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Eigenvalues for the Robin Laplacian in domains with variable curvature


Authors: Bernard Helffer and Ayman Kachmar
Journal: Trans. Amer. Math. Soc. 369 (2017), 3253-3287
MSC (2010): Primary 35P15, 47F05
DOI: https://doi.org/10.1090/tran/6743
Published electronically: September 13, 2016
MathSciNet review: 3605971
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Abstract: We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $ -\infty $. The two first terms in the expansion have been obtained by K. Pankrashkin in the 2D-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as candidate for the ground state energy.


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Bernard Helffer
Affiliation: Université de Paris-Sud, Bât 425, 91405 Orsay, France – and – Laboratoire Jean Leray, Université de Nantes, 44300 Nantes, France
Email: bernard.helffer@math.u-psud.fr

Ayman Kachmar
Affiliation: Department of Mathematics, Lebanese University, Hadath, Lebanon
Email: ayman.kashmar@gmail.com

DOI: https://doi.org/10.1090/tran/6743
Received by editor(s): January 13, 2015
Received by editor(s) in revised form: April 29, 2015
Published electronically: September 13, 2016
Article copyright: © Copyright 2016 American Mathematical Society