相关论文: Dynamical Green Functions and Discrete Schr\"oding…
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an…
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…
The central role played by pseudodifferential operators in relativistic dynamics is very well know. In this work, operators as the Schrodinger one (e.g: square root) are treated from the point of view of the non-local pseudodifferential…
In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators.…
We discuss 1-dimensional Schrodinger operators with complex and locally integrable potentials that may have an arbitrary behavior at (finite or infinite) endpoints. The main tool of our analysis are Green's operators, that is, their various…
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…
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…
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term…
In this short note we review some facts about elliptic differential operators on Riemannian manifolds.
A Schr\"odinger equation may be transformed by unitary operators into dynamical equations in different interaction pictures which share with it a common physical frame, i.e., the same underlying interactions, processes and dynamics. In…
We investigate discretizations of the integrable discrete nonlinear Schr\"odinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary…
We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq…
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.
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…
A geometrical interpretation of Schr\"odinger's kinetic and potential energy operators is proposed, allowing for a covariant momentum space formulation of the dynamics that is relevant for the theories with the deformation of the momentum…
We investigate the spectral properties of self-adjoint Schr\"odinger operators with attractive $\delta$-interactions of constant strength $\alpha > 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum…
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…
Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schr\"{o}dinger…
The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is represented by a conserved surface layer integral.…
We study Schr\"odinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the…