Related papers: Asymptotic number of scattering resonances for gen…

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials…

We study the resonances of Schr\"odinger operators on the infinite product $X=\mathbb{R}^d\times \mathbb{S}^1$, where $d$ is odd, $\mathbb{S}^1$ is the unit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that at high…

Let P be the operator $-\Delta+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical…

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…

The present paper addresses questions on 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 + V…

We prove upper bounds on the number of resonances and eigenvalues of Schr\"odinger operators $-\Delta+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the…

We study the stationary scattering for $(-\Delta)^{\frac 12} + V(x)$ on $\mathbb{R}^3$. For poly-homogeneous potentials decaying at infinity, we prove that the asymptotics of the potential can be recovered from the scattering matrix at a…

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use…

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 +…

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…

We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of…

We consider the semiclassical Schr\"odinger operator $-h^2\partial_x^2+V(x)$ on a half-line, where $V$ is a compactly supported potential which is positive near the endpoint of its support. We prove that the eigenvalues and the purely…

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away…

We study the scattering poles of $\sqrt{-\Delta} + V$, where $V$ is a compactly supported, bounded and complex valued potential. We show that the resolvent operator $ \chi R_V \chi$ has a meromorphic continuation to the whole Riemannian…

We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…

We prove the complete asymptotic expansion of the integrated density of states of a Schrodinger operator H = -\Delta + b acting in R^d when the potential b is either smooth periodic, or generic quasi-periodic (finite linear combination of…

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…

We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle…

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 prove the complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrodinger operator $H=-\Delta+b$ acting in $R^d$ when the potential $b$ is real and either smooth periodic, or…