A General Trotter-Kato Formula for a Class of Evolution Operators with Examples

International audienceIn this article we prove new results concerning the existence and various properties of an evolution system UA+B(t; s)0 <= s <= t <= T generated by the sum -(A(t) + B(t)) of two linear, time-dependent and generally unbounded operators dened on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t; s)0 <= s <= t <= T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D \incl t\in[0;T ] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct UA(t; s)0 <= s <= t <= T very simply in terms of the semigroup generated by -A(t). We then illustrate our results by considering various examples of non-autonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturba- tions of self-adjoint operators. We nally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics