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Spectral transitions for Aharonov-Bohm Laplacians on conical layers

Part of: Equations of mathematical physics and other areas of application Spectral theory and eigenvalue problems Elliptic equations and systems

Published online by Cambridge University Press:  26 January 2019

D. Krejčiřík ,
V. Lotoreichik  and
T. Ourmières-Bonafos
D. Krejčiřík*
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic ([email protected])
V. Lotoreichik
Affiliation:
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 250 68 Řež near Prague, Czech Republic ([email protected])
T. Ourmières-Bonafos
Affiliation:
CNRS & Université Paris-Dauphine, PSL Research University, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France ([email protected])
*
*Corresponding author.
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Abstract

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux, we establish a Hardy-type inequality. In the regime with an infinite discrete spectrum, we obtain sharp spectral asymptotics with a refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

Keywords

Schrödinger operatorquantum layersexistence of bound statesspectral asymptoticsconical geometries

MSC classification

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 35Q40: PDEs in connection with quantum mechanics 35Q60: PDEs in connection with optics and electromagnetic theory 35J10: Schrödinger operator
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1663 - 1687
Copyright
Copyright © Royal Society of Edinburgh 2019 

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