Quasi-Feynman formulas -- a method of obtaining the evolution operator for the Schroedinger equation

TL;DR

This paper introduces a novel method to represent the Schrödinger evolution operator using Chernoff-tangent families of bounded operators, broadening the tools available for quantum dynamics analysis.

Contribution

It develops a general framework expressing the evolution operator via Chernoff-tangent families, extending previous approaches and allowing for more flexible operator choices.

Findings

Provides a new representation of the evolution operator in terms of bounded operators.

Proves the main theorem within the context of semigroups of bounded operators.

Includes two practical examples demonstrating the method's application.

Abstract

For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if $\psi(0,x)=\psi_0(x)$ then $\psi(t,x)=(\exp(-it\mathcal{H})\psi_0)(x)$ for $x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$ defined on an abstract space $Q$, the configuration space of a quantum system, and $\mathcal{H}$ is the Hamiltonian of the system. In this paper the operator $\exp(-it\mathcal{H})$ for all real values of $t$ is expressed in terms of the family of self-adjoint bounded operators $S(t), t\geq 0$, which is Chernoff-tangent to the operator $-\mathcal{H}$. One can take $S(t)=\exp(-t\mathcal{H})$, or use other, simple families $S$ that are listed in the paper. The main theorem is proven on the level of semigroups of bounded operators in $\mathcal{F}$ so it can be used in a wider context due to its generality. Two examples of application are provided.