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Spectral transitions for Aharonov-Bohm Laplacians on conical layers

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

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 (david.krejcirik@fjfi.cvut.cz)
V. Lotoreichik
Affiliation:
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 250 68 Řež near Prague, Czech Republic (lotoreichik@ujf.cas.cz)
T. Ourmières-Bonafos
Affiliation:
CNRS & Université Paris-Dauphine, PSL Research University, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France (ourmieres-bonafos@ceremade.dauphine.fr)
*
*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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References

1Behrndt, J., Exner, P. and Lotoreichik, V.. Schrödinger operators with δ-interactions supported on conical surfaces. J. Phys. A: Math. Theor. 47 (2014), 355202, (16pp).Google Scholar
2Bernardi, C., Dauge, M. and Maday, Y.. Spectral Methods for Axisymmetric Domains, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaïez.Google Scholar
3Bonnaillie-Noël, V. and Raymond, N.. Magnetic Neumann Laplacian on a sharp cone. Calc. Var. Partial Differ. Equ. 53 (2015), 125–147.Google Scholar
4Bonnaillie-Noël, V., Dauge, M., Popoff, N. and Raymond, N.. Magnetic Laplacian in sharp three dimensional cones. Oper. Theory: Adv. Appl. 254 (2016), 37–56.Google Scholar
5Bruneau, V., Pankrashkin, K. and Popoff, N.. Eigenvalue counting function for Robin Laplacians on conical domains. J. Geom. Anal. 28 (2018), 123–151.Google Scholar
6Carron, G., Exner, P. and Krejčiřík, D.. Topologically nontrivial quantum layers. J. Math. Phys. 45 (2004), 774–784.Google Scholar
7Cazacu, C. and Krejčiřík, D.. The Hardy inequality and the heat equation with magnetic field in any dimension. Comm. Partial Differ. Equ. 41 (2016), 1056–1088.Google Scholar
8Chenaud, B., Duclos, P., Freitas, P. and Krejčiřík, D.. Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23 (2005), 95–105.Google Scholar
9Dauge, M., Lafranche, Y. and Raymond, N.. Quantum waveguides with corners. ESAIM: Proceedings 35 (2012), 14–45.Google Scholar
10Dauge, M., Ourmières-Bonafos, T. and Raymond, N.. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Commun. Pure Appl. Anal. 14 (2015), 1239–1258.Google Scholar
11Devyver, B., Pinchover, Y. and Psaradakis, G.. Optimal Hardy inequalities in cones. Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), 89–124.Google Scholar
12Duclos, P. and Exner, P.. Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995), 73–102.Google Scholar
13Duclos, P., Exner, P. and Krejčiřík, D.. Bound states in curved quantum layers. Comm. Math. Phys. 223 (2001), 13–28.Google Scholar
14Ekholm, T., Kovařík, H. and Krejčiřík, D.. A Hardy inequality in twisted waveguides. Arch. Ration. Mech. Anal. 188 (2008), 245–264.Google Scholar
15Exner, P. and Němcová, K.. Leaky quantum graphs: approximations by point-interaction Hamiltonians. J. Phys. A, Math. Gen. 36 (2003), 10173–10193.Google Scholar
16Exner, P. and Tater, M.. Spectrum of Dirichlet Laplacian in a conical layer. J. Phys. A 43 (2010), 474023.Google Scholar
17Exner, P., Šeba, P., Tater, M. and Vaněk, D.. Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys. 37 (1996), 4867–4887.Google Scholar
18Hassell, A. and Marshall, S.. Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree −2. Trans. Amer. Math. Soc. 360 (2008), 4145–4167.Google Scholar
19Kato, T.. Perturbation theory for linear operators (Berlin: Springer-Verlag, 1995).Google Scholar
20Kirsch, W. and Simon, B.. Corrections to the classical behavior of the number of bound states of Schrödinger operators. Ann. Physics 183 (1988), 122–130.Google Scholar
21Krejčiřík, D.. Twisting versus bending in quantum waveguides. in: Analysis on graphs and its applications. Selected papers based on the Isaac Newton Institute for Mathematical Sciences programme, Cambridge, UK, 2007. Proc. Symp. Pure Math. 77 (2008), 523–564. See arXiv:0712.3371v2 [math-ph] for a corrected version.Google Scholar
22Krejčiřík, D.. The improved decay rate for the heat semigroup with local magnetic field in the plane. Calc. Var. Partial Differ. Equ. 47 (2013), 207–226.Google Scholar
23Krejčiřík, D. and Lu, Z.. Location of the essential spectrum in curved quantum layers. J. Math. Phys. 55 (2014), 083520.Google Scholar
24Lotoreichik, V. and Ourmières-Bonafos, T.. On the bound states of Schrödinger operators with δ-interactions on conical surfaces. Comm. Partial Differ. Equ. 41 (2016), 999–1028.Google Scholar
25McLean, W.. Strongly elliptic systems and boundary integral equations (Cambridge: Cambridge University Press, 2000).Google Scholar
26Najar, H. and Raissi, M.. A quantum waveguide with Aharonov-Bohm magnetic field. Math. Methods. Appl. Sci. 39 (2016), 92–103.Google Scholar
27Pankrashkin, K.. On the discrete spectrum of Robin Laplacians in conical domains. Math. Model. Nat. Phenom. 11 (2016), 100–110.Google Scholar
28Reed, M. and Simon, B.. Methods of modern mathematical physics. IV. Analysis of operators (New York: Academic Press, 1978).Google Scholar
29Reed, M. and Simon, B.. Methods of modern mathematical physics. I: Functional analysis (New York: Academic Press, 1980).Google Scholar
30Saint-James, D., Sarma, G. and Thomas, E. J.. Type-II superconductivity (Oxford, New York: Pergamon Press, 1969).Google Scholar
31Simon, B.. Schrödinger operators in the twentieth century. J. Math. Phys. 41 (2000), 3523–3555.Google Scholar
32Yu, I.. Popov, Asymptotics of bound state for laterally coupled waveguides. Rep. Math. Phys. 43 (1999), 427–437.Google Scholar