The strong convergence of Schrödinger propagators

Alan D. Sloan
1981 Transactions of the American Mathematical Society  
Time dependent versions of the Trotter-Kato theorem are discussed using nonstandard analysis. Both standard and nonstandard results are obtained. In particular, it is shown that if a sequence of generators converges in the strong resolvent topology at each time to a limiting generator and if the sequence of generators and limiting generator uniformly satisfy Kisynski type hypotheses then the corresponding Schrodinger propagators converge strongly. The results are used to analyze time dependent,
more » ... form bounded perturbations of the Laplacian. I. Introduction. The time dependent Schrodinger equation x(s) = x" (I) may be advantageously discussed in terms of unitary propagators. Equation (1) is set in a complex Hubert space, %, and the time variables, t, s, are to range in some closed and possibly infinite interval, J, of real numbers. For each such /, A(t) is a selfadjoint operator on DC while x(t) is an element of % as is xs. A(-) is called the generator of (1). Definition. A jointly strongly continuous map (i, s) -> U(t, s) from J X J into the unitary operators on % is a unitary propagator providing (a) U(t, t) = Z, (b) U(t,s)U(s,r) = U(t,s), hold for all r, s, t in J. Given a generator A the relevant propagator is expected to have the property that x(t) = U(t, s)xs is the "solution" to equation (1). The precise sense in which x(-) is a solution and the exact collection of initial states xs for which such a solution exists remain to be specified later. At times, to avoid ambiguity, we will denote the propagator related to equation (1) by UA. The most satisfactory results are known in case A(t) = A for each / in J where A is some fixed selfadjoint operator. This is the time independent case. The related propagator may be explicitly given as UA(t, s) = e'('~s)A. Then x(t) = U(t, s)xs is differentiable in norm and satisfies equation (1) for all xs in D(A), the domain of A. Once a solution is known to exist, an analysis of stability properties of the equation may begin; this is perturbation theory. The basic question in this theory is: If two generators are close, are the corresponding solutions close? In the time
doi:10.1090/s0002-9947-1981-0603781-x fatcat:2arrb64p2rcbvdo5wsidkfwyde