Schrodinger semigroups for functional spaces on ramified Riemannian manifolds

One discusses the Feynman approximations for some Schrodinger semigroups, generated by extensions of differential operators (in particular, by self-adjoint extensions) in spaces of functions on ramified Riemannian manifolds. Such semigroups represent the solutions to heat-type evolution equations for corresponding differential operators. Each extension of differential operator is defined by proper boundary and matching conditions. The Feynman formulas are the representations of some objects, related to differential or pseudodifferential operators, (usually the solutions to some evolution equations) by the limit of integrals over Cartesian powers of come space when the power tends to infinity. The integrals over Cartesian powers from Feynman formula are called the Feynman approximations to corresponding object.