Dimension of spaces of solutions of the Schrodinger equation on noncompact riemannian manifolds

A. A. Grigor'yan ; A. G. Losev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 34-42.

Résumé

Let $M$ be a smooth connected noncompact Riemannian manifold, $\Delta$ the Laplace — Beltrami operator on $M,$ $q(x)$ a smooth nonnegative function on $M,$ not identically zero. Considering the Schrodinger equation on $M,$ $$ \Delta u - q(x)u=0, $$ we ask the following question: For what manifold $M$ and potentials $q(x)$ does this equation have a unique solutions $u=0?$ If this is the case we say that Liouville's theorem is true for Schrodinger equation. We say that $u$ is $q$-harmonic function if it satisfies the Schrodinger equation. The classical Liouville theorem says that any bounded harmonic function in $R^n$ is identically constant. We note that recently there has been a trend towards a more general approach to theorems of Liouville type, namely, they are estimated dimensionality of various solution spaces of linear equations elliptic type. In particular, Grigor'yan (1990) proved an exact estimate dimensionality of spaces of bounded harmonic functions on non-compact Riemannian manifolds in terms of massive sets. The aim of this paper is to prove a similar result for bounded solutions of the stationary equation Schrodinger. Dimensions of spaces of harmonic functions and $q$-harmonic functions has been studied in numerous articles for various classes of manifolds. Among them are the works of M.T. Anderson, T.H. Colding, A. Grigor'yan, P. Li, A.G. Losev, L.-F. Tam, D. Sullivan, S.-T. Yau and many other authors. In contrast to the mentioned articles (and many others), we do not restrict the manifold $M$ a priori in any way. Let $LB(M)$ be the space of bounded $q$-harmonic functions on $M.$ We define $q$-massive subsets of $M$ and prove that $LB(M)$ is equal to the maximal number of pairwise non-intersecting $q$-massive subsets of $M.$ We now state the exact formulations. A continuous function $u$ defined on some open set $\Omega \subset M$ is called $q$-subharmonic if for every domain $G\Subset \Omega$ and a $q$-harmonic function $$v\in C(\overline{G}), \quad u|_{\partial G}=v|_{\partial G} $$ implies $u\leq v$ in $G.$ An open proper subset $\Omega \subset M$ is called $q$-massive if there is a non-trival $q$-subharmonic function $u\in C(\overline{\Omega})$ such that $u|_{\partial \Omega}=0, \quad 0\leq u \leq 1. $ The main result of the paper is the following statement. Theorem. Let $m\geq 1$ be a natural number. The following statements are equivalent: 1) $\dim LB(M)\geq m;$ 2) there exist $m$ pairwise non-intersection $q$-massive subsets of $M.$ Note that in the case $ q (x) \equiv 0, $ this assertion is true only for $ m \geq 2. $

Détail

Keywords: stationary Schrodinger equation, Liouville-type theorems, non- compact Riemannian manifolds, massive sets
Mots-clés : dimension of spaces of solutions.

Bibliographie

[1] A.\;A. Grigoryan, “On Liouville theorems for harmonic functions with finite Dirichlet integral”, Sbornik: Mathematics, 132:4 (1987), 496–516 | Zbl

[2] A.\;A. Grigoryan, “Dimension of spaces of harmonic functions”, Mathematical Notes, 48:5 (1990), 55–60 | MR

[3] A.\;A. Grigoryan, “On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds”, Sbornik: Mathematics, 128:3 (1985), 354–363 | MR | Zbl

[4] S.\;A. Korolkov, A.\;G. Losev, “Solutions of elliptic partial differential equations on Riemannian manifolds with ends”, Science Journal of Volgograd State University. Mathematics. Physics, 2011, no. 1 (14), 23–40

[5] A.\;G. Losev, E.\;A. Mazepa, “Bounded solutions of the Schrodinger equation on Riemannian products”, St. Petersburg Mathematical Journal, 13:1 (2001), 84–110 | MR

[6] S.\;Y. Cheng, S.\;T. Yau, “Differential equationson Riemannian manifolds and their geometric applications”, Comm. Pure and Appl. Math., 28:3 (1975), 333–354 | DOI | MR | Zbl

[7] A. Grigor'yan, “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”, Bulletin of Amer. Math. Soc., 1999, no. 36, 135–249 | DOI | MR | Zbl

[8] S.\;A. Korolkov, A.\;G. Losev, “Generalized Harmonic Functions of Riemannian Manifolds with Ends”, Mathematische Zeitschrift, 272:1–2 (2012), 459–472 | DOI | MR | Zbl

[9] P. Li, L.\;-F. Tam, “Harmonic functions and the structure of complete manifolds”, J. Diff. Geom., 35:2 (1992), 359–383 | DOI | MR | Zbl

[10] C.\;-J. Sung, L.\;-F. Tam, J. Wang, “Spaces of harmonic functions”, J. London Math. Soc. (2), 2000, no. 3, 789–806 | DOI | MR | Zbl