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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Semiclassical analysis for highly degenerate potentials
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by P. Álvarez-Caudevilla and J. López-Gómez PDF
Proc. Amer. Math. Soc. 136 (2008), 665-675 Request permission

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
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2358508