Tunneling for the Robin Laplacian in smooth planar domains

Bernard Helffer (1, 2) , Ayman Kachmar (3) , Nicolas Raymond (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]
4 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Laplacian; Robin boundary condition; estimates of eigenvalues; Tunneling; Semiclassical asymptotics

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Bernard Helffer, Ayman Kachmar, Nicolas Raymond. Tunneling for the Robin Laplacian in smooth planar domains. Communications in Contemporary Mathematics, 2017, 19 (1), ⟨10.1142/S0219199716500309⟩. ⟨hal-01198585v3⟩