The Liouville type theorems for solution of stationary Schr\"odinger equation with finite Dirichlet integral

A. G. Losev ; V. V. Filatov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2016), pp. 13-23

Résumé

In this article we learn some property of solutions of stationary Shrödinger equation \begin{equation} Lu=\Delta u-c(x)u=0, \tag{1} \end{equation} where $c(x)\geq0$ smooth function, with finite Dirichlet integral \begin{equation} \int\limits_{M}|u|^2+c(x)u^2dx \tag{2} \end{equation} on non compact Riemannian manifolds. We prove an analog of Ahlfors's theorem on existing of non-trivial boundary harmonic function with finite energy integral. Main result of this article is the next theorem. Let $M$ be non-compact Riemannian manifold. Theorem 1. If non-trivial solution of equation (1) with finite integral (2) exists on $M$ (this solution may be not bounded), then there exists bounded solution of equation (1) with finite energy integral (2). To proof this theorem we use the following lemmas. Lemma 1. (Maximum principle) Let $B$ be precompact open set in $M$ with smooth boundary. If $$Lu=0,\ x\in B,$$ then $$\sup\limits_{B}|u|=\sup\limits_{\partial B}|u|.$$ Lemma 2. Let $B\subset M$ precompact open subset on $M$, $\{\phi_i\}_{i=1}^\infty$ is uniformly bounded on $B$ family of solutions (1), $\phi_i\in C^{2,\alpha}(B).$ Then the family $\{\phi_i\}_{i=1}^\infty$ is compact in class $C^2(B')$, where $B'\subset B$. Let $F$ be set of functions from class $C^2(B)$ with finite Dirichlet integral $$\int\limits_{B} |\nabla y|^2+c(x)y^2dx.$$ Lemma 3. $F$ is linear space, also on $F$ can be defined dot product as $$\langle a,b\rangle=\int\limits_{B} \left(\langle\nabla a, \nabla b\rangle +c(x)ab\right)dx,\quad \forall a,b\in F.$$ and norm for this dot product as $$\|a\|=\langle a,a\rangle^{\frac{1}{2}}=\left(\int\limits_{B}|\nabla a|^2+c(x)a^2dx\right)^{\frac{1}{2}}.$$ Lemma 4. (Dirichlet principle). Let $B\subset M$—precompact open subset on $M$ with smooth boundary. If for functions $u,v\in C^2(B)$ $$ \left \{ \begin{array}{c} \Delta u-c(x)u=0, x\in B, \\ u|_{\partial B}=v|_{\partial B}, \end{array} \right. $$ then $$\int\limits_{B}|\nabla u|^2+c(x)u^2dx\leq\int\limits_{B}|\nabla v|^2+c(x)v^2dx.$$

Détail

Keywords: Dirichlet integral, stationary Schrödinger equation, Liouville type theorems, Ahlfors's theorem, riemannian manifolds.