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We consider the Schr\"odinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple…

We consider the Schroedinger operator L_{\alpha} on the half-line with a periodic background potential and the Wigner-von Neumann potential of Coulomb type: csin(2\omega x+d)/(x+1). It is known that the continuous spectrum of the operator…

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in…

We consider the Schr\"odinger operator $H = -\Delta + V$ in a layer or in a $d$-dimensional cylinder. The potential $V$ is assumed to be periodic with respect to some lattice. We establish the absolute continuity of $H$, assuming $V \in…

The present paper is devoted to the study of resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_{L}= -\Delta +…

This article investigates the wave equation for the Schr\"{o}dinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-\Delta+V$, where $\Delta$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove…

We consider the magnetic Schr\"odinger operator $H=(i \nabla +A)^2- V$ with a non-negative potential $V$ supported over a strip which is a local deformation of a straight one, and the magnetic field $B:=\mathrm{rot}(A)$ is assumed to be…

Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

We consider the self-adjoint operator $H=H_0+V$, where $H_0$ is the free semi-classical Dirac operator on $R^3$. We suppose that the smooth matrix-valued potential $V=O(<x>^{-\delta}), \delta>0,$ has an analytic continuation in a complex…

We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and…

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…

We construct a potential $V$ on $\RR^d$, smooth away from one pole, and a sequence of quasi-modes for the operator $-\Delta+V$, which concentrate on this pole. No smoothing effect, Strichartz estimates nor dispersive inequalities hold for…

For the Schr\"odinger operator $-\Delta_\rm{g}+V$ on a complete Riemannian manifold with real valued potential $V$ of compact support, we establish a sharp equivalence between Sobolev regularity of $V$ and the existence of finite-order…

We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both…

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 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 fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to…

We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily…

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field and subject to an electric potential $v_0$ depending only on the variable along the magnetic field $x_3$. The operator $H_0$ has infinitely many eigenvalues of…