Related papers: Spectral enclosures for non-self-adjoint discrete …

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as…

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 consider a nonlocal differential--difference Schr\"odinger operator on a segment with the Neumann conditions and two translations in the free term. The values of the translations are denoted by $\alpha$ and $\beta$ and are treated as…

In this paper, we consider discrete Schr\"odinger operators of the form, \begin{equation*} (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n). \end{equation*} We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$.…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schroedinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L^1 norms and L ln L norms of the…

We prove an upper bound on the sum of the distances between the eigenvalues of a perturbed Schr\"odinger operator $H_0-V$ and the lowest eigenvalue of $H_0$. Our results hold for operators $H_0=-\Delta-V_0$ in one dimension with single-well…

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

We investigate the spectral properties of the Schr\"odinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$…

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number…

We prove a spectral inequality (a specific type of uncertainty relation) for Schr\"odinger operators with confinement potentials, in particular of Shubin-type. The sensor sets are allowed to decay exponentially, where the precise allowed…

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schr\"odinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the…

Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n > 4$ and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for…

We study the eigenvalues of Schr\"odinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where $V$ decays exponentially at infinity.

We study asymptotic spectral properties of the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$…

We use a classical result of Hildebrandt to establish simple conditions for the absence of eigenvalues of non-selfadjoint discrete and continuous Schr\"odinger operators on the boundary of their numerical range.

Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schr\"odinger operators are extended to allow singular potentials such as certain $L^p$-functions. The proof is based on accordingly adapted Carleman…