In this talk I will discuss some spectral properties of Hamiltonians related to a quantum particle constrained to live in a neighborhood of a surface with hard-wall boundary conditions. Such systems are well studied and I will focus on the specific case of conical surfaces. These geometries proved to have an interesting behavior: when the cone is smooth (except in its vertex), the essential spectrum is a half-line and there is an infinite number of bound states accumulating to the threshold of the essential spectrum.

In a first time, I will review recent results quantifying this accumulation then, in a second time, I will detail how the smoothness of the cone impacts the number of eigenvalues.


[4]  - Dirichlet spectrum of the Fichera layer, with Monique Dauge and Yvon Lafranche, 33p., submitted, arXiv:1711.08439, 2017.

[3] - Spectral transitions for Aharonov-Bohm Laplacians on conical layers, with David Krejcirík and Vladimir Lotoreichik, to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 27 p., arXiv:1607.02454, 2016.

[2] - Discrete spectrum of interactions concentrated near conical surfaces, with Konstantin Pankrashkin, to appear in Applicable Analysis, 22 p., arXiv:1612.01798, 2016.

[1] - Spectral asymptotics of the Dirichlet Laplacian in a conical layer, with Monique Dauge and Nicolas Raymond, Communications on Pure and Applied Analysis, vol. 14, Issue 3, pp. 1239-1258, 2015.