Trotter’s limit formula for the Schrödinger equation with singular potential

We discuss the Schrödinger equation with singular potentials. Our focus is non-relativistic Schrödinger operators H with scalar potentials V defined on $R$^d, hence covering such quantum systems as atoms, molecules, and subatomic particles whether free, bound, or localized. By a “singular potential” V, we refer to the case when the corresponding Schrödinger operators H, with their natural minimal domain in L²(⁠$R$^d), are not essentially self-adjoint. Since V is assumed real valued, the corresponding Hermitian symmetric operator H commutes with the conjugation in L²(⁠$R$^d), and so (by von Neumann’s theorem), H has deficiency indices (n, n). The case of singular potentials V refers to when n > 0. Hence, by von Neumann’s theory, we know the full variety of all the self-adjoint extensions. Since the Trotter formula is restricted to the case when n = 0, and here n > 0, two questions arise: (i) existence of the Trotter limit and (ii) the nature of this limit. We answer (i) affirmatively. Our answer to (ii) is that when n > 0, the Trotter limit is a strongly continuous contraction semigroup; so it is not time-reversible.