# Schrödinger Operators with a Switching Effect

@article{Exner2020SchrdingerOW,
title={Schr{\"o}dinger Operators with a Switching Effect},
author={Pavel Exner},
journal={arXiv: Mathematical Physics},
year={2020},
pages={13-31}
}
• P. Exner
• Published 6 February 2020
• Mathematics
• arXiv: Mathematical Physics
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 Schrodinger 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…

## References

SHOWING 1-10 OF 32 REFERENCES
Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition
• Mathematics
• 2015
We analyze two-dimensional Schrodinger operators with the potential $|xy|^p - \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$, which exhibit an abrupt change of its spectral properties
Smilansky's model of irreversible quantum graphs: II. The point spectrum
• Mathematics
• 2005
In the model suggested by Smilansky (2004 Waves Random Media 14 143–53) one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators
Lieb–Thirring inequalities for generalized magnetic fields
Following an approach by Exner et al. (Commun Math Phys 26:531–541, 2014), we establish Lieb–Thirring inequalities for general self-adjoint and second-degree differential operators with matrix valued
On the Absolutely Continuous Spectrum in a Model of an Irreversible Quantum Graph
• Mathematics
• 2005
A family Aα of differential operators depending on a real parameter α ⩾ 0 is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely
On the limiting behaviour of the spectra of a family of differential operators
We study a family of self-adjoint partial differential operators H ω , where ω is a large parameter. In the simplest case each operator acts in L 2 ((a, b) x R) as H ω = -∂ 2 x + ω(-∂ 2 y + Q(y)),
Smilansky's model of irreversible quantum graphs: I. The absolutely continuous spectrum
• Mathematics
• 2005
In the model suggested by Smilansky (2004 Waves Random Media 14 143–53) one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators
On a mathematical model of irreversible quantum graphs
The “irreversible quantum graph” model, suggested by U. Smilansky, is considered. Mathematically, the problem is in the investigation of the spectrum of the operator Aα determined by an infinite
A magnetic version of the Smilansky–Solomyak model
• Mathematics
• 2017
We analyze spectral properties of two mutually related families of magnetic Schr\"{o}dinger operators, $H_{\mathrm{Sm}}(A)=(i \nabla +A)^2+\omega^2 y^2+\lambda y \delta(x)$ and \$H(A)=(i \nabla