کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6415269 | 1335093 | 2009 | 45 صفحه PDF | دانلود رایگان |

In 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 â(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 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 âA(t) and âB(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 Dââtâ[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 âA(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.
Journal: Journal of Functional Analysis - Volume 257, Issue 7, 1 October 2009, Pages 2246-2290