arXiv subjects
Publications and source records attributed to Alessandra A. Verri.
Let $Ω\subset \mathbb R^3$ 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 operator in $Ω$ is non-empty and finite. Furthermore, we show a particular geometry for $Ω$ which implies that the total multiplicity of the discrete spectrum is equals 1.
Let $Ω$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $Ω$ with Dirichlet and Neumann boundary conditions on opposite sides of $Ω$. We prove some results on the existence and absence of the discrete spectrum of the operator; which are influenced by the twisted and bent effects of $Ω$. Provided that $Ω$ is thin enough, we show an asymptotic behavior of the eigenvalues. The interest in those considerations lies on the difference from the purely Dirichlet case. Finally, we perform an appropriate dilatation in $Ω$ and we compare the results.
We study the spectrum of the Dirichlet Laplacian operator in a two-dimensional twisted strip embedded in $\mathbb R^d$ with $d \geq 2$. It is shown that a local twisting perturbation can create discrete eigenvalues for the operator. In particular, we also study the case where the twisted effect "grows" at infinity while the width of the strip goes to zero. In this situation, we find an asymptotic behavior for the eigenvalues.
Let $Ω\subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-Δ_Ω^D$ the Dirichlet Laplacian operator in $Ω$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-Δ_Ω^D$. Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-Δ_Ω^D$; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.
We study the Neumann Laplacian $-Δ^N$ restricted to a periodic waveguide. In this situation its spectrum $σ(-Δ^N)$ presents a band structure. Our goal and strategy is to get spectral information from an analysis of the asymptotic behavior of these bands provided that the waveguide is sufficiently thin.
Consider the Dirichlet Laplacian operator $-Δ^D$ in a periodic waveguide $Ω$. On the condition that $Ω$ is sufficiently thin, we show that its spectrum $σ(-Δ^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $σ(-Δ^D)$ and locate it.
We study the Neumann Laplacian operator $-Δ_Ω^N$ restricted to a twisted waveguide $Ω$. The goal is to find the effective operator when the diameter of $Ω$ tends to zero. However, when $Ω$ is "squeezed" there are divergent eigenvalues due to the transverse oscillations. We show that each one of these eigenvalues influences the action of the effective operator in a different way. In the case where $Ω$ is periodic and sufficiently thin, we find information about the absolutely continuous spectrum of $-Δ_Ω^N$ and the existence and location of band gaps in its structure.
We restrict a quantum particle under a coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three dimensional tubes along the x-axis and diameter $\varepsilon$, and study the confining limit $\varepsilon\to0$. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.
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 $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.
We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in $R^n$, n = 1, 2, 3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.