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Semiclassical analysis for highly degenerate potentials

Authors:	P Á lvarez-Caudevilla J Ló pez-Gó mez

Institution:	Departamento de Matemáticas, Universidad Católica de Ávila, Ávila, Spain ; Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain

Abstract:	This paper characterizes the semi-classical limit of the fundamental energy, $\displaystyle E(h):= \sigma_1-h^2\Delta+a(x);\Omega],$ and ground state $\psi_h$ of the Schrödinger operator $-h^2\Delta+a$ in a bounded domain $\Omega$ , in the highly degenerate case when $a\geq 0$ and $a^{-1}(0)$ consists of two components, say $\Omega_{0,1}$ and $\Omega_{0,2}$ . The main result establishes that $\displaystyle \lim_{h\downarrow 0} \frac{E(h)}{h^2}= \min\left\{\sigma_1-\Delta;\Omega_{0,i}], i=1,2\,\right\}$ and that $\psi_h$ approximates in $H_0^1(\Omega)$ the ground state of $-\Delta$ in $\Omega_{0,i}$ if $\displaystyle \sigma_1-\Delta;\Omega_{0,i}]< \sigma_1-\Delta;\Omega_{0,j}],\qquad j \in \{1,2\}\setminus\{i\}.$

Keywords:	Fundamental energy ground state highly degenerate potentials classical conjecture of B Simon compact Riemann manifolds

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