Formulas that represent Cauchy problem solution for momentum and position Schr\"{o}dinger equation

TL;DR

This paper derives explicit formulas for solutions to specific Schrödinger equations in momentum and position space, using functional analysis and operator approximation methods.

Contribution

It introduces formulas for solving Cauchy problems of Schrödinger equations with polynomial and locally square integrable potentials, expanding solution techniques.

Findings

Derived formulas for Schrödinger equation solutions.

Constructed translation operators approximating solution semigroups.

Applied Chernoff product formula for functional analysis approach.

Abstract

In the paper we derive two formulas representing solutions of Cauchy problem for two Schr\"{o}dinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.