Abstract
We consider the scattering problem for a class of strongly singular Schrödinger operators in \(L^2({\mathbb R}^3)\) which can be formally written as \(H_{\alpha ,\varGamma }= -\varDelta + \delta _\alpha (x-\varGamma )\), where \(\alpha \in {\mathbb R}\) is the coupling parameter and \(\varGamma \) is an infinite curve which is a local smooth deformation of a straight line \(\varSigma \subset {\mathbb R}^3\). Using Kato–Birman method, we prove that the wave operators \(\varOmega _\pm (H_{\alpha ,\varGamma }, H_{\alpha ,\varSigma })\) exist and are complete.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005)
Behrndt, J., Frank, R.L., Kühn, Ch., Lotoreichik, V., Rohleder, J.: Spectral theory for Schrödinger operators with \(\delta \)-interactions supported on curves in \(\mathbb{R}^{3}\). Ann. H. Poincaré 18, 1305–1347 (2017)
Brasche, J.F., Exner, P., Kuperin, YuA, Šeba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)
Brasche, J.F., Teta, A.: Spectral analysis and scattering for Schrödinger operators with an interaction supported by a regular curve. In: Albeverio, S., Fenstadt, J.E., Holden, H., Lindstrøm, T. (eds.) Ideas and Methods in Quantum and Statistical Physics, pp. 197–211. Cambridge University Press, Cambridge (1992)
Cacciapuoti, A., Fermi, D., Posilicano, A.: Scattering from local deformations of a semitransparent plane. J. Math. Anal. Appl. 473, 215–257 (2019)
Exner, P., Ichinose, T.: A product formula related to quantum Zeno dynamics. Ann. H. Poincaré 6, 195–215 (2005)
Exner, P., Ichinose, T., Kondej, S.: On relations between stable and Zeno dynamics in a leaky graph decay model. In: Proceedings of the Conference “Operator Theory and Mathematical Physics” (Bȩdlewo 2004); Operator Theory: Advances and Applications, vol. 174, pp. 21–34. Basel, Birkhäuser (2007)
Exner, P., Ichinose, T., Neidhardt, H., Zagrebnov, V.A.: Zeno product formula revisited. Int. Eq. Oper. Theory 57, 67–81 (2007)
Exner, P., Kondej, S.: Curvature-induced bound states for a delta interaction supported by a curve in \(R^{3}\). Ann. H. Poincaré 3, 967–981 (2002)
Exner, P., Kondej, S.: Scattering by local deformations of a straight leaky wire. J. Phys. A: Math. Gen. 38, 4865–4874 (2005)
Exner, P., Kovařík, H.: Quantum Waveguides, p. xxii+382. Springer International, Heidelberg (2015)
Fujita, H., Okamoto, H.: Tosio Kato as an applied mathematician: a historical study of a Japanese mathematician, pp. 15–16. ICIAM Newsletter, October (2018)
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)
Kato, T.: Fundamental properties of Hamiltonian of the Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951)
Kato, T.: On finite dimensional perturbations of self-adjoint operators. J. Math. Soc. Jpn. 9, 239–249 (1957)
Kato, T.: Perturbations of continuous spectra by trace class operators. Proc. Jpn. Acad. 33, 260–264 (1957)
Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in Functional Analysis (Essays Dedicated to M.G. Krein on the Occasion of his 70th Birthday). Advances in Mathematics: Supplementary Studies, vol. 3, pp. 185–195. Academic, New York (1978)
Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)
Mantile, A., Posilicano, A.: Asymptotic completeness and S-matrix for singular perturbations. J. Math. Pures Appl., to appear; arXiv:1711.07556
Posilicano, A.: A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory IV. Analysis of Operators. Academic Press, New York (1979)
Rosenblum, M.: Perturbation of the continuous spectrum and unitary equivalence. Pac. J. Math. 7, 997–1010 (1957)
Simon, B.: Tosio Kato’s work on non-relativistic quantum mechanics: part 1. Bull. Math. Sci. 8, 121–232 (2018)
Simon, B.: Tosio Kato’s work on non-relativistic quantum mechanics: part 2. Bull. Math. Sci., to appear
Teta, A.: Quadratic forms for singular perturbations of the Laplacian. Publ. RIMS 26, 803–817 (1990)
Acknowledgements
The research was supported by the Czech Science Foundation (GAČR) within the project 17-01706S and the EU project CZ.02.1.01/0.0/0.0/16_019/0000778.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Exner, P., Kondej, S. (2019). Scattering on Leaky Wires in Dimension Three. In: Rassias, T.M., Zagrebnov, V.A. (eds) Analysis and Operator Theory . Springer Optimization and Its Applications, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-030-12661-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-12661-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-12660-5
Online ISBN: 978-3-030-12661-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)