Related papers: Decay estimates of discretized Green's functions f…
We are using the finite-gap approach for the construction of the Schr\"{o}dinger operator discretization on a quad graph. The latter is represented by a two-dimensional integer sublattice in a $d$-dimensional space. The Green's function of…
Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schr\"{o}dinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific…
The Green's function of the discrete Sch\"odinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's…
In this article, we investigate systems of generalized Schr\"odinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay…
We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schr\"odinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an…
The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular,…
We study the spatial asymptotics of Green's function for the 1d Schrodinger operator with operator-valued decaying potential. The bounds on the entropy of the spectral measures are obtained. They are used to establish the presence of a.c.…
We discuss recent results on spectral properties of discrete alloy-type random Schr\"odinger operators. They concern Wegner estimates and bounds on the fractional moments of the Green's function.
We provide quantitative inductive estimates for Green's functions of matrices with (sub)expoentially decaying off diagonal entries in higher dimensions. Together with Cartan's estimates and discrepancy estimates, we establish explicit…
We investigate the behavior of the Green functions of Schroedinger operators near the diagonal. The only non-trivial cases, where the on-diagonal singularities are non-zero and do not depend on the spectral parameter, are two and three…
In the presence of a confining potential $V$, the eigenfunctions of a continuous Schr\"odinger operator $-\Delta +V$ decay exponentially with the rate governed by the part of $V$ which is above the corresponding eigenvalue; this can be…
We consider the expansion of wave packets governed by the free Schr\"odinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments and in particular in the field of matter-wave…
We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete Schr\"odinger operators. Due to this, the set of their $m$-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral…
We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives…
A numerical method is developed for calculating the real time Green's functions of very large sparse Hamiltonian matrices, which exploits the numerical solution of the inhomogeneous time-dependent Schroedinger equation. The method has a…
We are concerned with the properties of weak solutions of the stationary Schr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$. Assuming $f$ to…
We consider non-self-adjoint electromagnetic Schr\"odinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic…
We give a sharp estimate of the number of zeros of analytic functions in the unit disc belonging to analytic quasianalytic Carleman--Gevrey classes. As an application, we estimate the number of the eigenvalues for discrete Schr\"odinger…
The Schr\"odinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an…
We consider scattering matrix for Schr\"odinger-type operators on $\mathbb{R}^d$ with perturbation $V(x)=O(\langle x\rangle^{-1})$ as $|x|\to\infty$. We show that the scattering matrix (with time-independent modifiers) is a…