On resonances and bound states of Smilansky Hamiltonian
https://doi.org/10.17586/2220-8054-2016-7-5-789-802
Abstract
We consider the self-adjoint Smilansky Hamiltonian Hξ in L2(R2) associated with the formal differential expression -∂2x – ½(∂2y +y2)- √2 ξyδ(x) in the sub-critical regime, ξ ϵ(0, 1). We demonstrate the existence of resonances for Hξ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small ξ>0. In addition, we refine the previously known results on the bound states of Hξ in the weak coupling regime (ξ→ 0+). In the proofs we use Birman-Schwinger principle for Hξ ,elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.
Keywords
Smilansky Hamiltonian,
resonances,
resonance free region,
weak coupling asymptotics,
Riemann surface,
bound states
Supporting agencies: This research was supported by the Czech Science Foundation (GAˇ CR) within the project 14-06818S.
About the Authors
P. Exner
Nuclear Physics Institute, Czech Academy of Sciences
Czech Republic
Czech Republic
25068 Rez, Czech Republic
V. Lotoreichik
Nuclear Physics Institute, Czech Academy of Sciences
Czech Republic
Czech Republic
25068 Rez, Czech Republic
M. Tater
Nuclear Physics Institute, Czech Academy of Sciences
Czech Republic
Czech Republic
25068 Rez, Czech Republic
References
1. Smilansky U. Irreversible quantum graphs. Waves Random Media, 2004, 14 (1), S143–S153.
2. Evans W.D., Solomyak M. Smilansky’s model of irreversible quantum graphs. I: The absolutely continuous spectrum. J. Phys. A: Math. Gen., 2005, 38 (21), P. 4611–4627.
3. Evans W.D., Solomyak M. Smilansky’s model of irreversible quantum graphs. II: The point spectrum. J. Phys. A: Math. Gen., 2005, 38 (35), P. 7661–7675.
4. Naboko S., Solomyak M. On the absolutely continuous spectrum in a model of an irreversible quantum graph. Proc. Lond. Math. Soc. (3), 2006, 92 (1), P. 251–272.
5. Rozenblum G., Solomyak M. On a family of differential operators with the coupling parameter in the boundary condition. J. Comput. Appl. Math., 2007, 208 (1), P. 57–71.
6. Solomyak M. On the discrete spectrum of a family of differential operators. Funct. Anal. Appl., 2004, 38 (3), P. 217–223.
7. Solomyak M. On a mathematical model of irreversible quantum graphs. St. Petersburg Math. J., 2006, 17 (5), P. 835–864.
8. Solomyak M. On the limiting behaviour of the spectra of a family of differential operators. J. Phys. A: Math. Gen., 2006, 39 (33), P. 10477–10489.
9. Guarneri I. Irreversible behaviour and collapse of wave packets in a quantum system with point interactions. J. Phys. A: Math. Theor., 2011, 44 (48), 485304, 22 p.
10. Aslanyan A., Parnovski L., and Vassiliev D. Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Math., 2000, 53 (3), P. 429–447.
11. Duclos P., Exner P., Meller B. Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields. Rep. Math. Phys., 2001, 47 (2), P. 253–267.
12. Edward J. On the resonances of the Laplacian on waveguides. J. Math. Anal. Appl., 2002, 272 (1), P. 89–116.
13. Exner P., Gawlista R., ˇ Seba P., Tater M. Point interactions in a strip. Ann. Phys., 1996, 252 (1), P. 133–179.
14. Exner P., Kovaˇ r´ ık H. Quantum waveguides. Springer, Cham., 2015, 382 p.
15. Christiansen T. Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends. Ann. Henri Poincar´e, 2002, 3 (5), P. 895–920.
16. Christiansen T. Asymptotics for a resonance-counting function for potential scattering on cylinders. J. Funct. Anal., 2004, 216 (1), P. 172–190.
17. Barseghyan D., Exner P. Spectral estimates for a class of Schr¨ odinger operators with infinite phase space and potential unbounded from below. J. Phys. A: Math. Theor., 2012, 45 (7), 075204, 14 p.
18. Barseghyan D., Exner P. A regular version of Smilansky model. J. Math. Phys., 2014, 55 (4), 042104, 13 p.
19. Barseghyan D., Exner P., Khrabustovskyi A., and Tater M. Spectral analysis of a class of Schr¨ odinger operators exhibiting a parameter dependent spectral transition. J. Phys. A: Math. Theor., 2016, 49 (16), 165302.
20. Abramowitz M., Stegun I. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, D.C:, 1964. 1046 p.
21. Chihara T. An introduction to orthogonal polynomials. Gordon and Breach Science Publishers, New York-London-Paris., 1978, 249 p.
22. Kato T. Perturbation theory for linear operators. Springer-Verlag, Berlin, 1995, 619 p.
23. Reed M., Simon B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York, 1978, 396 p.
24. Motovilov A. Analytic continuation of S matrix in multichannel problems. Theoret. Math. Phys., 1993, 95 (3), P. 692–699.
25. Simon B. Basic complex analysis. A comprehensive course in analysis, part 2A. American Mathematical Society, Providence, 2015, 641 p.
26. Simon B. Trace ideals and their applications. 2nd ed. American Mathematical Society, Providence, 2005, 150 p.
27. Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1969, 378 p.
28. Reed M., Simon B. Methods of modern mathematical physics. I: Functional analysis. Academic Press, New York, 1980, 400 p.
29. 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.
30. Br¨uning J., Geyler V., Pankrashkin K. Spectra of self-adjoint extensions and applications to solvable Schr¨ odinger operators. Rev. Math. Phys., 2008, 20 (1), P. 1–70.
31. Derkach V., Malamud M. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 1991, 95 (1), P. 1–95
Review
For citations:
Exner P., Lotoreichik V., Tater M. On resonances and bound states of Smilansky Hamiltonian. Nanosystems: Physics, Chemistry, Mathematics. 2016;7(5):789-802. https://doi.org/10.17586/2220-8054-2016-7-5-789-802
Views:
3
Email this article 