×

The Landau Hamiltonian with \(\delta \)-potentials supported on curves. (English) Zbl 1465.35132

Summary: The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian \(\mathsf{A}_\alpha = (\mathsf{i} \nabla + \mathbf{A})^2+\alpha \delta_{\Sigma}\) in \(L^2(\mathbb{R}^2)\) with a \(\delta \)-potential supported on a finite \(C^{1, 1} \)-smooth curve \(\Sigma\) are studied. Here \(\mathbf{A}=\frac{ 1}{ 2} B (-x_2, x_1)^{\text{T}}\) is the vector potential, \(B>0\) is the strength of the homogeneous magnetic field, and \(\alpha\in L^\infty(\Sigma)\) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve \(\Sigma \). After a general discussion of the qualitative spectral properties of \(\mathsf{A}_\alpha\) and its resolvent, one of the main objectives in the present paper is a local spectral analysis of \(\mathsf{A}_\alpha\) near the Landau levels \(B(2q+1), q\in \mathbb{N}_0\). Under various conditions on \(\alpha \), it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of \(\alpha \). Furthermore, the use of Landau Hamiltonians with \(\delta \)-perturbations as model operators for more realistic quantum systems is justified by showing that \(\mathsf{A}_\alpha\) can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, M. and Stegun, I. A. (eds.), Handbook of Mathematical Functions (Dover, New York, 1964). · Zbl 0171.38503
[2] Agranovich, M. S. and Amosov, B. A., Estimates of s-numbers and spectral asymptotics for integral operators of potential type on nonsmooth surfaces, Funct. Anal. Appl.30 (1996) 75-89. · Zbl 0895.47014
[3] Agranovich, M. S. and Selitskii, A. M., Fractional powers of operators corresponding to coercive problems in Lipschitz domains, Funct. Anal. Appl.47(2) (2013) 83-95. · Zbl 1287.47030
[4] Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H., Solvable Models in Quantum Mechanics, , 2nd edn. (American Mathematical Society, Chelsea Publishing, Providence, RI, 2005). · Zbl 1078.81003
[5] Albeverio, S. and Koshmanenko, V., On form-sum approximations of singularly perturbed positive self-adjoint operators, J. Funct. Anal.169 (1999) 32-51. · Zbl 0944.47008
[6] Avron, J., Herbst, I. and Simon, B., Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J.45 (1978) 847-883. · Zbl 0399.35029
[7] Behrndt, J., Exner, P., Holzmann, M. and Lotoreichik, V., Approximation of Schrödinger operators with \(\delta \)-interactions supported on hypersurfaces, Math. Nachr.290(8-9) (2017) 1215-1248. · Zbl 1376.35010
[8] Behrndt, J., Grubb, G., Langer, M. and Lotoreichik, V., Spectral asymptotics for resolvent differences of elliptic operators with \(\delta\) and \(\delta^\prime \)-interactions on hypersurfaces, J. Spectr. Theory5 (2015) 697-729. · Zbl 1353.47090
[9] Behrndt, J. and Langer, M., Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal.243 (2007) 536-565. · Zbl 1132.47038
[10] Behrndt, J. and Langer, M., Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, in Operator Methods for Boundary Value Problems, , Vol. 404 (Cambridge University Press, Cambridge, 2012), pp. 121-160. · Zbl 1331.47067
[11] Behrndt, J., Langer, M. and Lotoreichik, V., Schrödinger operators with \(\delta\) and \(\delta^\prime \)-potentials supported on hypersurfaces, Ann. Henri Poincaré14 (2013) 385-423. · Zbl 1275.81027
[12] Behrndt, J., Langer, M. and Lotoreichik, V., Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators, J. Lond. Math. Soc. (2)88(2) (2013) 319-337. · Zbl 1296.35097
[13] Behrndt, J., Langer, M., Lotoreichik, V. and Rohleder, J., Spectral enclosures for non-self-adjoint extensions of symmetric operators, J. Funct. Anal.275(7) (2018) 1808-1888. · Zbl 1472.47018
[14] Behrndt, J. and Rohleder, J., An inverse problem of Calderón type with partial data, Comm. Partial Differential Equations37 (2012) 1141-1159. · Zbl 1244.35164
[15] Berkolaiko, G. and Kuchment, P., Introduction to Quantum Graphs (American Mathematical Society, Providence, RI, 2013). · Zbl 1318.81005
[16] Birman, M. Sh. and Solomjak, M. Z., Spectral Theory of Self-Adjoint Operators in Hilbert Spaces (D. Reidel Publishing Co., Dordrecht, 1987).
[17] Bony, J.-F., Bruneau, V. and Raikov, G., Resonances and spectral shift function for magnetic quantum Hamiltonians, RIMS Hokyuroku Bessatsu B45 (2014) 77-100. · Zbl 1301.35074
[18] Brasche, J. F., Exner, P., Kuperin, Yu. A. and Šeba, P., Schrödinger operators with singular interactions, J. Math. Anal. Appl.184 (1994) 112-139. · Zbl 0820.47005
[19] Brasche, J. and Ožanová, K., Convergence of Schrödinger operators, SIAM J. Math. Anal.39 (2007) 281-297. · Zbl 1131.81004
[20] Bruneau, V. and Miranda, P., Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian, J. Funct. Anal.274 (2018) 2499-2531. · Zbl 1395.35158
[21] Bruneau, V., Miranda, P. and Raikov, G., Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians, Rev. Math. Phys.26 (2014) 1450003. · Zbl 1286.35180
[22] Bruneau, V., Pushnitski, A. and Raikov, G., Spectral shift function in strong magnetic fields, Algebra i Analiz16 (2004) 207-238; translation in St. Petersburg Math. J.16 (2005) 181-209. · Zbl 1082.35115
[23] Bruneau, V. and Sambou, D., Counting function of magnetic resonances for exterior problems, Ann. Henri Poincaré17 (2016) 3443-3471. · Zbl 1354.81013
[24] Brüning, J., Geyler, V. and Pankrashkin, K., Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys.20 (2008) 1-70. · Zbl 1163.81007
[25] Costabel, M., Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal.19(3) (1988) 613-626. · Zbl 0644.35037
[26] Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B., Schrödinger Operators — With Applications to Quantum Mechanics and Global Geometry (Springer, Berlin, 1987). · Zbl 0619.47005
[27] Delfour, M. C. and Zolésio, J. P., Shape analysis via oriented distance functions, J. Funct. Anal.123 (1994) 129-201. · Zbl 0814.49032
[28] Demuth, M., Hansmann, M. and Katriel, G., Eigenvalues of non-selfadjoint operators: A comparison of two approaches, Oper. Theory Adv. Appl.232 (2013) 107-163. · Zbl 1280.47005
[29] Derkach, V. A. and Malamud, M. M., Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal.95 (1991) 1-95. · Zbl 0748.47004
[30] Derkach, V. A. and Malamud, M. M., The extension theory of Hermitian operators and the moment problem, J. Math. Sci.73 (1995) 141-242. · Zbl 0848.47004
[31] Dodonov, V. V., Malkin, I. A. and Man’ko, V. I., The Green function of the stationary Schrödinger equation for a particle in a uniform magnetic field, Phys. Lett. A51 (1975) 133-134.
[32] P. Exner, Leaky quantum graphs: A review, 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., Vol. 77 (Amer. Math. Soc., 2008), pp. 523-564. · Zbl 1153.81487
[33] Exner, P. and Ichinose, T., Geometrically induced spectrum in curved leaky wires, J. Phys. A34 (2001) 1439-1450. · Zbl 1002.81024
[34] Exner, P. and Kondej, S., Bound states due to a strong \(\delta\) interaction supported by a curved surface, J. Phys. A36 (2003) 443-457. · Zbl 1050.81009
[35] Exner, P. and Kondej, S., Aharonov and Bohm versus Welsh eigenvalues, Lett. Math. Phys.108 (2018) 2153-2167. · Zbl 1396.81090
[36] Exner, P. and Kovařík, H., Quantum Waveguides (Springer, Heidelberg, 2015). · Zbl 1314.81001
[37] Exner, P., Lotoreichik, V. and Pérez-Obiol, A., On the bound states of magnetic Laplacians on wedges, Rep. Math. Phys.82(2) (2018) 161-185. · Zbl 1441.35181
[38] Exner, P. and Tater, M., Spectra of soft ring graphs, Waves Random Media14 (2004) 47-60. · Zbl 1063.81534
[39] Exner, P. and Yoshitomi, K., Persistent currents for 2D Schrödinger operator with a strong \(\delta \)-interaction on a loop, J. Phys. A35 (2002) 3479-3487. · Zbl 1010.81012
[40] Filonov, N. and Pushnitski, A., Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains, Comm. Math. Phys.264 (2006) 759-772. · Zbl 1106.81040
[41] Fournais, S. and Helffer, B., Spectral Methods in Surface Superconductivity (Birkhäuser, Boston, 2010). · Zbl 1256.35001
[42] Fu, J. H. G., Curvature measures and generalized Morse theory, J. Differ. Geom.30 (1989) 619-642. · Zbl 0722.53064
[43] G. E. Galkowski, Distribution of resonances in scattering by thin barriers, UC Berkeley Electronic Theses and Dissertations (2015).
[44] Garnett, J. B. and Marshall, D. E., Harmonic Measure (Cambridge University Press, Cambridge, 2005). · Zbl 1077.31001
[45] Gesztesy, F. and Mitrea, M., A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math113 (2011) 53-172. · Zbl 1231.47044
[46] Goffeng, M., Kachmar, A. and Persson Sundqvist, M., Clusters of eigenvalues for the magnetic Laplacian with Robin condition, J. Math. Phys.57 (2016) 063510. · Zbl 1355.81073
[47] Goffeng, M. and Schrohe, E., Spectral flow of exterior Landau-Robin Hamiltonians, J. Spectr. Theory7 (2017) 847-879. · Zbl 1405.58005
[48] Gorbachuk, V. I. and Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equations (Kluwer Academic Publ., Dordrecht, 1991). · Zbl 0751.47025
[49] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic Press, New York, 1980). · Zbl 0521.33001
[50] Haase, M., The Functional Calculus for Sectorial Operators (Birkhäuser, Basel, 2006). · Zbl 1101.47010
[51] Helffer, B., Spectral Theory and its Applications (Cambridge University Press, Cambridge, 2013). · Zbl 1279.47002
[52] Honnouvo, G. and Hounkonnou, M. N., Asymptotics of eigenvalues of the Aharonov-Bohm operator with a strong \(\delta \)-interaction on a loop, J. Phys. A37 (2004) 693-700. · Zbl 1057.81018
[53] Hornberger, K. and Smilansky, U., Magnetic edge states, Phys. Rep.367 (2002) 249-385.
[54] Kato, T., Perturbation Theory for Linear Operators (Springer, Berlin, 1995). · Zbl 0836.47009
[55] Kato, T., Variation of discrete spectra, Comm. Math. Phys.111 (1987) 501-504. · Zbl 0632.47002
[56] Kirsch, A. and Grinberg, N., The Factorization Method for Inverse Problems (Oxford University Press, Oxford, 2008). · Zbl 1222.35001
[57] Klopp, F. and Raikov, G., The fate of the Landau levels under perturbations of constant sign, Int. Math. Res. Notices2009 (2009) 4726-4734. · Zbl 1181.35149
[58] Kovařík, H. and Pankrashkin, K., On the \(p\)-Laplacian with Robin boundary conditions and boundary trace theorems, Calc. Var. Partial Differ. Equ.56 (2017), Art. 49, 29 pp. · Zbl 1375.49063
[59] Landkof, N. S., Foundations of Modern Potential Theory (Springer, Berlin-Heidelberg-New York, 1972). · Zbl 0253.31001
[60] Lieb, E. and Loss, M., Analysis (American Mathematical Society, Providence, RI, 2001). · Zbl 0966.26002
[61] Luecking, D., Finite rank Toeplitz operators on the Bergman space, Proc. Am. Math. Soc.136(5) (2008) 1717-1723. · Zbl 1152.47021
[62] Mantile, A., Posilicano, A. and Sini, M., Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces, J. Differ. Equations261 (2016) 1-55. · Zbl 1337.35032
[63] Marschall, J., The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math.58 (1987) 47-65. · Zbl 0605.46024
[64] McLean, W., Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000). · Zbl 0948.35001
[65] Melgaard, M. and Rozenblum, G., Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank, Comm. Partial Differential Equations28 (2003) 697-736. · Zbl 1044.35038
[66] Ožanová, K., Approximation by point potentials in a magnetic field, J. Phys. A39 (2006) 3071-3083. · Zbl 1091.81028
[67] Persson, M., Eigenvalue asymptotics for the even-dimensional exterior Landau-Neumann Hamiltonian, Adv. Math. Phys. (2009) 873704. · Zbl 1201.81055
[68] Popov, I. Yu., The operator extension theory, semitransparent surface and short range potential, Math. Proc. Camb. Phil. Soc.118 (1995) 555-563. · Zbl 0845.47007
[69] Pushnitski, A., Raikov, G. and Villegas-Blas, C., Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian, Comm. Math. Phys.320 (2013) 425-453. · Zbl 1277.47059
[70] Pushnitski, A. and Rozenblum, G., Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain, Doc. Math.12 (2007) 569-586. · Zbl 1132.35424
[71] Pushnitski, A. and Rozenblum, G., On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign, J. Anal. Math.114 (2011) 317-340. · Zbl 1253.47018
[72] Raikov, G., Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behavior near the essential spectrum tips, Comm. Partial Differential Equations15 (1990) 407-434. · Zbl 0739.35055
[73] Raikov, G. and Warzel, S., Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials, Rev. Math. Phys.14 (2002) 1051-1072. · Zbl 1033.81038
[74] Raymond, N., Bound States of the Magnetic Schrödinger Operator, , Vol. 27 (European Mathematical Society (EMS), Zurich, 2017). · Zbl 1370.35004
[75] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975). · Zbl 0308.47002
[76] Rozenblum, G. and Shirokov, N., Finite rank Bergman-Toeplitz and Bargmann-Toeplitz operators in many dimensions, Complex Anal. Oper. Theory4(4) (2010) 767-775. · Zbl 1202.47034
[77] G. Rozenblum and A. Sobolev, Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential, in Spectral Theory of Differential Operators. M. Sh. Birman 80th Anniversary Collection, Amer. Math. Soc. Transl. Ser. 2, Vol. 225 (American Mathematical Society, 2008), pp. 169-190. · Zbl 1167.35028
[78] Rozenblum, G. and Tashchiyan, G., On the spectral properties of the perturbed Landau Hamiltonian, Comm. Partial Differential Equations33(6) (2008) 1048-1081. · Zbl 1158.47029
[79] Schmüdgen, K., Unbounded Self-Adjoint Operators on Hilbert Space (Springer, Dordrecht, 2012). · Zbl 1257.47001
[80] Shimada, S., The approximation of the Schrödinger operators with penetrable wall potentials in terms of short range Hamiltonians, J. Math. Kyoto Univ.32 (1992) 583-592. · Zbl 0776.35044
[81] Sobolev, A. V., Asymptotic behavior of energy levels of a quantum particle in a homogeneous magnetic field perturbed by an attenuating electric field. I, (Russian) in Linear and Nonlinear Partial Differential Equations. Spectral Asymptotic Behavior, , Vol. 9 (Leningrad. Univ., Leningrad, 1984), pp. 67-84.
[82] Stahl, H. and Totik, V., General Orthogonal Polynomials (Cambridge University Press, Cambridge, 2010). · Zbl 1187.33008
[83] Stollmann, P. and Voigt, J., Perturbation of Dirichlet forms by measures, Potential Anal.5 (1996) 109-138. · Zbl 0861.31004
[84] Tamura, H., Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields, Osaka J. Math.25 (1988) 633-647. · Zbl 0731.35073
[85] Teschl, G., Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators (American Mathematical Society, Providence, 2014). · Zbl 1342.81003
[86] Wolff, T. H., Recent work on sharp estimates in second-order elliptic unique continuation problems, J. Geom. Anal.3 (1993) 621-650. · Zbl 0787.35017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.