Related papers: Spectral Asymptotics for Waveguides with Perturbed…

We consider a twisted quantum waveguide i.e. a domain of the form \Omega_{\theta} : = r_\theta \omega \times R, where \omega \subset R^2 is a bounded domain, and r_\theta = r_\theta(x_3) is a rotation by the angle \theta(x_3) depending on…

In this work, we analyze the Dirichlet Laplacian $-\Delta_{\Omega}^D$ in an unbounded waveguide $\Omega \subset \mathbb R^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the…

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite…

We consider the Dirichlet Laplacian in a three-dimensional waveguide that is a small deformation of a periodically twisted tube. The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling…

Let $\Omega \subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the…

Let $\Omega \subset \mathbb R^3$ be a broken sheared waveguide, i.e., it is built by translating a cross-section in a constant direction along a broken line in $\mathbb R^3$. We prove that the discrete spectrum of the Dirichlet Laplacian…

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in…

Consider the Dirichlet Laplacian operator $-\Delta^D$ in a periodic waveguide $\Omega$. On the condition that $\Omega$ is sufficiently thin, we show that its spectrum $\sigma(-\Delta^D)$ is absolutely continuous (in each finite region). In…

We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are…

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty…

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. We prove some…

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an…

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union…

We consider a family $\{\Omega^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-\Delta_{\Omega^\varepsilon}$ on $\Omega^\varepsilon$. The…

It is well known that the spectrum of the Dirichlet Laplacian for a two-dimensional waveguide, which is a local deformation of a straight strip, is unstable with respect to waveguide boundary deformations. This means that, when the…

In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain…

Let $\Omega$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overline{\Omega};\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega)$-realization $A$ of…

We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on…

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an…

We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $\Pi^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry…