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

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrödinger-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 e ^it? or a similar object (in our case, this concerns the semigroup e ^t? , which is often referred to in the literature as the Schrödinger 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.In the paper, Feynman formulas for Schrödinger semigroups e ^t? are obtained, where the role of ? is played by the operator ? _a +V which is a perturbation of the self-adjoint extension of the Laplace operator (parametrized by some a ∈ (?∞, ∞]).