Related papers: Finite Sections of Periodic Schr\"odinger Operator…
We discuss discrete one-dimensional Schr\"odinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the…
In this work we investigate a class of degenerate Schr\"odinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. We establish…
Recently it was pointed out that the solutions found in literature for the space fractional Schr\"odinger equation in a piecewise manner are wrong, except the case with the delta potential. We reanalyze this problem and show that an exact…
We prove conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite. We use a method for bounding eigenvalues based on estimates of the Hilbert-Schmidt norm of semigroup differences…
For certain one-dimensional Schroedinger-type difference operators with a complex potential, a "complete" set of exponentially decaying eigenvectors is shown to exist. "Completeness" entails that the parameters involved are obtained through…
This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where $A$ is a random Jacobi…
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…
Let $H_0$ be a discrete periodic Schr\"odinger operator on $\ell^2(\mathbb{Z}^d)$: $$H_0=-\Delta+V,$$ where $\Delta$ is the discrete Laplacian and $V: \mathbb{Z}^d\to \mathbb{C}$ is periodic. We prove that for any $d\geq3$, the Fermi…
We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes.…
The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like…
For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions…
This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schr\"odinger operators.
We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schr\"odinger operator with a periodic potential plus a finitely supported perturbation. We describe all…
Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schr\"odinger operator is Fermi isospectral to the zero potential. This provides a negative…
In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the…
Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral…
Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schr\"odinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined…
We consider perturbations of quasi-periodic Schr\"odinger operators on the integer lattice with analytic sampling functions by decaying potentials and seek decay conditions under which various spectral properties are preserved. In the…
Our paper investigates one-dimensional Schr\"odinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We…
We discuss a method based on a segmentary approximation of solutions of the Schr\"odinger by quadratic splines, for which the coefficients are determined by a variational method that does not require the resolution of complicated algebraic…