Abstract
This paper summarizes the contents of a plenary talk given at the 14th Biennial Conference of Indian SIAM in Amritsar in February 2018. We discuss here the effect of an abrupt spectral change for some classes of Schrödinger operators depending on the value of the coupling constant, from below bounded and partly or fully discrete, to the continuous one covering the whole real axis. A prototype of such a behavior can be found in the Smilansky–Solomyak model devised to illustrate that an irreversible behavior is possible even if the heat bath to which the systems are coupled has a finite number of degrees of freedom and analyze several modifications of this model, with regular potentials or a magnetic field, as well as another system in which \(x^py^p\) potential is amended by a negative radially symmetric term. Finally, we also discuss resonance effects in such models.
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References
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS Chelsea Publishing, Providence, 2005)
D. Barseghyan, P. Exner, A regular version of Smilansky model. J. Math. Phys. 55, 042104 (2014)
D. Barseghyan, P. Exner, A regular analogue of the Smilansky model: spectral properties. Rep. Math. Phys. 80, 177–192 (2017)
D. Barseghyan, P. Exner, A magnetic version of the Smilansky-Solomyak model. J. Phys. A: Math. Theor. 50, 485203 (24pp) (2017)
D. Barseghyan, P. Exner, A. Khrabustovskyi, M. Tater, Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition. J. Phys. A: Math. Theor. 49, 165302 (2016)
W.D. Evans, M. Solomyak, Smilansky’s model of irreversible quantum graphs. I: the absolutely continuous spectrum. J. Phys. A: Math. Gen. 38, 4611–4627 (2005)
W.D. Evans, M. Solomyak, Smilansky’s model of irreversible quantum graphs. II: the point spectrum. J. Phys. A: Math. Gen. 38, 7661–7675 (2005)
P. Exner, D. Barseghyan, Spectral estimates for a class of Schrödinger operators with infinite phase space and potential unbounded from below. J. Phys. A: Math. Theor. 45, 075204 (14pp) (2012)
P. Exner, J. Lipovský, Smilansky-Solomyak model with a \(\delta ^{\prime }\)-interaction. Phys. Lett. A 382, 1207–1213 (2018)
P. Exner, V. Lotoreichik, M. Tater, Spectral and resonance properties of Smilansky Hamiltonian. Phys. Lett. A 381, 756–761 (2017)
P. Exner, V. Lotoreichik, M. Tater, On resonances and bound states of Smilansky Hamiltonian. Nanosyst.: Phys. Chem. Math. 7, 789–802 (2016)
L. Geisinger, T. Weidl, Sharp spectral estimates in domains of infinite volume. Rev. Math. Phys. 23, 615–641 (2011)
I. Guarneri, Irreversible behaviour and collapse of wave packets in a quantum system with point interactions. J. Phys. A: Math. Theor. 44, 485304 (22 pp) (2011)
I. Guarneri, A model with chaotic scattering and reduction of wave packets. J. Phys. A: Math. Theor. 51, 095304 (16 pp) (2018)
V. Jakšić, S. Molchanov, B. Simon, Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Funct. Anal. 106, 59–79 (1992)
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1995)
A. Laptev, T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184, 87–111 (2000)
O. Mickelin, Lieb-Thirring inequalities for generalized magnetic fields. Bull. Math. Sci. 6, 1–14 (2016)
S. Naboko, M. Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph. Proc. Lond. Math. Soc. 92(3), 251–272 (2006)
M. Reed, B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, IV. Analysis of Operators (Academic Press, New York, 1975, 1978)
G. Rozenblum, M. Solomyak, On a family of differential operators with the coupling parameter in the boundary condition. J. Comput. Appl. Math. 208, 57–71 (2007)
B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum. Ann. Phys. 146, 209–220 (1983)
U. Smilansky, Irreversible quantum graphs. Waves Random Media 14, S143–S153 (2004)
M. Solomyak, On the discrete spectrum of a family of differential operators. Funct. Anal. Appl. 38, 217–223 (2004)
M. Solomyak, On a mathematical model of irreversible quantum graphs. St. Petersbg. Math. J. 17, 835–864 (2006)
M. Solomyak, On the limiting behaviour of the spectra of a family of differential operators. J. Phys. A: Math. Gen. 39, 10477–10489 (2006)
M. Znojil, Quantum exotic: a repulsive and bottomless confining potential. J. Phys. A: Math. Gen. 31, 3349–3355 (1998)
Acknowledgements
Our recent results discussed in this survey are the result of a common work with a number of colleagues, in the first place Diana Barseghyan, Andrii Khrabustovskyi, Vladimir Lotoreichik, and Miloš Tater whom I am grateful for the pleasure of collaboration. The research was supported by the Czech Science Foundation (GAČR) within the project 17-01706S.
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Exner, P. (2020). Schrödinger Operators with a Switching Effect. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Mathematical Modelling, Optimization, Analytic and Numerical Solutions. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-0928-5_2
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