Related papers: Estimates for eigenvalues of Schr\"{o}dinger opera…
We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…
We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on…
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…
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{\"o}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to…
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, $\Im V \le 0$…
We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are…
The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…
We prove the bispectrality of some class of matrix Schr\"odinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the…
The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schr\"{o}dinger operators and more general operators with the spectral dimensions $d\leq 2$. The classical Cwikel-Lieb-Rosenblum…
The purpose of the present work is to establish decorrelation estimates at distinct energies for some random Schr\"odinger operator in dimension one. In particular, we establish the result for some random operators on the continuum with…
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…
We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order…
We consider Schr\"odinger operators with complex-valued decreasing potentials on the half-line. Such operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the…
Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is…
We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…
We prove decay estimates for generalized eigenfunctions of discrete Schr\"odinger operators on weighted infinite graphs in the spirit of Agmon.
We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any…
This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our…
We prove dispersive estimates for Schroedinger operators in dimension three without any assumptions on zero energy. Ie, we allows resonances and/or eigenvalues at zero energy.
We derive sharp quantitative bounds for eigenvalues of biharmonic operators perturbed by complex-valued potentials in dimensions one, two and three.