Convergence Rates for the Trotter Splitting for Unbounded Operators · arXivDesk
2407.04045Jul 4, 2024comments welcome; v2: changed the title to that of the published version (previous title: Convergence rates for the Trotter-Kato splitting)
Convergence Rates for the Trotter Splitting for Unbounded Operators
Simon Becker, Niklas Galke, Robert Salzmann, Lauritz van Luijk
We study convergence rates of the Trotter splitting eA+L=limn→∞(eL/neA/n)n
Nearby in the stack
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 Schrödinger 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 Schrödinger operator with
V(x)=±∣x∣−a
potential. In each case, our conditions are fully explicit.