Trotter-Kato product formula for unitary groups

TL;DR

This paper proves the Trotter-Kato product formula for unitary groups with imaginary times in the L^2-norm, extending the classical results and providing a canonical form for associated holomorphic Kato functions.

Contribution

It establishes the validity of the Trotter-Kato product formula for imaginary times in the L^2-norm and introduces a canonical representation for holomorphic Kato functions.

Findings

Trotter product formula holds for imaginary times in L^2-norm.

Extension to Trotter-Kato product formula with holomorphic Kato functions.

Provides a canonical representation for holomorphic Kato functions.

Abstract

Let $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm, that is, one has % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(e^{-itA/n}e^{-itB/n})^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % for any element $h$ of the Hilbert space and any $T > 0$. The result remains true for the Trotter-Kato product formula % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(f(itA/n)g(itB/n))^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % where $f(\cdot)$ and $g(\cdot)$ are so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.