Absolute Continuity and Band Gaps of the Spectrum of the Dirichlet Laplacian in Periodic Waveguides

Mamani, Carlos R.; Verri, Alessandra A.

doi:10.1007/s00574-017-0065-5

Cited by 1 publication

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References 24 publications

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“…We note that this result has been previously established by Verri [34] in the special setting of purely twisted strips. In fact, in recent years there has been an exponential growth of interest in effective models for thin waveguides under various geometric and analytic deformations, see [12,15,17,29,32,9,8,31,30,35,6,34,7] and further references therein. We refer to [27] for a unifying approach to this type of problems.…”

Section: Thin Stripsmentioning

confidence: 99%

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Quantum strips in higher dimensions

Krejčiřı́k¹,

Zahradova²

2020

Operators and Matrices

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We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the twodimensional surface by a one-dimensional Schrödinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.

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