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A Jensen

There is a large literature on the existence and completeness of wave operators for Schrodinger operators with time-dependent potentials, see for example [4, 6, 9, 10, 15], and references therein We introduce the notation

W { s , t ) =U { s , t ) Uo { t - s ) (14)

for all s, teWi The Fourier transform of V{t, x) with respect to the x-variable is denoted K(r, i) The bounded operators on L^iWi^) are denoted J'(L^) The finite complex regular measures on IR'^ are denoted 9Jl(IR^) The total variation norm on ä«(lR^) IS denoted ||-|Iîr(r^)

Assumption 1.1. V{t, x) is a realvalued function such that УеЬ^{Ж,Ш{^^))

Under this assumption FeL^(IR,L"°(lR'^)) and we can apply the results in [16, 18] to conclude that there exists a unique unitary propagator associated with the problem (1 2) Our main results are stated in the following theorem

Theorem 1.2. Let V satisfy Assumption 1 1 (i) Let seЖ The limits

W±is ) = hm W{s,t)

f - > ± 00

exist in operator norm in ^{L^) and are unitary

( ii ) The operators WjJ) extend to bounded operators on L^{^% 1 ^ P ^ oo One has sup^gR ||H^+(s)L(lp) < oo for each p The operators W+{s) are invertible in ^L^)

( ill ) For each s, telR the operator W{s,t) extends to a bounded operator on L^(R^), 1 ^ p ^ 00 We have

W^ { s ) = hm W{s,t)

f - * ±00

in operator norm in 0^{L^)

The proof of Theorem 1 2 is given in Sect 2 It turns out to be quite simple However, it seems to be the first result on L^-boundedness of the wave operators for Schrodinger operators with genuinely time-dependent potentials Previous results on scattering theory in a Banach space for a pair of operators (Я, Hq) have assumed that H and Hq generate (semi-) groups See for example [3, 11, 12] This excludes the free Schrodinger equation, since l/o(0 is known to be unbounded on LP{Wi% p Ф 2, or equivalently, the problem (1 1) is not well-posed in L^^ß.^ p^l Here we use the observation that W{s,t) given by (14) extends to a bounded operator on L^(IR'*) and satisfies an integral equation (see (2 9)) This equation has previously been used in [2, Theorem 2 16]

Even though Voit) is unbounded on the L^-spaces (p + 2), it has nice mapping properties from a Besov space which is a subspace of L^, to L^ In Sect 3 we use the result on the wave operators to give an analogous mapping property of the propagator U{t, s) in L^'-spaces Such results are based on the intertwining property

V { t , s ) W^ { s ) =WSWo { t - s )

Thus Theorem 1 2 can be used to transfer results on the free propagator in L^-spaces to properties of U{t, s) in the same or analogous spaces As another