Tunneling for the Robin Laplacian in smooth planar domains

TL;DR

This paper derives an explicit asymptotic formula for the splitting of the first two eigenvalues of the semiclassical Robin Laplacian in symmetric planar domains, revealing a tunneling effect caused by boundary curvature.

Contribution

It provides a rigorous derivation of the tunneling effect for the Robin Laplacian in smooth domains with specific boundary curvature properties, using a Born-Oppenheimer-like approach.

Findings

Explicit asymptotic formula for eigenvalue splitting

Identification of tunneling effect induced by boundary curvature

Weyl formula derived as a byproduct

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.