Product Approximations for Solutions to Time-Dependent Schrödinger Evolution Equations in Hilbert Space
by Pierre-A. Vuillermot (Institut Élie Cartan de Nancy)
In this talk we discuss approximation formulae for a class of unitary operators U(t,s) associated with linear non-autonomous evolution equations of Schrödinger type defined in a Hilbert space 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 equations we are interested in satisfy the equations in a very 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 L(H), the algebra of all linear bounded operators on H. Our results may be relevant to the numerical analysis of U(t,s) and we illustrate them with two evolution problems in quantum mechanics.