Geometrically Induced Spectral Effects in Tubes with a Mixed Dirichlet—Neumann Boundary

Fedor L. Bakharev, Pavel Exner

Research output

1 Citation (Scopus)

Abstract

We investigate spectral properties of the Laplacian in L2 (Q), where Q is a tubular region in ℝ3 of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.

Original languageEnglish
Pages (from-to)213-231
Number of pages19
JournalReports on Mathematical Physics
Volume81
Issue number2
DOIs
Publication statusPublished - 1 Apr 2018

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Tube
tubes
Twist
thresholds
eigenvalues
Essential Spectrum
Discrete Spectrum
boundary conditions
Spectral Properties
Dirichlet
cross sections
Cross section
Eigenvalue
Boundary conditions
Sufficient Conditions

Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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