Spectral and resonance properties of the Smilansky Hamiltonian

Exner, Pavel; Lotoreichik, Vladimir; Tater, Miloš

doi:10.1016/j.physleta.2016.12.053

Cited by 6 publications

(10 citation statements)

References 21 publications

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“…We are not going to give proofs of these claims referring to the papers quoted above, instead we will show how the the discrete spectrum can be found numerically following [10] which can provide additional insights. At the time, however, the method we use, rephrasing the task as a spectral problem for Jacobi matrices is the core of the proofs done by Solomyak et al providing thus a feeling of what is the technique involved.…”

Section: Smilansky-solomyak Modelmentioning

confidence: 99%

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Schrödinger Operators with a Switching Effect

Exner

2020

Mathematical Modelling, Optimization, Analytic and Numerical Solutions

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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 Smilansky-Solomyak model devised to illustrate that an an irreversible behavior is possible even if the heat bath to which the systems is 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 p y p potential is amended by a negative radially symmetric term. Finally, we also discuss resonance effects in such models.

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Section: Smilansky-solomyak Modelmentioning

confidence: 99%

“…The models we consider have other interesting properties. Let us return to the setting of Section 2 and show that the system exhibits a rich resonance structure; we refer to [10,11] for a detailed discussion of these phenomena. To begin with, we have to say which resonances we speak about.…”

Section: Resonances In Smilansky-solomyak Modelmentioning

confidence: 99%

Schrödinger Operators with a Switching Effect

Exner

2020

Mathematical Modelling, Optimization, Analytic and Numerical Solutions

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“…The asymptotic expansion of this eigenvalue Λ 1 can be found by an argument similar to that used in [14] for the original Smilansky model. The system of equations (14) can be after substitution Q n = (n + 1 2 ) 1/4 C n rewritten as…”

Section: Discrete Spectrum Of H βmentioning

confidence: 99%

“…The affirmative answer was provided in [11] where such a potential family was constructed, in [12] it was demonstrated that the effect persists even if the system is exposed to a homogeneous magnetic field (in which case the original Smilansky interpretation is ultimately lost). Moreover, it was shown that the original model has a rich resonance structure [13,14].…”

Section: Introductionmentioning

confidence: 99%

Smilansky–Solomyak model with a δ′-interaction

Exner¹,

Lipovský²

2018

Physics Letters A

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We investigate a strongly singular version of the model of irreversible dynamics proposed by Smilansky and Solomyak in which the interaction responsible for an abrupt change of the spectrum is of δ ′ type. We determine the spectrum in both the subcritical and supercritical regimes and discuss its character as well as its asymptotic properties of the discrete spectrum in terms of the coupling constant.

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“…While this is all true in many cases, it need not be true in general. This was demonstrated by Uzy Smilansky using a simple model [Sm04] which was subsequently analyzed in detail and generalized by Mikhail Solomyak and coauthors [So04a,So04b,ES05a,ES05b,So06a,So06b,NS06,RS07], see also [Gu11] and [ELT17].…”

Section: Introductionmentioning

confidence: 95%

A magnetic version of the Smilansky–Solomyak model

Barseghyan¹,

Exner²

2017

J. Phys. A: Math. Theor.

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Abstract. We analyze spectral properties of two mutually related families of magnetic Schrödinger operators,, with the parameters ω > 0 and λ < 0, where A is a vector potential corresponding to a homogeneous magnetic field perpendicular to the plane and V is a regular nonnegative and compactly supported potential. We show that the spectral properties of the operators depend crucially on the one-dimensional, respectively. Depending on whether the operators L and L(V ) are positive or not, the spectrum of H Sm (A) and H(V ) exhibits a sharp transition.

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Spectral and resonance properties of the Smilansky Hamiltonian

Cited by 6 publications

References 21 publications

Schrödinger Operators with a Switching Effect

Schrödinger Operators with a Switching Effect

Smilansky–Solomyak model with a δ′-interaction

A magnetic version of the Smilansky–Solomyak model

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