Fundamental Solutions of the Instationary Schrodinger Difference Operator

TL;DR

This paper investigates the existence and convergence of fundamental solutions for explicit and implicit backward time-dependent Schrödinger equations using discrete Fourier transform methods.

Contribution

It introduces a method to establish the existence of fundamental solutions for discrete Schrödinger operators and proves their convergence to continuous solutions in the l1-norm.

Findings

Discrete fundamental solutions exist for both explicit and implicit schemes.

These solutions converge to the continuous fundamental solution in the l1-norm.

The approach uses discrete Fourier transform and its symbol for the Laplace operator.

Abstract

In this paper we will study the existence of fundamental solutions for the explicit and implicit backward time dependent Schodinger equation, via discrete Fourier transform and its symbol for the Laplace operator. In both cases we will prove that the discrete fundamental solutions obtained converges to the continuous fundamental solution in the $l_1-$norm sense.