Boundary triples for Schrӧdinger operators with singular interactions on hypersurfaces
https://doi.org/10.17586/2220-8054-2016-7-2-290-302
Abstract
The self-adjoint Schr¨odinger operator Aδ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L2(Rn). The aim of this note is to construct a boundary triple for S∗ and a self-adjoint parameter Θδ,α in the boundary space L2(C) such that Aδ,α corresponds to the boundary condition induced by Θδ,α. As a consequence, the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of Aδ,α in terms of the Weyl function and Θδ,α.
Keywords
Boundary triple,
Weyl function,
Schrӧdinger operator,
singular potential,
δ-interaction,
hypersurface
Supporting agencies: J. Behrndt and V. Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund (FWF): Project P 25162-N26. J. Behrndt also wishes to thank Professor Igor Popov for the pleasant and fruitful research stay at the ITMO University in St. Petersburg in September 2015. V. Lotoreichik was also supported by the Czech Science Foundation (GA CR) under the project 14-06818S.
About the Authors
J. Behrndt
Institut fur Numerische Mathematik, Technische Universitat Graz
Austria
Austria
Steyrergasse 30, 8010 Graz
M. Langer
Department of Mathematics and Statistics, University of Strathclyde
United Kingdom
United Kingdom
26 Richmond Street, Glasgow G1 1XH
V. Lotoreichik
Department of Theoretical Physics, Nuclear Physics Institute CAS
Czech Republic
Czech Republic
250 68 ˇRez near Prague
References
1. Br¨uning J., Geyler V., Pankrashkin K. Spectra of self-adjoint extensions and applications to solvable Schr¨odinger operators. Rev. Math. Phys., 2008, 20, P. 1–70.
2. Derkach V. A., Hassi S., Malamud M. M., de Snoo H. Boundary relations and their Weyl families. Trans. Amer. Math. Soc., 2006, 358, P. 5351–5400.
3. Derkach V. A., Malamud M. M. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 1991, 95, P. 1–95.
4. Derkach V. A., Malamud M. M. The extension theory of Hermitian operators and the moment problem. J. Math. Sci., 1995, 73, P. 141–242.
5. Gorbachuk V. I., Gorbachuk M. L. Boundary Value Problems for Operator Differential Equations, Kluwer Academic Publ., Dordrecht, 1991.
6. Behrndt J., Langer M., Lotoreichik V. Schr¨odinger operators with δ and δ′-potentials supported on hypersurfaces. Ann. Henri Poincar´e, 2013, 14, P. 385–423.
7. Brasche J. F., Exner P., Kuperin Yu. A., ˇSeba P. Schr¨odinger operators with singular interactions. J. Math. Anal. Appl., 1994, 184, P. 112–139.
8. Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H. Solvable Models in Quantum Mechanics. With an appendix by Pavel Exner. AMS Chelsea Publishing, 2005.
9. Exner P., Kovaˇrik H. Quantum Waveguides, Theoretical and Mathematical Physics, Springer, 2015.
10. Exner P. Leaky quantum graphs: a review. Proc. Symp. Pure Math., 2008, 77, P. 523–564.
11. Albeverio S., Kostenko A., Malamud M., Neidhardt H. Spherical Schr¨odinger operators with δ-type interactions. J. Math. Phys., 2013, 54, 052103, 24 pp.
12. Behrndt J., Exner P., Lotoreichik V. Schr¨odinger operators with δ and δ′ interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys., 2014, 26, 1450015, 43 pp.
13. Duchˆene V., Raymond N. Spectral asymptotics of a broken delta interaction. J. Phys. A, 2014, 47, 155203, 19 pp.
14. Exner P. An isoperimetric problem for leaky loops and related mean-chord inequalities. J. Math. Phys., 2005, 46, 062105, 10 pp.
15. Exner P., Fraas M. On geometric perturbations of critical Schr¨odinger operators with a surface interaction. J. Math. Phys., 2009, 50, 112101, 12 pp.
16. Exner P., Ichinose T. Geometrically induced spectrum in curved leaky wires. J. Phys. A, 2001, 34, P. 1439– 1450.
17. Exner P., Kondej S. Bound states due to a strong δ interaction supported by a curved surface. J. Phys. A, 2003, 36, P. 443–457.
18. Exner P., Kondej S. Scattering by local deformations of a straight leaky wire. J. Phys. A, 2005, 38, P. 4865–4874.
19. Exner P., Pankrashkin K. Strong coupling asymptotics for a singular Schr¨odinger operator with an interaction supported by an open arc. Comm. Partial Differential Equations, 2014, 39, P. 193–212.
20. Exner P., Yoshitomi K. Asymptotics of eigenvalues of the Schr¨odinger operator with a strong δ-interaction on a loop. J. Geom. Phys., 2002, 41, P. 344–358.
21. Kondej S., Lotoreichik V. Weakly coupled bound state of 2-D Schr¨odinger operator with potential-measure. J. Math. Anal. Appl., 2014, 420, P. 1416–1438.
22. Kondej S., Veseliˇc I. Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Ann. Henri Poincar´e, 2007, 8, P. 109–134.
23. Mantile A., Posilicano A., Sini M. Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. ArXiv:1505.07236, 2015.
24. Popov I. Yu. The operator extension theory, semitransparent surface and short range potential. Math. Proc. Camb. Phil. Soc., 1995, 118, P. 555–563.
25. Stollmann P., Voigt J. Perturbation of Dirichlet forms by measures. Potential Anal., 1996, 5, P. 109–138.
26. Brown B. M., Grubb G., Wood I. G. M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr., 2009, 282, P. 314–347.
27. Grubb G. A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa (3), 1968, 22, P. 425–513.
28. Malamud M. M. Spectral theory of elliptic operators in exterior domains. Russ. J. Math. Phys., 2010, 17, P. 96–125.
29. Behrndt J., Langer M. Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal., 2007, 243, P. 536–565.
30. Behrndt J., Langer M. Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, in: Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Series, 2012, 404, P. 121–160.
31. Behrndt J., Langer M., Lotoreichik V. Spectral estimates for resolvent differences of self-adjoint elliptic operators. Integral Equations Operator Theory, 2013, 77, P. 1–37.
32. Gesztesy F., Mitrea M. Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr¨odinger operators on bounded Lipschitz domains. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2008, 79, P. 105–173.
33. Gesztesy F., Mitrea M. A description of all self-adjoint extensions of the Laplacian and Kre˘ın-type resolvent formulas on non-smooth domains. J. Anal. Math., 2011, 113, P. 53–172.
34. Grubb G. Extension theory for elliptic partial differential operators with pseudodifferential methods. London Math. Soc. Lecture Note Series, 2012, 404, P. 221–258.
35. Posilicano A. Self-adjoint extensions of restrictions. Oper. Matrices, 2008, 2, P. 1–24.
36. Posilicano A., Raimondi L. Krein’s resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators. J. Phys. A, 2009, 42, 015204, 11 pp.
37. Post O. Boundary pairs associated with quadratic forms. Math. Nachr., 2015, doi: 10.1002/mana.201500048.
38. Schm¨udgen K. Unbounded Self-Adjoint Operators on Hilbert Space, Springer, Dordrecht, 2012.
39. Behrndt J., Langer M. On the adjoint of a symmetric operator. J. Lond. Math. Soc. (2), 2010, 82, P. 563–580.
40. Lions J., Magenes E. Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, Berlin–Heidelberg–New York, 1972.
41. Grubb G. Distributions and Operators. Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.
42. Behrndt J., Grubb G., Langer M., Lotoreichik V. Spectral asymptotics for resolvent differences of elliptic operators with δ and δ′-interactions on hypersurfaces. J. Spectral Theory, 2015, 5, P. 697–729.
Review
For citations:
Behrndt J., Langer M., Lotoreichik V. Boundary triples for Schrӧdinger operators with singular interactions on hypersurfaces. Nanosystems: Physics, Chemistry, Mathematics. 2016;7(2):290-302. https://doi.org/10.17586/2220-8054-2016-7-2-290-302
Views:
7
Email this article 