Tunneling for the Robin Laplacian in smooth planar domains
Analysis of PDEs
2016-02-12 v3
Abstract
We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.
Keywords
Cite
@article{arxiv.1509.03986,
title = {Tunneling for the Robin Laplacian in smooth planar domains},
author = {Bernard Helffer and Ayman Kachmar and Nicolas Raymond},
journal= {arXiv preprint arXiv:1509.03986},
year = {2016}
}