Product approximations for solutions to a class of evolution equations in Hilbert space

Abstract

In this article we prove approximation formulae for a class of unitary evolution operators U(t,s)_{s,t\in [0,T] } associated with linear non-autonomous evolution equations of Schr\"{o}dinger type defined in a Hilbert space \mathcal{H} . An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for U(t,s) involve weak operator limits of products of suitable approximating functions taking values in \mathcal{L(H)} , the algebra of all linear bounded operators on \mathcal{H} . Our results may be relevant to the numerical analysis of U(t,s) and we illustrate them by considering two typical examples, including one related to the theory of time-dependent singular perturbations of self-adjoint operators.