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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
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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.