\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 123, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/123\hfil Magnetic Schr\"odinger equation] {Nonlinear subelliptic Schr\"odinger equations with external magnetic field} \author[K. Tintarev\hfil EJDE-2004/123\hfilneg] {Kyril Tintarev} \address{Kyril Tintarev \hfill\break Department of Mathematics \\ Uppsala University \\ P. O. Box 480, 751 06 Uppsala, Sweden} \email{kyril.tintarev@math.uu.se} \date{} \thanks{Submitted July 9, 2004. Published October 18, 2004} \thanks{Research done while visiting the Hebrew University in Jerusalem. \hfill\break\indent Supported by a grant from the Swedish Research Council} \subjclass[2000]{35H20, 35J60, 35Q60, 43A85, 58J05} \keywords{Homogeneous spaces; magnetic field; Schr\"odinger operator; \hfill\break\indent subelliptic operators; semilinear equations; weak convergence; concentration compactness} \begin{abstract} To account for an external magnetic field in a Hamiltonian of a quantum system on a manifold (modelled here by a subelliptic Dirichlet form), one replaces the the momentum operator $\frac 1i d$ in the subelliptic symbol by $\frac 1i d-\alpha$, where $\alpha\in TM^*$ is called a magnetic potential for the magnetic field $\beta=d\alpha$. We prove existence of ground state solutions (Sobolev minimizers) for nonlinear Schr\"odinger equation associated with such Hamiltonian on a generally, non-compact Riemannian manifold, generalizing the existence result of Esteban-Lions \cite{EstebanLions} for the nonlinear Schr\"odinger equation with a constant magnetic field on $\mathbb{R}^N$ and the existence result of \cite{FiesTin} for a similar problem on manifolds without a magnetic field. The counterpart of a constant magnetic field is the magnetic field, invariant with respect to a subgroup of isometries. As an example to the general statement we calculate the invariant magnetic fields in the Hamiltonians associated with the Kohn Laplacian and for the Laplace-Beltrami operator on the Heisenberg group. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} In this paper we study nonlinear Schr\"odinger equations with external magnetic field on (generally) non-compact Riemannian manifolds. A summary exposition on the magnetic Schr\"odinger operator can be found in \cite{AHS}. The scope of the paper includes subelliptic Hamiltonians. Let $M$ be a differentiable $n$-dimensional Riemannian manifold and let $\alpha$ be a 1-form on $M$. We consider the quadratic form \begin{equation} E_0=\int_M a\big(\frac{1}{i}du-u\alpha,\frac{1}{i}du-u\alpha\big)d\mu \end{equation} where $\mu$ is the Riemannian measure of $M$ and $a\in TM^{2,0}$ (called the {\em symbol} of the quadratic form), is a smooth Hermitian bilinear form with real-valued coefficients defined on fibers $TM^*_x$. The form $E$ is understood in physics as a generalized Hamiltonian for a quantum particle on $M$ in presence of the external magnetic field $\beta=d\alpha$. In general, a magnetic field is a closed 2-form that does not have to be exact. Quantization of systems with a non-potential magnetic field is more complicated (see \cite{Gruber} and references therein) and is not considered here. The potential $\alpha$ is defined by $\beta$ up to an arbitrary closed form and the energy is invariant under the gauge transformation $(\alpha,u)\mapsto (\alpha+d\varphi,e^{i\varphi}u)$. The (stationary) nonlinear Schr\"odinger equation for complex-valued functions on $M$ in the weak form is: \begin{equation} \label{SchrEquation} \int_M\big( a(\frac{1}{i}du-u\alpha,\frac{1}{i}dv-v\alpha)+\lambda uv- |u|^{q-2}uv \big)d\mu=0, \end{equation} $v\in C_0^\infty(M)$. In what follows we will use the notation $a[\alpha]:=a(\alpha,\alpha)$, $E_0[u]:=E_0(u,u)$ etc. for quadratic forms. Let $H^1(M)$ be the Hilbert space defined as the closure of $C_0^\infty(M;\mathbb{C})$ with respect to the Hilbert norm $\left(\int_M (|du|^2+|u|^2)d\mu\right)^{1/2}$. For an open set $\Omega\subset M$ the subspace $H^1(\Omega)$ will be the closure of $C_0^\infty(\Omega)$ in $H^1(M)$. We assume that the symbol $a$ and the number $2^*$ are related via the Sobolev inequality for the real-valued functions $u\in H^1(M)$: \begin{equation}\label{ii} \int_M (a[du]+|u|^2)d\mu \ge c\|u\|^2_{L^q(M,d\mu)}, \quad q\in[2,2^*], \end{equation} and that, in restriction to $H^1(\Omega)$ with any bounded $\Omega\subset M$, and with $q\in(2,2^*)$, this imbedding is compact. This is true, for example, when $a[\xi]\ge c|\xi|^2$ with some $c>0$ (the uniformly elliptic case) and when $M$ satisfies the assumption (\ref{iii}) below. In this case $2^*=\frac{2n}{n-2}$ for $n>2$, and $2^*=\infty$ for $n=2$. The relation (\ref{ii}) holds as well when $M$ is a Lie group and the symbol of $E_0$ is $a=\sum_j X_j\otimes X_j$, where $X_j\in TM$, $j=1,\dots,m$, are left-invariant vector fields. If the subsequent commutators of $X_j$ span the whole tangent space of $M$ (H\"ormander condition), then there exists a $N\ge n$, called homogeneous dimension, such that (\ref{ii}) holds with $2^*=\frac{2N}{N-2}$ (\cite{FollandStein,Folland,Varopoulos} and references therein). Let now $H^1_\alpha(M)$ (resp. $H^1_\alpha(\Omega)$) be the closure of $C_0^\infty(M;\mathbb{C})$ (resp. $C_0^\infty(\Omega;\mathbb{C})$) in the metric of \begin{equation} E[u]:=E_0[u]+\|u\|^2_{L^2(M,d\mu)}.\end{equation} The following inequality is an elementary generalization of the diamagnetic inequality, well known for the Euclidean case (see e.g. \cite{LL}). \begin{lemma} Let $\alpha\in TM^*$ and let $a$ be as above. The following inequality is true for every $u\in C_0^\infty(M;\mathbb{C})$ at every point where $u\neq 0$: \begin{equation} \label{pol} a[du-iu\alpha]\ge a[d|u|]. \end{equation} \end{lemma} \begin{proof} Let $v$, $w$ be the real and the imaginary parts of $u$. The assertion follows from the following chain of identities that use the bilinearity of $a$ and the chain rule: \begin{align*} a[du-iu\alpha]-a[d|u|]&=a[du]+|u|^2a[\alpha]-2va(\alpha,d w)+2wa(\alpha,d v)\\ &\quad -|u|^{-2}\left\{v^2a[dv]+w^2a[dw]+2vwa(dv,dw)\right\}\\ &=|u|^{-2}\left\{ a[v dw - w dv]+2|u|^2a(\alpha,w dv-v dw)+|u|^4a[\alpha] \right\}\\ &=|u|^{-2}a[w dv-v dw+|u|^2\alpha] \ge 0. \end{align*} \end{proof} \begin{proposition} \label{demag} The following inequality holds: \begin{equation} \label{pol2} E(u)\ge \||u|\|^2_{H^1(M)}, \quad u\in H^1_\alpha(M). \end{equation} Moreover, the space $H^1_\alpha(M)$ is continuously imbedded into $L^q(M,\mu)$, $q\in(2,2^*$, and for any bounded open $\Omega\subset M$ the imbedding of $H^1_\alpha(\Omega)$ into $L^q(\Omega,\mu)$ is compact. \end{proposition} \begin{proof} Using approximation operators $T_\epsilon: C_0^1(M)\to C_0^1(M)$, $T_\epsilon u:=(u^2+\epsilon^2)^{1/2}-\epsilon$, one can immediately deduce from Lemma~\ref{pol} (see for details the proof of Lemma 7.6 in \cite{GilbTrud}) that $u\in H^1_\alpha(M)\Rightarrow |u|\in H^1(M)$ with $E_0(u)\ge \||u|\|^2_{H^1(M)}$. Thus, by (\ref{ii}) applied to $|u|$, the space $H^1_\alpha(M)$ is continuously embedded into $L^q(M,\mu)$ and for any open bounded $\Omega$, the subspace $H^1_\alpha(\Omega)$ is compactly embedded into $L^q(\Omega,\mu)$. \end{proof} Critical points of the map $u\mapsto (E(u),\int|u|^qd\mu)$, $H\to\mathbb{R}^2$ provide solutions of the equation (\ref{SchrEquation}) (up to a scalar multiple). We look here for solutions of the ground state type, that is, the minimizers in the problem \begin{equation} \label{theproblem} c_q:=\inf_{\int_M|u|^qd\mu=1}E[u], q\in(2,2^*). \end{equation} By analogy with the semilinear elliptic problem for the Laplacian on $\mathbb{R}^n$ without a magnetic field, the minimum in the problem (\ref{theproblem}) is not expected to exist without substantial additional assumption. Existence of a minimizer is known for (\ref{theproblem}) in the Euclidean case with a constant magnetic field (\cite{EstebanLions}). If the field is not constant, or a potential term is added to the equation, existence of minimum has been derived from various penalty conditions at infinity, typically involving a potential term $\int V(x)|u|^2$ in the energy (see \cite{Kurata}). One may also observe absence of minimizer if the penalty condition is appropriately reversed (\cite{EstebanLions}). In this paper we consider invariant (which, in case of a discrete group, means space-periodic) magnetic fields on manifolds that are co-compact with respect to their isometry groups, a class that includes homogeneous Riemannian spaces and in particular, Lie groups. Let $I$ be a subgroup of the isometry group of $M$, closed in the CO-topology. We assume that there is a compact set $K\subset M$ such that \begin{equation} \label{iii} \bigcup_{\eta\in I}\eta K=M. \end{equation} We assume that the symbol $a$ is invariant with respect to the transformations $\eta\in I$. This is true, in particular, if it is the symbol of the Laplace-Beltrami operator or of an invariant subelliptic operator as defined above. Consider now the condition of invariance of the magnetic field $\beta$. The invariance relation $\forall\eta,\;\eta\beta=\beta$, where $\eta:TM^{0,2}_{\eta \cdot}\to TM^{0,2}_\cdot$ is the natural action of the isometry $\eta\in I$ on 2-forms, written in terms of the magnetic potential $\alpha$ is equivalent to $d(\eta\alpha-\alpha)=0$ where $\eta:TM^*_{\eta x}\to TM^*_x$ is the natural action of $\eta\in I$ on the cotangent bundle of $M$. For a technical reason (existence of global magnetic shifts) we put a somewhat stronger condition on $\alpha$, namely that \begin{equation} \label{iv}\forall\eta\in I, \eta\alpha-\alpha \mbox{ is exact}. \end{equation} This will allow to construct global magnetic shifts relative to $\eta\in I$ in the next section. The main result of this paper is \begin{theorem} \label{main} Let $a$ and $\mu$ be invariant under the action of the group $I$. Assume \eqref{ii}, \eqref{iii}, \eqref{iv}. Then the problem \eqref{theproblem} has a point of minimum which, up to the constant multiple is a non-trivial solution of \eqref{SchrEquation}. \end{theorem} \begin{remark} \rm The statement of the theorem remains true if one replaces in the energy the term $\int |u|^2$ in $E[u]$ with $\int V(x)|u|^2d\mu$, $V\in L^1_{loc}(M,\mu)$, $\inf_MV>0$ provided that $V\circ\eta=V$, $\eta\in I$. This generalization does not require any essential changes in the proof. \end{remark} The proof of the existence of the minimum in (\ref{theproblem}) is based on the concentration compactness principle (see \cite{PLL1a,PLL1b} for a fundamental exposition for the subcritical case). One can use here the approach of \cite{BrezisLieb,Struwe}, and we give an essentially equivalent proof, using a general ``multi-bump" expansion for bounded sequences (in the spirit of \cite{Lions86}) from \cite{acc}. In what follows we assume conditions of Theorem~\ref{main}. \section{Concentration compactness with magnetic shifts} By (\ref{iv}), for every $\eta\in I$ there exists a $\psi_\eta\in C^\infty(M)$ such that \begin{equation} \label{psi} \eta\alpha-\alpha=d\psi_\eta. \end{equation} This implies that $d\psi_{\rm id}=0$, so that $\psi_{\rm id}$ is constant on connected components of $M$. Since the relation (\ref{psi}) is satisfied by $\psi_\eta-\psi_{\rm id}$, we normalize $\psi_\eta$ by setting \begin{equation} \label{zero} \psi_{\rm id}(x) = 0, \quad x\in M. \end{equation} Let \begin{equation} \label{shifts} g_\eta u=e^{i\psi_\eta}u\circ\eta, \quad u\in C_0^\infty(M). \end{equation} The action $g_\eta$ on $u\in C_0^\infty(M)$ (as well as its continuous extension below) is called a magnetic shift. We set \begin{equation} D:=\{g_\eta\}_{\eta\in I}.\end{equation} \begin{lemma} Every operator $g\in D$ extends by continuity to a unitary operator on $H^1_\alpha(M)$. The (renamed) set $D$ of extended operators is a multiplicative operator group on $H^1_\alpha(M)$. \end{lemma} \begin{proof} It suffices to prove that \begin{gather} \label{inv} g_{\eta^{-1}}=g_\eta^{-1},\\ \label{conj} g_{\eta^{-1}}=g_\eta^* \end{gather} for every $\eta\in I$. To prove (\ref{inv}), note that from (\ref{psi}) and (\ref{zero}) it follows immediately that \begin{equation} \label{inverse-eta}\psi_\eta=-(\psi_{\eta^{-1}}\circ\eta). \end{equation} Then solving the equation $g_\eta u=v$, one has $v=e^{-i\psi_\eta\circ\eta^{-1}}u\circ\eta^{-1} =e^{i\psi_{\eta^{-1}}}u\circ\eta^{-1}$. In order to prove (\ref{conj}), consider the following calculations, taking into account invariance properties of $a$ and $\mu$, (\ref{inverse-eta}) and (\ref{iv}): \begin{align*} E_0(u,g_\eta v) &=\int_M e^{-i\psi_\eta}a\left(du+iu\alpha, d(v\circ\eta)-id\psi_\eta v\circ\eta+i(v\circ\eta)\alpha\right)d\mu\\ &=\int_M e^{-i\psi_\eta\circ\eta^{-1}} a\left((du)\circ\eta^{-1}+i(u\circ\eta^{-1})\eta^{-1}\alpha,dv+iv\alpha\right)d\mu\\ &=\int_M e^{i\psi_{\eta^{-1}}} a\left(d(u\circ\eta^{-1})+i(u\circ\eta^{-1})(\alpha+d\psi_{\eta^{-1}}),dv +iv\alpha\right)d\mu\\ &= E_0(g_{\eta^{-1}}u,v), \; u,v\in C_0^\infty(M). \end{align*} \end{proof} \begin{lemma} The group $D$ on $H^1_\alpha(M)$ is a set of dislocations according to \cite{acc}, i.e. a set of unitary operators on a separable Hilbert space satisfying the condition: \begin{itemize} \item[(*)] Any sequence $g_k \in D$ that does not converge to zero weakly has a strongly convergent subsequence. \end{itemize} \end{lemma} We recall that a sequence of operators $g_k$ in a Banach space $E$ is called strongly convergent if for every $x\in E$, $g_kx$ converges. \begin{proof} Assume that $g_{\eta_k}\not\rightharpoonup 0$. Then there exist $u,v\in C_0^\infty(M)$ and a renamed subsequence of $\eta_k$, such that $(g_{\eta_k}u,v)\not\to 0$, so that $\eta_k^{-1}(\mathop{\rm supp} u)\cap\mathop{\rm supp} v\neq\emptyset$. Let $x_k\in\mathop{\rm supp} u$ be such that $\eta_kx_k\in\mathop{\rm supp} v$. Since $\mathop{\rm supp} u$ is compact, a renamed subsequence of $x_k$ converges to some $x\in\mathop{\rm supp} u$. Since $\mathop{\rm supp} v$ is compact and $\eta_k$ are isometries, a renamed subsequence of $\eta_kx$ converges, and therefore $\eta_k$ converges to some $\eta\in I$ in the compact-open topology (cf. \cite{Kob}) and therefore uniformly on compact sets. Then $g_{\eta_k} v$ converges for any $v\in C_0^\infty(M)$ by convergence of integrals under uniform convergence. Since operators in $D$ are unitary, it suffices to verify the strong operator convergence on $C_0^\infty(M)$, which in turn follows from convergence of integrals under uniform convergence. \end{proof} \begin{definition} \label{def:cwconvergence} \rm Let $u, u_{k} \in H^1_\alpha(M)$. We will say that $u_{k} $ converges to $u$ {\em $D$-weakly}, which we will denote as $u_{k} \stackrel{D}{\rightharpoonup} u$, if for all $\varphi \in H^1_\alpha(M)$, \begin{equation} \label{eq:cwconvergence} \lim_{k \to \infty} \sup_{g \in D} (g(u_{k} -u), \varphi) = 0. \end{equation} \end{definition} \begin{lemma}\label{BrezisLieb} Let $u_k\in H_\alpha^{1}(M)$ be a bounded sequence. Then \begin{equation} u_k\stackrel{D}{\rightharpoonup} 0 \Rightarrow u_k\to 0 \quad \mbox{in } L^q(M,\mu), q\in(2,2^*). \end{equation} \end{lemma} \begin{proof} If $g_{\eta_k}u_k\rightharpoonup 0$, then due to the inequality (\ref{pol2}), $|u_k|\circ\eta_k\rightharpoonup 0$ in $H^1(M)$. Then $|u_k|\to 0$ in $L^q(M,\mu)$ by \cite[Lemma 3.7]{BiSchiTi} (when $a$ is uniformly elliptic, one can also refer to \cite[Lemma 2.6]{FiesTin}). \end{proof} \begin{theorem}[\cite{acc}] \label{abstractcc} Let $u_{k} \in H$ be a bounded sequence. Then there exist $w^{(n)} \in H$, $g_{k} ^{(n)} \in D$, $k,n \in \mathbb{N}$, such that for a renumbered subsequence \begin{gather} \label{separates} g_k^{(1)}=id,\; {g_{k} ^{(n)}} ^{-1} g_{k} ^{(m)}\rightharpoonup 0 \quad \mbox{ for } n \neq m, \\ \label{dwl} w^{(n)}=\mathop{\rm w-lim} {g_{k} ^{(n)}}^{-1}u_k \\ \label{norms} \sum_{n \in {\mathbb{N}}} \|w ^{(n)}\|^2 \le \limsup \|u_k\|^2 \\ \label{BBasymptotics} u_{k} - \sum_{n\in{\mathbb{N}}} g_{k} ^{(n)} w^{(n)} \stackrel{D}{\rightharpoonup} 0. \end{gather} \end{theorem} \begin{lemma} Let $D$ be the group of magnetic shifts in $H^1_\alpha(M)$, let $u_k$ be a bounded sequence in $H^1(M)$ and let $w^{(n)}$ be as in Theorem~\ref{abstractcc}. Then the corresponded renamed subsequence $u_k$ satisfies \begin{equation} \label{q} \int_M|u_k|^qd\mu =\sum_{n\in{\mathbb{N}}} \int_M |w^{(n)}|^q d\mu, \quad q\in(2,2^*).\end{equation} \end{lemma} \begin{proof} Apply Theorem~\ref{abstractcc} for the bounded (by (\ref{pol2}) sequence $|u_k|$ in $H^1(M)$ equipped with the dislocation group $D_0:=\{v\to v\circ\eta, \eta\in I$. Since the weak convergence in both spaces $H^1$ and $H^1_\alpha$ implies convergence in measure, the weak limits (\ref{dwl}) in the $(H^1,D_0)$-case, written in terms of those in the $(H^1_\alpha,D)$-case, are $|w^{(n)}|$. Note now that $g_{\eta_k}\rightharpoonup 0$ (in $(H^1,D_0)$) implies that for any compact set $K\subset M$, $d(\eta_kK,0)\to\infty$. Indeed, if $\eta_kx_k$ were bounded for some $x_k\in K$, then, since $\eta_k$ are isometries, $\eta_k$ converges in the CO topology (cf. \cite{Kob}). Then the assertion of the lemma follows elementarily from restriction of $|w^{(n)}|$ to disjoint balls of arbitrarily large radius. \end{proof} \section{Magnetic Schr\"odinger equation on the Heisenberg group} In this section we give an example of a manifold with a subelliptic energy form and a potential magnetic field to which Theorem~\ref{main} applies. Let $\mathbb{H}^3$ be the space $\mathbb{R}^3$, whose elements we denote as $\eta=(x,y,t)$, equipped with the group operation \begin{equation} \eta\circ\eta'=(x+x',y+y',t+t' +2(xy'-yx')). \end{equation} This group multiplication endows $\mathbb{H}^3$ with the structure of a Lie group with $e=0$. Two invariant vector fields $X=\partial_{x}+2y\partial_t$ and $Y=\partial_{y}-2x\partial_t$ satisfy the bracket condition, namely, together with $T=[X,Y]$ they form the basis in the tangent space, which yields the homogeneous dimension $N=4$ and $2^*=4$. The Riemannian structure is fixed by setting the scalar product at $T\mathbb{H}^3$ so that the given basis $X,Y,T$ is orthonormal. The Riemannian measure and the left and the right Haar measure on $\mathbb{H}^3$ coincide with the Lebesgue measure. The Sobolev inequality (\ref{ii}) holds with the subelliptic symbol $a=X\otimes X+Y\otimes Y$ for $22$, $\sum t_n^{2/q}=1$ only if all but one of $t_n$, say for $n=n_0$, equals zero. We conclude that $w^{(n_0)}$ is the minimizer for (\ref{theproblem}). \end{proof} \begin{remark} \rm We note that from the proof of Theorem~\ref{main} follows that that for any minimizing sequence $u_k$ for (\ref{theproblem}) there is a sequence $\eta_k$, such that $g_{\eta_k}u_k$ converges to the minimizer in $H^1_\alpha(M)$. 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