Regularization of Schrödinger groups and semigroups (Q1761059)

A Schrödinger semigroup is a semigroup of bounded operators acting on a real or complex Hilbert space and generated by a self-adjoint operator. A Schrödinger group is a group of unitary operators on a complex Hilbert space whose generator is the product of a self-adjoint operator and the imaginary unit. In both cases, the generating self-adjoint operator is called a Hamiltonian. This paper considers Schrödinger groups and semigroups generated by self-adjoint extensions of second-order differential operators on spaces of square integrable functions defined on the real line or half line. In the case of the half line, it is assumed that the coefficient of the leading derivative is constant, while in the case of the entire real line, it is assumed that this coefficient depends on the spatial coordinate. The authors show that the Schrödinger semigroup and group generated by any self-adjoint extension of the corresponding differential operator on the space of functions on the half-line can be obtained by applying appropriate limit processes to the semigroups and groups generated by some differential operators on the space of functions defined on the entire real line. It follows, in particular, that diffusion on the half line with elastic reflection on the boundary is the limit of diffusions on the entire real line with appropriate drift and absorption parameters. They also show that, in the case of a jump-like dependence of the coefficient of the leading derivative in the differential operator on the coordinate, not all self-adjoint extensions generate Schrödinger semigroups obtained by limit processes from semigroups generated by symmetrizations of operators with smoothed coefficients. Among all self-adjoint extensions of operators with discontinuous (or vanishing on some sets) coefficients they distinguish extensions generating those semigroups which can be obtained by using such limit processes.