Related papers: Resolvent estimates for one-dimensional Schr\"odin…
We prove a family of dispersive estimates for the higher order Schr\"odinger equation $iu_t=(-\Delta)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions…
In this paper, we consider the dispersive estimates for Schr\"odinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-\mu}$ for $|x|\gg 1$ with $0<\mu<2$, in one dimension. As an application, we establish both the…
We prove a quantitative unique continuation principle for Schr\"odinger operators $H=-\Delta+V$ on $\mathrm{L}^2(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^d$ and $V$ is a singular potential: $V \in \mathrm{L}^\infty(\Omega)…
We study resolvent estimates for non-selfadjoint semiclassical pseudodifferential operators with double characteristics. Assuming that the quadratic approximation along the double characteristics is elliptic, we obtain polynomial upper…
We consider a linear Schr\"odinger operator $H = -\Delta + V$ with a strongly singular potential $V$ not bounded from below on a non-compact incomplete Riemannian manifold $M$. We assume that the negative part of potential $V_{-}$ is…
We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schr\"odinger operators $P=\Delta+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the…
We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schr\"odinger operators with non-negative matrix-valued potentials, i.e., operators acting on $\psi\in L^2(\mathbb{R}^n,\mathbb{C}^d)$…
We study the one-dimensional Schr\"odinger operators $$ S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the…
We propose a rigorous method for computing two-sided eigenvalue bounds of the Schr\"odinger operator $H=-\Delta+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the…
In this paper we consider the Schr\"odinger operator $\mathcal L_V= -\Delta + V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schr\"odinger operator $\log \mathcal L_V$ proving its main…
On a Lie group $G$, we investigate the discreteness of the spectrum of Schr\"odinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable function…
We consider a semi-classical Schr\"odinger operator, -h^2\Delta + V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the…
We investigate a two-dimensional Schr\"odinger operator, $-h^2 \Delta +iV(x)$, with a purely complex potential $iV(x)$. A rigorous definition of this non-selfadjoint operator is provided for bounded and unbounded domains with common…
Estimates for eigenvalues of Schr\"{o}dinger operators on the half-line with complex-valued potentials are established. Schr\"{o}dinger operators with potentials belonging to weak Lebesque's classes are also considered. The results cover…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
We study the inverse spectral problem for Bessel type operators with potential (v(x)): (H_\kappa=-\partial_x^2+\frac{k}{x^2}+v(x)). The potential is assumed smooth in ((0,R)) and with an asymptotic expansion in powers and logarithms as…
We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher…
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…
If the dimension $d$ is even, the resonances of the Schr\"odinger operator $-\Delta +V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show…
In this article, we prove the finiteness of the number of eigenvalues for a class of Schr\"odinger operators $H = -\Delta + V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$…