Semiclassical analysis for highly degenerate potentials

Authors:
P. Álvarez-Caudevilla and J. López-Gómez

Journal:
Proc. Amer. Math. Soc. **136** (2008), 665-675

MSC (2000):
Primary 35B25, 35P15, 35J10, 31C12

DOI:
https://doi.org/10.1090/S0002-9939-07-09076-4

Published electronically:
November 2, 2007

MathSciNet review:
2358508

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper characterizes the semi-classical limit of the fundamental energy, \begin{equation*} E(h):= \sigma _1[-h^2\Delta +a(x);\Omega ], \end{equation*} 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 \begin{equation*} \lim _{h\downarrow 0} \frac {E(h)}{h^2}= \min \left \{\sigma _1[-\Delta ;\Omega _{0,i}], \; i=1,2 \right \} \end{equation*} and that $\psi _h$ approximates in $H_0^1(\Omega )$ the ground state of $-\Delta$ in $\Omega _{0,i}$ if \begin{equation*} \sigma _1[-\Delta ;\Omega _{0,i}]< \sigma _1[-\Delta ;\Omega _{0,j}],\qquad j \in \{1,2\}\setminus \{i\}. \end{equation*}

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Additional Information

**P. Álvarez-Caudevilla**

Affiliation:
Departamento de Matemáticas, Universidad Católica de Ávila, Ávila, Spain

Email:
pablocaude@eresmas.com

**J. López-Gómez**

Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain

Email:
Lopez_Gomez@mat.ucm.es

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

Received by editor(s):
January 19, 2007

Published electronically:
November 2, 2007

Additional Notes:
This work was partially supported by the Ministry of Education and Science of Spain under research grants REN2003–00707 and CGL2006-00524/BOS

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.