Geometrically induced discrete spectrum in curved tubes

Dirac operator spectrum in tubes and layers with a zigzag type boundary

We derive a number of spectral results for Dirac operators in geometrically nontrivial regions in $\mathbb{R}^2$ and $\mathbb{R}^3$ of tube or layer shapes with a zigzag type boundary using the corresponding properties of the Dirichlet Laplacian.

Location of the nodal set for thin curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrodinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigen…

负曲率流形中的管与 Steklov 特征值

Tubes and Steklov eigenvalues in negatively curved manifolds

本文研究带全测地边界的紧致夹负曲率流形上的 Steklov 特征值，通过边界邻域 Steklov-Dirichlet 估计和管状邻域定理，建立第一非零特征值的体积型几何下界。

The nature of the essential spectrum in curved quantum waveguides

We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to…

Location of the essential spectrum in curved quantum layers

We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimensions, the result applies to the location of propagating states in nanostructures under physically natural conditions.

Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive $δ$ interaction supported by a periodic curve in $\mathbb{R}^d, d=2,3$.

On the spectrum of curved quantum waveguides

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the distance between the essential spectrum and the spectral threshold…

关于由核与管组成的集合的谱

On the spectrum of sets made of cores and tubes

本文采用纯变分方法研究了由核与无界管组成的无界开集的谱性质，获得了变分本征函数的指数衰减，并利用奇异摄动技术推翻了Hersch关于“书皮”集合给出Makai-Hayman不等式最佳常数的猜想。