Related papers: Eigenstates with Infinite Position Moments

We review recent and give some new results on the spectral properties of Schroedinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials…

We consider Schr\"odinger operators with smooth periodic potentials in Euclidean spaces of dimension bigger than 1 and prove a uniform lower bound on the density of states for large values of the spectral parameter.

We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the…

We give a necessary and sufficient condition for the maximum principle of Schr\"{o}dinger operators in terms of the bottom of the spectrum of time-changed processes. As a corollary, we obtain a sufficient condition for the Liouville…

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…

In this paper, we study 1-d random Schr\"odinger operators on a finite interval with Dirichlet boundary conditions. We are interested in the approximation of the ground state energy using the minimum of the effective potential. For the 1-d…

We obtain the spectrum of bound states for a modified P\"oschl-Teller and square potential wells in the nonlinear Schr\"odinger equation. For a fixed norm of bound states, the spectrum for both potentials turns out to consist of a finite…

The discrete one-dimensional Schr\"odinger operator is studied in the finite interval of length $N=2 M$ with the Dirichlet boundary conditions and an arbitrary potential even with respect to the spacial reflections. It is shown, that the…

We determine approximate eigenvalues and eigenfunctions shapes for bound states in the $3D$ shallow spherical ultrarelativistic well. Existence thresholds for the ground state and first excited states are identified, both in the purely…

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of…

We study the stationary scattering theory for the matrix Schr\"odinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption…

Let H be a Schr\"odinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schr\"odinger equation (H-E)u(x,E)=0, and let B_R denote a ball of…

In this paper, we prove the energy stable property for time-dependent (generalized) Schr\"{o}dinger operators by using Hardy inequality. Such property acts very important roles in quantum scattering theory and nonlinear problem. As an…

We discuss spectral properties of the self-adjoint operator \[ -d^2/dt^2 + (t^{k+1}/(k+1)-\alpha)^2 \] in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over $\alpha$ of the ground state energy of this operator is…

The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type $$ H_\lambda=-\frac{d^2}{dx^2}+U(x)+\lambda\alpha_\lambda V(\alpha_\lambda x) $$ is considered. The potentials $U$ and $V$ are real-valued bounded…

We define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson…

We consider a discrete Schroedinger operator whose potential is the sum of a Wigner-von Neumann term and a summable term. The essential spectrum of this operator equals to the interval [-2,2]. Inside this interval, there are two critical…

We present a new approach to solve a Schr\"odinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity.…

We prove that the wave operators for $n \times n$ matrix Schr\"odinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying…

We show persistence of both Anderson and dynamical localization in Schr\"odinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\"odinger operator with a non-positive ergodic random…