Trotter-Kato product formula for unitary groups

2009/07/07 by Pavel Exner, Hagen Neidhardt, Exner, Pavel +1
Mathematics · #32A40 #47A55 #47B25 #47D03 #81Q30 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0907.1199

openalex publication_date 2009/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L²-norm, that is, one has % % lim_n→+∞∫^T₀ ‖(e^-itA/ne^-itB/n)ⁿh - e^-itCh‖²dt = 0 % % for any element h of the Hilbert space and any T > 0. The result remains true for the Trotter-Kato product formula % % lim_n→+∞∫^T₀ ‖(f(itA/n)g(itB/n))ⁿh - e^-itCh‖²dt = 0 % % where f(⋅) and g(⋅) are so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.

Citations

• On the product of semi-groups of operators

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