Trotter-Kato product formula for unitary groups
Pavel Exner, Hagen Neidhardt

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 and be non-negative self-adjoint operators in a separable Hilbert space such that its form sum is densely defined. It is shown that the Trotter product formula holds for imaginary times in the -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 of the Hilbert space and any . 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 and are so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.
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Taxonomy
TopicsSpectral Theory in Mathematical Physics · Advanced Algebra and Geometry · Advanced Operator Algebra Research
