Related papers: Schr\"odinger operators on a half-line with invers…

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…

We prove asymptotic formulas of Szeg\H{o} type for the periodic Schr\"odinger operator $H=-\frac{d^2}{dx^2}+V$ in dimension one. Admitting fairly general functions $h$ with $h(0)=0$, we study the trace of the operator…

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…

We study dynamical properties of random Schr\"odinger operators $H^{(\omega)}$ defined on the Hilbert space $\ell^2(\bbZ^d)$ or $L^2(\bbR^d)$. Building on results from existing multi-scale analyses, we give sufficient conditions on…

In the first part of the paper we consider the Schr\"odinger operator $ -\Delta-V(x),\quad V>0. $ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we…

In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with $\delta'$-like potentials used to represent localized dipoles. These operators arise as norm resolvent…

We study the convergence of 1D Schr\"odinger ope\-rators $H_\varepsilon$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$ \alpha \delta'(x)+\beta…

For any positive real number $s$, we study the scattering theory in a unified way for the fractional Schr\"{o}dinger operator $H=H_0+V$, where $H_0=(-\Delta)^\frac s2$ and the real-valued potential $V$ satisfies short range condition. We…

For a bounded real-valued function $V$ on ${\Bbb R}^d$, we consider two Schr\"odinger operators $H_+=-\Delta+V$ and $H_-=-\Delta-V$. We prove that if the negative spectra $H_+$ and $H_-$ are discrete and the negative eigenvalues of $H_+$…

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable…

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-\Delta)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2(\mathbb{R}^d/\Gamma)$, with $\Gamma$ a maximal dimension lattice in $\mathbb{R}^d$ and…

We study the low-energy asymptotics of the spectral shift function for Schr\"odinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue…

The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_\lambda(\omega) = T + U + \lambda V(\omega)$ acting in $\ell^2(\T)$,…

We study long-time behaviors for branching-diffusion process corresponding to the drifted Schr\"odinger operator $\mathcal{L} = \frac{1}{2} \Delta + \langle \nabla V,\nabla \rangle - K$, where $K$ represents the reduction rate of a…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in…

We investigate some bounds for the density of states in the pure point regime for the random Schr\"{o}dinger operators $H^{\omega}=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nq_n(\omega)$, acting on $\ell^2(\mathbb{Z}^d)$, where…

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 the discrete Schr\"odinger operator $H = -\Delta + V$ on $\ell^2(\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in\mathbb{N}^*$, where $\Delta$ is the standard discrete Laplacian and $V_n =…