On the scattering operator for the Schrödinger equation with a time-dependent potential

Abstract

In this paper we give some results on the scattering operator for the Schrodinger equation with a time-dependent potential. We consider the free Schrodinger equa-tion $$ i{\partial _t}u(t,x) = - {\Delta _x}u(t,x),{\text{ }}u(s,x) = {u_o}(x)$$ (1) and the full schrodinger equation. $$ i{\partial _t}v(t,v) = - \Delta v(t,x) + V(t,x)v(t,x),{\text{ }}v(s,x) = vo(x)$$ (2) Here V is a potential depending explicitly on time. The solution to (1) is given by \(u(t) = {e^{ - i(t - s){H_o}}}{u_o},Where({H_O}) = - \Delta x\) with the domain being the usual Sobolev space of order 2, D(Ho) = H 2(R d). If we assume V(t, x) a real-valued function, such that V ∈ L l(R; L ∞(R d)), then associated with (2) is a unitary propagator on L 2(R d), denoted by U(t, s), such that the solution to (2) is given by υ(t) = U(t, s)υ o, see for example [8, 9] and references therein. More precisely, υ(t) solves the equation in the sense that v satisfies the integral equation $$ v(t) = Uo(t - s)vo - i\int_s^t {Uo(t - \tau )v(\tau )} d\tau $$ (3) i.e. v is a mild solution to the Cauchy problem (2). The propagator satisfies U(t, t) = 1 and U(t, s)U(s, r) = U(t, r) for all t, s,r ∈ R. Furthermore,(t,s)→U(t,s)is strongly continuous.