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.
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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.
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References:
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[4]  - Dirichlet spectrum of the Fichera layer, with Monique Dauge and Yvon
Lafranche, 33p., submitted, arXiv:1711.08439, 2017.
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[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.
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[2] - Discrete spectrum of interactions concentrated near conical surfaces,
with Konstantin Pankrashkin, to appear in Applicable Analysis, 22 p.,
arXiv:1612.01798, 2016.
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[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.