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Semiclassical analysis for highly degenerate potentials
Authors:P. Á  lvarez-Caudevilla   J. Ló  pez-Gó  mez
Affiliation: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^2Delta+a(x);Omega], $

and ground state $ psi_h$ of the Schrödinger operator $ -h^2Delta+a$ in a bounded domain $ Omega$, in the highly degenerate case when $ ageq 0$ and $ a^{-1}(0)$ consists of two components, say $ Omega_{0,1}$ and $ Omega_{0,2}$. The main result establishes that

$displaystyle lim_{hdownarrow 0} frac{E(h)}{h^2}= minleft{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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