Spectral Theory
We prove a unique continuation principle or uncertainty relation valid for Schr\"odinger operator eigenfunctions, or more generally solutions of a Schr\"odinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution…
In this note we prove Minami's estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami's estimate to prove Poisson statistics for the energy level spacing. Our result…
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…
An isometric operator $V$ in a Pontryagin space $\mathcal{H}$ is called standard, if its domain and the range are nondegenerate subspaces in $\mathcal{H}$. Generalized resolvents of standard isometric operators were described in the paper…
We develop a spectral analysis of a class of block Jacobi operators based on the conjugate operator method of Mourre. We give several applications including scalar Jacobi operators with periodic coefficients, a class of difference operators…
By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, M. M\"oller and the author found a factorization of the Selberg zeta function into a product of Fredholm determinants of…
We investigate the order $\rho$ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for $\rho$ which is explicit in terms of the parameters of the canonical system…
We prove a lower bound on the eigenvalues $\lambda_k$, $k\in\mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $\Omega\subset\mathbb{R}^n$ of volume $V$: $$ \lambda_k \geq C_n\bigg( \delta\frac{k}{V}\bigg)^{2/n} $$ where $\delta$…
Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with…
We study bound states generated by a unique potential minimum in the situation where the system is strongly confined to a bounded region containing the minimum (by imposing Dirichlet boundary conditions). In this case the eigenvalues of the…
A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean…
We prove a Pleijel-type theorem for the asymptotic behaviour of the number of nodal domains of eigenfunctions of the quantum harmonic oscillator in any dimension.
We study the $p$-spectrum of a locally symmetric space of constant curvature $\Gamma\backslash X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(\Gamma\backslash G)_{\tau_p}$, where…
In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the…
The inverse scattering problem is studied for the matrix Sturm-Liouville equation on the line. Necessary and sufficient conditions for the scattering data are obtained.
In this paper we present how spectral properties of certain linear operators vary when operators are considered in different Hilbert spaces having common dense domain as the space of polynomials in one real variable with complex…
We study the survival probability associated with a semi-classical matrix Shr\"odinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual…
We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to small random perturbations. Furthermore, we let the coupling constant $\delta$ be $\exp\{-\frac{1}{Ch}\}\leq \delta…
This is a first paper by the authors dedicated to the distribution of eigenvalues for random perturbations of large bidiagonal Toeplitz matrices.
A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q &…