,l}$ with possible accumulations at $\pm\infty$ only. The function $\lambda_{n,m}'(p)$ is monotone on every interval $ (p_{n,m,l}, p_{n,m,l+1})$ and takes there all values between $\lambda_{n,m}'(p_{n,m,l})=:\alpha_{n,m,l}$ and $\lambda_{n,m}'(p_{n,m,l+1})=:\beta_{n,m,l}$. %Let us denote by $\alpha_{n,m,l}$ and $\beta_{n,m,l}$ the smaller and the bigger of these two numbers. We consider the asymptotics of integral \eqref{eq:TE} on each of the subspaces $L^2 (p_{n,m,l}, p_{n,m,l+1})$ separately. Let us set $\gamma=x_{3} t^{-1}$. First we suppose that $f\in C_{0}^\infty (p_{n,m,l}, p_{n,m,l+1})$. The stationary points of integral \eqref{eq:TE} are determined by the equation \begin{equation}\label{eq:stp} \lambda_{n,m}^\prime(p)=\gamma. \end{equation} If $\gamma\not\in (\alpha_{n,m,l}, \beta_{n,m,l})$, it does not have solutions from the interval $(p_{n,m,l}, p_{n,m,l+1})$. Therefore integrating directly by parts, we find that function \eqref{eq:TE} decays in this region of $x_{3}/t$ faster than any power of $(|x_{3}|+|t|)^{-1}$ (and $r$). If $\gamma\in (\alpha_{n,m,l}, \beta_{n,m,l})$, then on the interval $(p_{n,m,l}, p_{n,m,l+1})$ equation \eqref{eq:stp} has a unique solution which we denote by $\nu_{n,m,l} (\gamma)$. Let us set %\begin{equation}\label{eq:TE2} \[ \Phi_{n,m,l}(\gamma)=\nu_{n,m,l} (\gamma)\gamma -\lambda_{n,m}(\nu_{n,m,l}(\gamma)) \] and denote by $\chi_{n,m,l}$ the characteristic function of the interval $ (\alpha_{n,m,l}, \beta_{n,m,l})$. For $\gamma$ from this interval, we apply the stationary phase method to integral \eqref{eq:TE} which yields \begin{eqnarray}\label{eq:TE1} u (r,x_{3},t)= \tau_{n,m,l}^{(\pm)} e^{i\Phi_{n,m,l}(\gamma)t } \psi_{n,m} (r,\nu_{n,m,l}(\gamma)) |\lambda_{n,m}'' (\nu_{n,m,l} (\gamma)) |^{-1/2} \nonumber\\ \times f(\nu_{n,m,l}(\gamma)) \chi_{n,m,l} (\gamma) |t|^{-1/2} + u_\infty(r,x_{3},t), \q \gamma=x_{3} t^{-1}, \q t\to\pm \infty, \end{eqnarray} where $\tau_{n,m,l}^{(\pm)}= e^{ \mp \pi i \sgn (\lambda''_{n,m}(p))/4}$ for $p\in (p_{n,m,l}, p_{n,m,l+1})$ and \begin{equation}\label{eq:TE1r} \lim_{t\rightarrow\pm\infty } \| u_\infty(\cdot,t)\|= 0. \end{equation} Since the norm in the space ${\goth H}$ of the first term in the right-hand side of (\ref{eq:TE1}) equals the norm of $f$ in the space $ L^2 (p_{n,m,l}, p_{n,m,l+1})$, asymptotics (\ref{eq:TE1}) extends to all functions (\ref{eq:InV}) with an arbitary $f \in L^2 (p_{n,m,l}, p_{n,m,l+1})$. Thus, we have proven \begin{theorem}\label{time} Assume \eqref{kr8} and \eqref{j1}. Let $u(t)=\exp(-i {\bf H}_m t) u_0$ where $u_0$ satisfies $(\ref{eq:InV})$ with $f\in L^2 (p_{n,m,l}, p_{n,m,l+1})$. Then the asymptotics as $t\rightarrow \pm \infty$ of this function is given by relations $(\ref{eq:TE1})$, $(\ref{eq:TE1r})$. % where $\Phi_{n,m,l}$ is phase function $(\ref{eq:TE2})$. \end{theorem} Of course asymptotics $(\ref{eq:TE1})$, $(\ref{eq:TE1r})$ extends automatically to all $f \in L^2({\Bbb R})$ with compact support and to linear of functions $\psi_{n,m} (r,p)f_{n}(p)$ over different $n$. By virtue of formulas $(\ref{eq:TE1})$, $(\ref{eq:TE1r})$ a quantum particle in magnetic field \eqref{magnfi} remains localized in the $(x_{1},x_{2})$-plane but propagates in the $x_{3}$-direction. If $f\in L^2 (p_{n,m,l}, p_{n,m,l+1})$, then a particle ``lives" as $|t|\to\infty$ in the region where $x_{3}\in (\alpha_{n,m,l} t, \beta_{n,m,l} t)$. In particular, if $ \lambda' (p)>0$ ($ \lambda' (p)<0$) for $p\in (p_{n,m,l}, p_{n,m,l+1})$, then a particle propagates in the positive (negative) direction as $t\to+\infty$. Thus, according to Corollary~\ref{t41c} if $b(r)=b_{0}r^{-\delta}$, $\delta\in [0,1]$, $b_{0}>0$, then a particle with the magnetic quantum number $m\neq 0$ propagates always in the positive direction of the $x_{3}$-axis. If $\delta =1$, then this result remains true from all $m$. On the contrary, if $\delta =0$ and $m=0$, then a particle will propagate in a negative direction for some interval of momenta $p$. \medskip {\bf 5.2.} Theorem~\ref{time} implies the existence of asymptotic velocity in the $x_{3}$-direction. The corresponding operator is defined by the equation (cf. \eqref{UI}) %\bel{UIx} \[ {\bf H}_{m}' =\bigoplus_{n\in {\Bbb N}} {\cal F}_{m} ^* \Psi _{n,m} \Lambda_{n,m}' \Psi_{n,m}^* {\cal F}_{m} , \] where $\Lambda_{n,m}' $ are the operators of multiplication by the functions $\lambda_{n,m}' (p)$. To put it differently, the operator ${\bf H}_{m}' $ acts as multiplication by $\lambda_{n,m}' (p)$ in the spectral representation of the operator ${\bf H}_{m} $ where it acts as multiplication by the functions $\lambda_{n,m}(p)$. \begin{pr}\label{asve} Assume \eqref{kr8} and \eqref{j1}. Then, for an arbitrary bounded function ${\mathcal Q}$, \bel{UIy} \slim_{|t| \to \infty} \exp{(i{\bf H}_m t)} {\mathcal Q}(x_{3}/t) \exp{(-i{\bf H}_m t)} = {\mathcal Q}\left({\bf H}'_m\right) \ee $($in particular, the strong limit in the left-hand side exists$)$. \end{pr} \begin{proof} We shall check that for all $u_{0}\in{\cal H}_{m}$ \bel{UIz} \lim_{|t| \to \infty} \| {\mathcal Q}(x_{3}/t) \exp{(-i{\bf H}_m t)} u_{0} - \exp{(-i{\bf H}_m t)} {\mathcal Q} ({\bf H}'_m) u_{0} \| =0 \ee which is equivalent to relation \eqref{UIy}. Remark that if $u_{0}$ satisfies \eqref{eq:InV}, then \bel{UIzz} ( \Psi_{n,m} {\mathcal Q} ({\bf H}'_m) u_0) (r,p) =\psi_{n,m} (r,p){\mathcal Q} (\lambda_{n,m}' (p)) f(p). \ee It suffices to prove \eqref{UIz} on a dense set of elements $u_{0}$ such that equality \eqref{eq:InV} is true with $f\in L^2 (p_{n,m,l}, p_{n,m,l+1})$. Applying the operator ${\mathcal Q}(x_{3}/t)$ to asymptotic relation \eqref{eq:TE1}, we see that the asymptotics of ${\mathcal Q}(x_{3}/t) \exp{(-i{\bf H}_m t)} u_{0}$ is given again by formula \eqref{eq:TE1} where the function $f(\nu_{n,m,l}(\gamma))$ in the right-hand side is replaced by the function ${\mathcal Q} (\gamma)f(\nu_{n,m,l}(\gamma))$. Similarly, it follows from Theorem~\ref{time} and relation \eqref{UIzz} that the asymptotics of $ \exp{(-i{\bf H}_m t)} {\mathcal Q} ({\bf H}'_m) u_{0} $ is given by formula \eqref{eq:TE1} where the function $f(\nu_{n,m,l}(\gamma))$ in the right-hand side is replaced by the function ${\mathcal Q} (\lambda'_{n,m} (\nu_{n,m,l}(\gamma)))f(\nu_{n,m,l}(\gamma))$. So for the proof of \eqref{UIz}, it remains to take equation \eqref{eq:stp} into account. \end{proof} Relation \eqref{UIy} shows that ${\bf H}_m'$ can naturally be interpreted as the operator of asymptotic velocity in the $x_{3}$-direction. Similar results concerning the Iwatsuka model (see \cite{i} or \cite{CFKS}) have been obtained in \cite{mp}. Numerous useful discussions with Georgi Raikov as well as a financial support by the Chilean Science Foundation {\em Fondecyt} under Grant 7050263 are gratefully acknowledged. \begin{thebibliography} {[10]} \frenchspacing \baselineskip=12 pt plus 1pt minus 1pt \bibitem {AB} {\sc Y. 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