Bounded solutions of the stationary Schr\"{o}dinger equation with finite energy integral on model manifolds

A. G. Losev ; V. V. Filatov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 24 (2021) no. 3, pp. 5-17.

Résumé

Conditions for the existence of nontrivial bounded solutions of the stationary Schrodinger equation with a finite energy integral on model varieties are obtained. A condition for the existence of nontrivial bounded solutions with a finite integral of energy in the exterior of a compactum on arbitrary Riemannian manifolds is also obtained. Let $D=(0;+\infty)\times S,$ where $S$ is compact Riemannian manifold. Metrics on $D$ is following $$ds^2=dr^2+g^2(r)d\theta^2.$$ Where $g(r)$ is positive, smooth on $(0,+\infty)$ function, $d\theta^2$ is metrics on $S.$ We will study solutions of the stationary Schrodinger equation $$\Delta u-c(r)u=0$$ on $D$. Let $r_0=\mathrm{const} >0, n=\dim D$. Theorem 1. If one of the following conditions is fulfilled on $ D $: $ \mu) $ $ R \infty $; $ \eta) $ $ R = \infty, $ $ K = \infty $; $ \xi) $ $ J = \infty, $ $ K \infty $ then the Liouville function of the end $ D $ has a finite energy integral. If one of the conditions $ \omega) $ $ R = \infty $, $ I \infty; $ $ \rho) $ $ R = \infty, $ $ I = \infty, $ $ J \infty $ then The Liouville Function of end $ D $ has a divergent energy integral. Theorem 2. On an arbitrary Riemannian manifold $ M $, the convergence of the energy integral of the Liouville function of the exterior of the compact (Liouville function of the end) implies the convergence of the energy integral of the Liouville function.

Détail

Keywords: energy integral, stationary Schrödinger equation, massive sets, Riemannian manifolds.
Mots-clés : Liouville function