Further Development of Trotter Product Formulas with Problems on Path Integrals

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In two papers in Commun.Math.Phys. 2001, Ichinose, with Hideo Tamura, Hiroshi Tamura and V.A.Zagrebnov, proved norm convergence of Trotter product formula for the operator sum of two nonnegative selfadjoint operators with optimal error bound. The present research has been done to go beyond this result, keeping in mind its relation to path integral and examples/applications of the theory of Schrodinger operators, and brought the following results.(1)Further development of Trotter product formula in norm-It is known that the norm convergence of Trotter product formula does not hold in general for the form sum of two selfadjoint operators. However, it was proved in J. Functional Analysis 2004 with some condition by Ichinose together with H. Neidhardt and V.A. Zagrebnov. It is an open problem whether the condition may be relaxed. To be unexpected,Ichinose and Hideo Tamura also discovered, in Lett. Math. Phys. 2004, norm convergence of the unitary product formula to hold for the Dirac and the relativistic SchrOdinger operator with nontrivial scalar potentials.(2)Problems of path integral and of convergence of integral kernels of the Trotter product: Fujiwara succeeded to improve the error estimate of the stationary phase method of the oscillatory integral in large dimensions, and used it to determine the second term of the semiclassical approximation to Feynman path integral. In this connection, Trotter product formula is thought to give a kind of time-sliced approximation. The good norm-convergence of Trotter product formula might suggest convergence of the integral kernels, as Ichinose and Hideo Tamura anticipated. In fact, we proved it 2004 in two papers in Commn. PDE and J. Reine Angew. Math.(3)Zeno product formula: Ichinose proved 2005 an intermediate result with P.Exner.