Finitely many δ interactions with supports on concentric spheres

Shabani, Juma

doi:10.1063/1.528005

Cited by 43 publications

(16 citation statements)

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“…Schrödinger operators with interactions supported by manifolds of a lower dimension were studied first in examples with a particular symmetry [AGS87,Sha88], a more systematical investigation began with the papers [BT92,BEKŠ94].…”

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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“…A thorough analysis including the δ ′ extension can be found in [9]; an extension to multiple spheres is given in [10] and other related results are reviewed in [3]. From the mathematical point of view such models are described by means of spherically symmetric self-adjoint extensions of the symmetric operatoṙ…”

mentioning

confidence: 99%

“…The spherical symmetry allows us to reduced the analysis to a family of halfline problems by partial wave decomposition [10]). Using the isometry U :…”

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confidence: 99%

Resonance asymptotics in the generalized Winter model

Exner

Fraas

2006

Physics Letters A

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We consider a modification of the Winter model describing a quantum particle in presence of a spherical barrier given by a fixed generalized point interaction. It is shown that the three classes of such interactions correspond to three different types of asymptotic behaviour of resonances of the model at high energies.Models with contact interactions are popular because they allow us to study properties of quantum systems in a framework which makes explicit solutions possible. It was found in the beginning of the eighties that the usual δ interaction on the line has a counterpart, named not quite fortunately δ ′ , and a little later the complete four-parameter class of the generalized point interactions (GPI's) was introduced [1,2]. Properties of these interactions are now well understood -see [3] for a rather complete bibliography.The GPI's fall into different classes according to their behaviour at low and high energies. The most simple manifestation can be found in scattering. While a δ-type barrier behaves as a "usual" regular potential becoming transparent at high energies, the δ ′ -like one on the contrary decouples asymptotically in the same limit. In addition, there is an intermediate class for which both the reflection and transmission amplitudes have nonzero limits as the energy tends to zero or infinity [4]. Furthermore, in Kronig-Penney-type models describing periodic arrays of such interactions, the indicated classes differ by the gap behaviour at high energies; this has important consequences for spectral nature of the corresponding Wannier-Stark systems [5,6].

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“…Then the identity (16) follows from Eq.s (35) and (36), by exchanging order of integration and by taking into account the identity 1…”

Section: Since By Dominated Convergence One Hasmentioning

confidence: 99%

Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

Cacciapuoti

Fermi

Posilicano

2017

Advances in Quantum Mechanics

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ABSTRACT. We study the relative zeta function for the couple of operators A 0 and A α , where A 0 is the free unconstrained Laplacian in L 2 (R d ) (d ≥ 2) and A α is the singular perturbation of A 0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−∆ ), where −∆ is the free Laplacian in L 2 (R d−1 ). Thus α may depend on the components of the wave vector parallel to hyperplane; in this sense A α describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever α = χ I (−∆ ), where χ I is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.

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Finitely many δ interactions with supports on concentric spheres

Cited by 43 publications

References 1 publication

Quantum Waveguides

Quantum Waveguides

Resonance asymptotics in the generalized Winter model

Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

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