Representation of solutions of the Cauchy problem for a one-dimensional Schrödinger equation with a smooth bounded potential by quasi-Feynman formulae

Abstract

We consider the Cauchy problem for a Schrödinger equation whose Hamiltonian is the difference of the operator of multiplication by the potential and the operator of taking the second derivative. Here the potential is a real differentiable function of a real variable such that this function and its derivative are bounded. This equation has been studied since the advent of quantum mechanics and is still a good model case for various methods of solving partial differential equations. We find solutions of the Cauchy problem in the form of quasi-Feynman formulae by using Remizov’s theorem. Quasi-Feynman formulae are relatives of Feynman formulae containing multiple integrals of infinite multiplicity. Their proof is easier than that of Feynman formulae but they give longer expressions for the solutions. We provide detailed proofs of all theorems and deliberately restrict the spectrum of our results to the domain of classical mathematical analysis and elements of real analysis trying to avoid general methods of functional analysis. As a result, the paper is long but accessible to readers who are not experts in the field of functional analysis.