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 $\mathcal{L}(\mathcal{H})$, 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.