Remarks on the Trotter-Kato Product Formula for Unitary Groups
Résumé
Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L 2-norm, that is, one has limn→+∞∫−TT∥∥(e−itA/ne−itB/n)nh−e−itCh∥∥2dt=0 for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: ϕ(⋅),ψ(⋅):R+⟶C , where R+:=[0,∞) , satisfying in addition Re(ϕ(y))≥0,Jm(ϕ(y)≤0 and Jm(ψ(y))≤0 for y∈R+ , we prove that \rm s-limn→∞(ϕ(tA/n)ψ(tB/n))n=e−itC holds true uniformly on [0,T]∋t for any T > 0.
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