Convergence Rates for the Trotter Splitting for Unbounded Operators

TL;DR

This paper investigates the convergence rates of the Trotter splitting method for unbounded operators, providing explicit conditions and results applicable to various Schrödinger and Dirac operators, including singular and many-body Hamiltonians.

Contribution

It offers a comprehensive analysis of Trotter splitting convergence rates for unbounded operators on Banach and Hilbert spaces, using complex interpolation and energy constraints.

Findings

Derived explicit convergence rates for Schrödinger operators with singular potentials.

Established convergence rates for a broad class of quantum Hamiltonians.

Provided new techniques applicable to unitary dynamics and unbounded operators.

Abstract

We study convergence rates of the Trotter splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schr\"odinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schr\"odinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.