Feynman formulas generated by self-adjoint extensions of the Laplace operator

Abstract

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrodinger-type equation (in a special case, for a heat-type equation) using the limit of integrals of finite multiplicity over Cartesian powers of the phase space (in the special case of the configuration space). The limit thus obtained, defining an explicit representation of a one-parameter unitary group eitĤ or a similar object (in our case, this concerns the semigroup etĤ, which is often referred to in the literature as the Schrodinger semigroup) by integral operators, is interpreted by using Feynman integrals, whereas the expression thus obtained is referred in turn as the Feynman formula. As a rule, the Chernoff theorem, which is a generalization of the well known Trotter formula, is used in the derivation of the Feynman formula.