Related papers: Counterexample to the second eigenfunction having …

We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman-Schwinger type operators $K_{\lambda}(\mu)$ and their associated 2-modified perturbation…

We represent in this note the solutions of the electronic Schr\"odinger equation as traces of higher-dimensional functions. This allows to decouple the electron-electron interaction potential but comes at the price of a degenerate elliptic…

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral…

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive…

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean…

Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schr\"odinger operator with the truncated Toeplitz matrices. We…

An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{\"o}dinger…

This paper analyzes the scattering matrix for two unbounded self-adjoint operators: the standard Laplace operator in three-dimensional space and a second operator that differs from the first by an infinite sum of zero-range potentials.

We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These…

By applying new functional analysis tools in the framework of Fourier interpolation formulas, such as sc-Fredholm operators and Schauder frames, we are able to improve and refine several properties of these aforementioned formulas on the…

The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue…

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…

We consider the problem of minimizing the eigenvalues of the Schr\"{o}dinger operator $H=-\Delta+\alpha F(\ka)$ ($\alpha>0$) on a compact $n-$manifold subject to the restriction that $\ka$ has a given fixed average $\ka_{0}$. In the…

We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space…

Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the…

A local perturbation theory for the spectral analysis of the Schr\"odinger operator with two periodic potentials whose periods are commensurable has been constructed. It has been shown that the perturbation of the periodic 1D Hamiltonian by…

We consider discrete one-dimensional Schroedinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of…

We investigate the effect of small diffusion on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions in one dimensional space. The asymptotic behaviors of the principal eigenvalues, as…

We investigate the eigenvalues of perturbed spherical Schr\"odinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of…