The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.
Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.
A. G. Tokarev w Definitions and terminology Throughout the paper N = {1, 2, ... } ; R ~ is the countable product of lines; R (~176 , the vector space 9 OO of finite real number sequences; {e,}i=l , the standard basis in R(~176 and Iixl[ = [zll-~-.---~-I~gn[, the norm in R n , n E N. We denote by Jm the m x m matrix whose entries under the main diagonal are ones and all the rest are zeros; by jT the transpose of J,n ; by L1 ~ L2, the sum of locally convex spaces LI and L2, and by L1 • L2, their product.If E is a locally convex space (LCS), then s
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