Feynman integral for functional Schrödinger equations

We consider functional Schrödinger equations associated with a wide class of Hamiltonians in all Fock representations of the bosonic canonical commutation relations, in particular the Cook-Fock, Friedrichs-Fock, and Bargmann-Fock models. An infinite-dimensional symbolic calculus allows to prove the convergence of the corresponding Hamiltonian Feynman integrals for propagators of coherent states. Dedication The roots of my appreciation of M. I. Vishik and his PDE Seminar go back to the golden 1960’s of Moscow mathematics (cf. [33]). In the Spring of 1960 A. Volpert gave a lecture at the Moscow Gelfand seminar about his index formula of elliptic boundary problems for PDE systems in two variables. His approach was based on the Muskhelishvili index formula of onedimensional singular integral operators. I. M. Gelfand interpreted the Volpert index formula in terms of characteristic cohomological classes and suggested to generalize it to elliptic boundary problems in higher dimensions and on compact manifolds using a homotopy of the elliptic data. The Vishik seminar began in the Spring of 1961 with a response to the Gelfand challenge. It started with a review of still recent Calderon-Zygmund algebra of multivariable singular integral operators and their applications to the Cauchy problem for elliptic PDE’s. Soon the Calderon-Zygmund algebra was extended to the algebra of singular integro-differential operators (see [10]). The elliptic homotopy of the extended algebra is much simpler because its symbols are continuous rather than polynomial in cotangent directions. Indeed, this extension was essential for the first general solution of the Gelfand index problem, given by M. Atiyah and I. Singer [1]. 1991 Mathematics Subject Classification. Primary 58D30 ; Secondary 7G30, 81Q05.