Spectral Asymptotics of the Dirichlet Laplacian on a Generalized Parabolic Layer

Exner, Pavel; Lotoreichik, Vladimir

doi:10.1007/s00020-020-2571-x

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…We noted in the introduction a recent result [EKP20] showing that the spectrum of a cylindrical potential layer 2 has the same behavior. One can conjecture that also potential layers with a different geometry might behave as their hard-wall counterparts, such as treated in [EL20], irrespective of the channel profile. (j) Another subject of interest concerns the influence of external fields.…”

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confidence: 99%

Spectral properties of soft quantum waveguides

Exner¹

2020

J. Phys. A: Math. Theor.

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We investigate properties of a particle confined to a hard-wall spiralshaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to 'less curved' waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity exceeds a critical size. For more general spiral regions the spectral nature depends on whether they are 'expanding' or 'shrinking'. The most interesting situation occurs in the asymptotically Archimedean case where the existence of bound states depends on the direction from which the asymptotics is reached.

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confidence: 99%

Spectral properties of soft quantum waveguides

Exner¹

2020

J. Phys. A: Math. Theor.

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“…We noted in the introduction a recent result [EKP20] showing that the spectrum of a cylindrical potential layer § has the same behavior. One can conjecture that also potential layers with a different geometry might behave as their hard-wall counterparts, such as treated in [EL20], irrespective of the channel profile. (x) Another subject of interest concerns the influence of external fields.…”

Section: Discussionmentioning

confidence: 99%

Spectral properties of soft quantum waveguides

Exner

2020

Preprint

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We consider a soft quantum waveguide described by a two-dimensional Schrödinger operators with an attractive potential in the form of a channel of a fixed profile built along an infinite smooth curve which is not straight but it is asymptotically straight in a suitable sense. Using Birman-Schwinger principle we show that the discrete spectrum of such an operator is nonempty if the potential well defining the channel profile is deep and narrow enough. Some related problems are also mentioned.

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“…4]. In particular, the existence of an infinite number of eigenvalues was established in the conical [15,9,30] and parabolic [14] layers.…”

Section: Introductionmentioning

confidence: 99%