A Trotter–Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t,s)0≤s≤t≤TUA+B(t,s)0≤s≤t≤T generated by the sum −(A(t)+B(t))−(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space BB. In particular, writing L(B)L(B) for the algebra of all linear bounded operators on BB, we can express UA+B(t,s)0≤s≤t≤TUA+B(t,s)0≤s≤t≤T as the strong limit in L(B)L(B) of a product of the holomorphic contraction semigroups generated by −A(t)−A(t) and −B(t)−B(t), thereby getting a product formula of the Trotter–Kato type under very general conditions which allow the domain D(A(t)+B(t))D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂∩t∈[0,T]D(A(t)+B(t))D⊂∩t∈[0,T]D(A(t)+B(t)) everywhere dense in BB. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrödinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication.