Results on Convergence in Norm of Exponential Product Formulas and Pointwise of the Corresponding Integral Kernels

TL;DR

This paper reviews the convergence properties, both in norm and pointwise, of exponential product formulas for Schrödinger operators, including error bounds and their optimality, over the past decade and a half.

Contribution

It provides a comprehensive review of norm and pointwise convergence results for exponential product formulas in Schrödinger operators, with detailed error bounds and their optimality.

Findings

Exponential product formulas converge in norm for Schrödinger operators.

Pointwise convergence of integral kernels is established.

Error bounds for convergence are derived and shown to be optimal.

Abstract

For the last one and a half decades it has been known that the exponential product formula holds also {\it in norm} in nontrivial cases. In this note, we review the results on its convergence in norm as well as pointwise of the integral kernels in the case for Schr\"odinger operators, with error bounds. Optimality of the error bounds is elaborated.