We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$, in $L^2(\R_+)$, where $q\in\bH_+=\{q', xq\in L^2(\R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in\bH_+$ is given.
We investigate Dirichlet Laplacian in a straight twisted tube of a non-circular cross section, in particular, its discrete spectrum coming from a local slowdown of the twist. We prove a Lieb-Thirring-type estimate for the spectral moments and present two examples illustrating how the bound depends on the tube cross section.
A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues $λ_n$, a complete asymptotic expansion for large $n$ is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularised sums of integer powers of eigenvalues $λ_n$ in terms of the heat invariants.
We prove the following. For any complex valued $L^p$-function $b(x)$, $2 \leq p < \infty$ or $L^\infty$-function with the norm $\| b | L^{\infty}\| < 1$, the spectrum of a perturbed harmonic oscillator operator $L = -d^2/dx^2 + x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in $L^2(\mathbb{R})$.
We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$ in $L^2(R_+)$, where $q\in H_+=\{q', xq\in L^2(R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in H_+$ is given. Moreover, we solve the similar inverse problem for the family of boundary conditions $ψ'(0)=b ψ(0)$, $b\in R$.
Consider the operator $ T=-{d^2dx^2}+x^2+q(x)$ in $L^2(\mathbb{R})$, where real functions $q$, $q'$ and $\int_0^xq(s)ds$ are bounded. In particular, $q$ is periodic or almost periodic. The spectrum of $T$ is purely discrete and consists of the simple eigenvalues $\{μ_n\}_{n=0}^\infty$, $μ_n<μ_{n+1}$. We determine their asymptotics $μ_n = (2n+1) + (2π)^{-1}\int_{-π}^πq(\sqrt{2n+1}\sinθ)dθ+ O(n^{-1/3})$.
In this paper, we consider the perturbations of the Harmonic Oscillator Operator by an odd pair of point interactions: $z (δ(x - b) - δ(x + b))$. We study the spectrum by analyzing a convenient formula for the eigenvalue. We conclude that if $z = ir$, $r$ real, as $r \to \infty$, the number of non-real eigenvalues tends to infinity.
In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the harmonic oscillator and for a generic bounded perturbation, the set of semiclassical measures for quasimodes of sufficiently small width is strictly smaller than the corresponding set of semiclassical measures for the unperturbed system. In this work we precise the description of this set taking into account the Diophantine properties of the vector of frequencies,…
We derive explicit inequalities for sums of eigenvalues of one-dimensional Schrödinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.
In this paper we study a semiclassical heat trace expansion for perturbations of the harmonic oscillator, by adapting to the semiclassical setting techniques developed by Hitrik and Polterovich in [HP]. We use the expansion to obtain certain inverse spectral results.
We consider the operator $ L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R}) $, where $ w(x) = s [ δ(x - b) - δ(x + b)], b \neq 0,$ real, $s \in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and $λ_n = (2n + 1) + s^2 (κ(n) / n) + ρ(n)$, where $ κ(n) = \frac{1}{2π} [(-1)^{n + 1} \sin ( 2 b \sqrt{2n} ) - \frac{1}{2} \sin ( 4 b \sqrt{2n} ) ]$ and $ |ρ(n) | \leq C (\log n) / (n^{3/2})$ If $s = i γ$, $γ$ real, the number $T(γ)$ of non-real eigenvalues is finite, and $T(γ) \leq [ C (1 + | γ|) \log (e + | γ|)]^2.$ The analogue of the above equations i…
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 discrete spectrum.
Since the publication of the important work of Rauch and Taylor (Potential and scattering theory on wildly perturbed domains, JFA, 1975) a lot has been done to analyse wild perturbations of the Laplace operator. Here we present results concerning the norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbe…
In this work we consider semi-classical Schrödinger operators with potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider certain classes of small random perturbations and show that with probability very close to 1, the number of resonances in rectangles $[a,b]-i[0,ch^{2/3}[$, is equal to the number of eigenvalues in $[a,b]$ of the Dirichlet realization of the unperturbed operator in ${\cal O}$ up to a small remainder.
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
We study the inverse scattering for Schr{ö}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on …
We study properties of the nodal sets of high frequency eigenfunctions and quasimodes for radial perturbations of the Harmonic Oscillator. In particular, we consider nodal sets on spheres of large radius (in the classically forbidden region) for quasimodes with energies lying in intervals around a fixed energy $E$. For well chosen intervals we show that these nodal sets exhibit quantitatively different behavior compared to those of the unperturbed Harmonic Oscillator. These energy intervals are defined via a careful analysis of the eigenvalue spacings for the perturbed operator, based on analy…
We consider a boundary value problem for a general second order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization we obtain a similar problem in a non-perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of th…
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 example of such a domain perturbation. Let $Ω$ be a (not {necessarily} bounded) domain in $\mathbb{R}^n$. We perturb it to $ Ω_\varepsilon=Ω\setminus \cup_{k=1}^m S_{k,\varepsilon},$ where $S_{k,\varepsilon}$ are closed surfaces with small suitably scaled holes (``windows'') through which the bounded domains enclosed by these surfaces (``resonators'') are connec…
In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.
Based on a previous paper [Chr17] on Neumann data for Dirichlet eigenfunctions on triangles, we extend the study in two ways. First, we investigate the (semi-classical) Neumann data mass on perturbed triangles. Specifically, we replace one side of a triangle by adding a smooth perturbation, and assume that the disparity between the perturbation and the original side is bounded by a small value $ε$. Second, we add a small $ε$ sized potential to the (semi-classical) Laplacian and see how the results change on triangles. In both cases, we find that the $L^2$ norm of Neumann data on each side is c…
We prove spectral stability results for the $curl curl$ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $H^2$-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are…
We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio $N$, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the…
