×

On the spectral properties of Dirac operators with electrostatic \(\delta\)-shell interactions. (English. French summary) Zbl 1382.81095

Summary: In this paper the spectral properties of Dirac operators \(A_\eta\) with electrostatic \(\delta\)-shell interactions of constant strength \(\eta\) supported on compact smooth surfaces in \(\mathbb R^3\) are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of \(A_\eta\) are investigated. In particular, it turns out that the discrete spectrum of \(A_\eta\) inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of \(A_\eta\) and the free Dirac operator \(A_0\) is trace class, and in the nonrelativistic limit \(A_\eta\) converges in the norm resolvent sense to a Schrödinger operator with an electric \(\delta\)-potential of strength \(\eta\).

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35Q40 PDEs in connection with quantum mechanics
35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
47A10 Spectrum, resolvent
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable Models in Quantum Mechanics (2005), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI, with an Appendix by Pavel Exner
[2] Arrizabalaga, N.; Mas, A.; Vega, L., Shell interactions for Dirac operators, J. Math. Pures Appl. (9), 102, 4, 617-639 (2014) · Zbl 1297.81083
[3] Arrizabalaga, N.; Mas, A.; Vega, L., Shell interactions for Dirac operators: on the point spectrum and the confinement, SIAM J. Math. Anal., 47, 2, 1044-1069 (2015) · Zbl 1314.81083
[4] Arrizabalaga, N.; Mas, A.; Vega, L., An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators, Commun. Math. Phys., 344, 2, 483-505 (2016) · Zbl 1341.81023
[5] Behrndt, J.; Exner, P.; Holzmann, M.; Lotoreichik, V., Approximation of Schrödinger operators with \(δ\)-interactions supported on hypersurfaces, Math. Nachr., 290, 8-9, 1215-1248 (2017) · Zbl 1376.35010
[6] Behrndt, J.; Langer, M., Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243, 2, 536-565 (2007) · Zbl 1132.47038
[7] Behrndt, J.; Langer, M., Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, (Lond. Math. Soc. Lect. Note Ser., vol. 404 (2012)), 121-160 · Zbl 1331.47067
[8] Behrndt, J.; Langer, M.; Lotoreichik, V., Schrödinger operators with \(δ\) and \(\delta^\prime \)-potentials supported on hypersurfaces, Ann. Henri Poincaré, 14, 2, 385-423 (2013) · Zbl 1275.81027
[9] Behrndt, J.; Langer, M.; Lotoreichik, V., Spectral estimates for resolvent differences of self-adjoint elliptic operators, Integral Equ. Oper. Theory, 77, 1, 1-37 (2013) · Zbl 1311.47059
[10] Behrndt, J.; Langer, M.; Lotoreichik, V., Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators, J. Lond. Math. Soc. (2), 88, 2, 319-337 (2013) · Zbl 1296.35097
[11] Berkolaiko, G.; Kuchment, P., Introduction to Quantum Graphs (2013), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1318.81005
[12] Brasche, J.; Exner, P.; Kuperin, Y.; Šeba, P., Schrödinger operators with singular interactions, J. Math. Anal. Appl., 184, 112-139 (1994) · Zbl 0820.47005
[13] Bruk, V. M., A certain class of boundary value problems with a spectral parameter in the boundary condition, Mat. Sb., 100, 142, 210-216 (1976) · Zbl 0334.47010
[14] Brüning, J.; Geyler, V.; Pankrashkin, K., Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys., 20, 1-70 (2008) · Zbl 1163.81007
[15] Carlone, R.; Malamud, M. M.; Posilicano, A., On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set, J. Differ. Equ., 254, 9, 3835-3902 (2013) · Zbl 1277.34120
[16] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H., Boundary relations and their Weyl families, Trans. Am. Math. Soc., 358, 5351-5400 (2006) · Zbl 1123.47004
[17] Derkach, V. A.; Malamud, M. M., Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95, 1, 1-95 (1991) · Zbl 0748.47004
[18] Derkach, V. A.; Malamud, M. M., The extension theory of Hermitian operators and the moment problem, J. Math. Sci., 73, 2, 141-242 (1995) · Zbl 0848.47004
[19] Dittrich, J.; Exner, P.; Šeba, P., Dirac operators with a spherically symmetric \(δ\)-shell interaction, J. Math. Phys., 30, 12, 2875-2882 (1989) · Zbl 0694.46053
[20] Exner, P., Spectral properties of Schrödinger operators with a strongly attractive \(δ\) interaction supported by a surface, (Proc. of the NSF Summer Research Conference. Proc. of the NSF Summer Research Conference, Mt. Holyoke, 2002. Proc. of the NSF Summer Research Conference. Proc. of the NSF Summer Research Conference, Mt. Holyoke, 2002, Contemp. Math., vol. 339 (2003), Amer. Math. Soc.), 25-36 · Zbl 1044.35037
[21] Exner, P., Leaky quantum graphs: a review, (Analysis on Graphs and Its Applications. Analysis on Graphs and Its Applications, Proc. Symp. Pure Math., vol. 77 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 523-564 · Zbl 1153.81487
[22] Exner, P.; Kovařík, H., Quantum Waveguides, Theor. Math. Phys. (2015), Springer · Zbl 1314.81001
[23] Gesztesy, F.; Holden, H.; Simon, B.; Zhao, Z., A trace formula for multidimensional Schrödinger operators, J. Funct. Anal., 141, 2, 449-465 (1996) · Zbl 0864.35030
[24] Gesztesy, F.; Mitrea, M.; Zinchenko, M., Variations on a theme of Jost and Pais, J. Funct. Anal., 253, 2, 399-448 (2007) · Zbl 1133.47010
[25] Gesztesy, F.; Šeba, P., New analytically solvable models of relativistic point interactions, Lett. Math. Phys., 13, 4, 345-358 (1987) · Zbl 0646.60107
[26] Gohberg, I. C.; Kreĭn, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18 (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0181.13504
[27] Gorbachuk, V. I.; Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equations (1991), Kluwer Academic Publ.: Kluwer Academic Publ. Dordrecht · Zbl 0751.47025
[28] Jonsson, A.; Wallin, H., Function spaces on subsets of \(R^n\), Math. Rep., 2, 1 (1984), xiv+221 pp · Zbl 0875.46003
[29] Kočubeĭ, A. N., On extensions of symmetric operators and symmetric binary relations, Mat. Zametki, 17, 41-48 (1975)
[30] Kato, T., Perturbation Theory for Linear Operators (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0836.47009
[31] Reed, M.; Simon, B., Methods of Modern Mathematical Physics III. Scattering Theory (1979), Academic Press: Academic Press New York, London · Zbl 0405.47007
[32] Thaller, B., The Dirac Equation, Texts. Monogr. Phys. (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0881.47021
[33] Wehling, T. O.; Black-Schaffer, A. M.; Balatsky, A. V., Dirac materials, Adv. Phys., 63, 1, 1-76 (2014)
[34] Weidmann, J., Lineare Operatoren in Hilberträumen, Teil I (2000), Teubner: Teubner Stuttgart · Zbl 0972.47002
[35] Weidmann, J., Lineare Operatoren in Hilberträumen, Teil II (2003), Teubner: Teubner Stuttgart · Zbl 1025.47001
[36] Yafaev, D. R., Mathematical Scattering Theory. Analytic Theory (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1197.35006
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.