Related papers: On the perturbed periodic Schr\"odinger operators …

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave…

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…

We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential…

In this paper, I consider one-dimensional periodic Schr{\"o}dinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one…

We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate…

We study singular Sturm-Liouville operators of the form \[ \frac{1}{r_j}\left(-\frac{\mathrm d}{\mathrm dx}p_j\frac{\mathrm d}{\mathrm dx}+q_j\right),\qquad j=0,1, \] in $L^2((a,b);r_j)$, where, in contrast to the usual assumptions, the…

We consider Schr\"odinger operators with periodic potentials in the positive quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any integer $N$ and any interval $I$ there exists a periodic potential such that the…

A model operator $H$ corresponding to a three-particle discrete Schr\"odinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_\alpha(p),$…

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and…

We study the Schr\"odinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and…

Let $Q(x)$ denote a periodic function on the real line. The Schr\"odinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we…

We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential spectrum has been known to be difficult, given that…

The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…

We provide examples of operators $T(D)+V$ with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the Fermi surfaces of constant kinetic energy $T$. We make the connection to…

We study the spectrum of a one-dimensional Schroedinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of…

We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and…

We obtain H\"older stability estimates for the inverse Steklov and Calder\'on problems for Schr\"odinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier…

The spectral properties of the Schr\"odinger operator $T_ty= -y''+q_ty$ in $L^2(\R)$ are studied, with a potential $q_t(x)=p_1(x), x<0, $ and $q_t(x)=p(x+t), x>0, $ where $p_1, p$ are periodic potentials and $t\in \R$ is a parameter of…

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…