Tunneling resonances in systems without a classical trapping

Борисов, Д. И.; Exner, Pavel; Головина, А. М.

doi:10.1063/1.4773098

Cited by 12 publications

(8 citation statements)

References 28 publications

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“…The method also works without this assumption, however, the argument is more complicated [BE07]. Moreover, by the same technique one can also treat resonances: an example of a guide in which there are two long but finite Dirichlet barriers separating the Neumann window from semi-infinite Neumann boundary segments has been worked out in [BEG13]. 2.…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

143

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

143

133

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“…The study of the resonances of the operator H ℓ emerging in the vicinity of the point λ 0 is reduced first to a special operator equation and then we show that these resonances coincide with the zeroes of certain holomorphic function. Our scheme is based on approach suggested in work [6], but there are serious modification due to a much more general formulation of the problem.…”

Section: Resonances As Zeroes Of a Holomorphic Functionmentioning

confidence: 99%

“…They can be situations, when only finitely many resonances emerge from a point in the essential spectrum. This was the case in [6], where a Laplacian was considered in the strip with a combination of Dirichlet and Neumann conditions imposed so that the final model could be regarded as an operator with three distant perturbations and the central perturbation had a simple isolated discrete eigenvalue λ 0 embedded into the essential spectrum of the left and right perturbations. It was shown that the original operator had one resonance converging to λ 0 and the leading terms in its asymptotics were found.…”

Section: Introductionmentioning

confidence: 99%

On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders

Борисов

Головина

2020

Preprint

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We consider a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations; this operator is obtained by "gluing" several single perturbation operators H (k) , k = 1, . . . , n, at large distances. The coefficients of each operator H (k) are periodic in the outlets of the cylinder; the structure of these periodic parts at different outlets can be different. We consider a point λ0 ∈ R in the essential spectrum of the operator with several distant perturbations and assume that this point is not in the essential spectra of middle operators H (k) , k = 2, . . . , n − 1, but is an eigenvalue of at least one of H (k) , k = 1, . . . , n. Under such assumption we show that the operator with several distant perturbations possesses finitely many resonances in the vicinity of λ0. We find the leading terms in asymptotics for these resonances, which turn out to be exponentially small. We also conjecture that the made assumption selects the only case, when the distant perturbations produce finitely many resonances in the vicinity of λ0. Namely, as λ0 is in the essential spectrum of at least one of operators H (k) , k = 2, . . . , n − 1, we do expect that infinitely many resonances emerge in the vicinity of λ0.

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“…In paper [9], there was studied a problem for the Laplacian in the strip, where the distant perturbations were replaced by the Dirichlet and Neumann boundary conditions. A similar problem but in a multi-dimensional cylinder was studied in [1].…”

Section: Introductionmentioning

confidence: 99%

“…In all studied models, in the case, when the resonances and/or eigenvalues emerged and accumulated along some segment in the essential spectrum, their asymptotics were also power with respect to the distance between the distant potentials [6], [12], [13], [21], [24]. In the case of perturbing the eigenvalues embedded into the essential spectrum, only finitely many resonances with exponential asymptotics arose [1], [9], [14]. This poses a natural question: whether this is true that the emergence of infinitely many resonances means that they always have power in the distance asymptotics, while in the case of emergence of finitely many resonances the corresponding asymptotics are always exponential?…”

Section: Introductionmentioning

confidence: 99%

On infinite system of resonance and eigenvalues with exponential asymptotics generated by distant perturbations

Борисов

Konyrkulzhaeva

2020

Ufimsk. Mat. Zh.

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We consider an one-dimensional Schrödinger operator with four distant potentials separated by large distance. All distances are proportional to a sam large parameter. The initial potentials are of kink shapes, which are glued mutually so that the final potential vanishes at infinity and between the second and third initial potentials and it is equal to one between the first and the second potentials as well as between the third and fourth potentials. The potentials are not supposed to be real and can be complex-valued. We show that under certain, rather natural and easily realizable conditions on the four initial potentials, the considered operator with distant potentials possesses infinitely many resonances and/or eigenvalues of form 𝜆 = 𝑘 2 𝑛 , 𝑛 ∈ Z, which accumulate along a given segment in the essential spectrum. The distance between neighbouring numbers 𝑘 𝑛 is of order the reciprocal of the distance between the potentials, while the imaginary parts of these quantities are exponentially small. For the numbers 𝑘 𝑛 we obtain the representations via the limits of some explicitly calculated sequences and the sum of infinite series. We also prove explicit effective estimates for the convergence rates of the sequences and for the remainders of the series.

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Tunneling resonances in systems without a classical trapping

Cited by 12 publications

References 28 publications

Quantum Waveguides

Quantum Waveguides

On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders

On infinite system of resonance and eigenvalues with exponential asymptotics generated by distant perturbations

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