Related papers: Resonances and Spectral Shift Function for the sem…

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We…

Let $H(\Om_0)=-\Delta+V$ be a Schr\"odinger operator on a bounded domain $\Om_0\subset \mathbb R^d$ with Dirichlet boundary conditions. Suppose that the $\Om_\ell$ ($\ell \in \{1,...,k\}$) are some pairwise disjoint subsets of $\Om_0$ and…

We prove a dispersive estimate for the evolution of Schroedinger operators $H = -\Delta + V(x)$ in ${\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\R^3)\cap L^q(\R^3)$, $p < \frac32 < q$, so that…

Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact self-adjoint operator $V,$ the functions $r_j(z) = - \sigma_j^{-1}(z),$ where $\sigma_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a lot of important…

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space $\Omega =\mathbb R^3_+$ given by the self-adjoint operator named $\delta$-interaction. We will assume Dirichlet or Neumann boundary…

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic…

We consider semi-classical Schr{\"o}dinger operator $ P(h)=-h^2\Delta +V(x)$ in ${\mathbb R}^n$ such that the analytic potential $V$ has a non-degenerate critical point $x_0=0$ with critical value $E_0$ and we can define resonances in some…

We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following…

We consider non-self-adjoint Schr\"{o}dinger operators $H_{{\rm c}}=-\Delta+V_{{\rm c}}$ (resp. $H_{{\rm r}}=-\Delta+V_{{\rm r}}$) acting in $L^2(\mathbb R^d)$, $d\ge 1$, with dilation analytic complex (resp. real) potentials. We were able…

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

We define Dirac operators on $\mathbb{S}^3$ (and $\mathbb{R}^3$) with magnetic fields supported on smooth, oriented links and prove self-adjointness of certain (natural) extensions. We then analyze their spectral properties and show, among…

We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in…

The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown…

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint…

We consider semiclassical Schroedinger operators on R^n, with C^\infty potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we…

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…

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic…

We consider the Schr\"odinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study…

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is…