### Abstract

We investigate spectral properties of the Laplacian in L^{2} (Q), where Q is a tubular region in ℝ^{3} 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 language | English |
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Pages (from-to) | 213-231 |

Number of pages | 19 |

Journal | Reports on Mathematical Physics |

Volume | 81 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

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### Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Reports on Mathematical Physics*, vol. 81, no. 2, pp. 213-231. https://doi.org/10.1016/S0034-4877(18)30038-7

**Geometrically Induced Spectral Effects in Tubes with a Mixed Dirichlet—Neumann Boundary.** / Bakharev, Fedor L.; Exner, Pavel.

Research output

TY - JOUR

T1 - Geometrically Induced Spectral Effects in Tubes with a Mixed Dirichlet—Neumann Boundary

AU - Bakharev, Fedor L.

AU - Exner, Pavel

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Dirichlet—Neumann boundary

KW - discrete spectrum

KW - Laplacian

KW - tube

UR - http://www.scopus.com/inward/record.url?scp=85046831182&partnerID=8YFLogxK

U2 - 10.1016/S0034-4877(18)30038-7

DO - 10.1016/S0034-4877(18)30038-7

M3 - Article

AN - SCOPUS:85046831182

VL - 81

SP - 213

EP - 231

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

SN - 0034-4877

IS - 2

ER -