Related papers: Equidistribution estimates for eigenfunctions and …
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner…
In this paper we consider an initial/boundary value problem for the Schr\"odinger equation with a right-hand side involving the fractional Sturm-Liouville operator with singular propagation and potential. To construct a solution, first…
In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of…
We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two…
In this paper, we consider an eigenvalue problem of the elliptic operator $$ L_r={\rm div}(T^r\nabla\cdot )$$ on compact submanifolds in arbitrary codimension of space forms $\mathbb{R}^N(c)$ with $c\geq0$. Our estimates on eigenvalues are…
We estimate the number of small eigenvalues of Schr\"odinger operators on Riemannian vector bundles over geometrically finite manifolds.
Neat stuff about eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrodinger operators
We derive a counting formula for the eigenvalues of Schr\"odinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue…
We prove dynamical upper bounds for discrete one-dimensional Schroedinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations. We demonstrate the applicability of our approach by a study of…
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the…
Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…
Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami…
We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…
In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given…
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$. Given two…
Eigenvalues of the Lam\'e operator are studied as complex-analytic functions in period $\tau$ of an elliptic function. We investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues…
We consider Schr\"odinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with…
This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schr\"odinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates…
We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of…
We generalize the Stein-Tomas [17] $L^2$-restricition theorem and the uniform Sobolev estimates of Kenig, Ruiz and the second author [11] by allowing critically singular potential. We also obtain Strichartz estimates for Schr\"odinger and…