A stationary approach to the existence and completeness of long-range wave operators

On solutions of the Schrödinger equation with radiation conditions at infinity: the long-range case

“…Note that functions a± and u in this formulation are related by the equalities Similarly to the short-range case [15], Theorem 5.2 can be reformulated in terms of solutions of equation (1.1) with asymptotics (1.6), (1.7) at infinity but we shall not dwell upon it here. Recall that the function ^:'^(A)a satisfies (3.18) and as shown in [7] the contribution to (5.6) of the term R{\ =p ^0)^±(A) disappears in the limit t -)• ±00. Taking into account that the function n±(rf, A) is given by (3.14) we obtain …”

Time Dependent Representations of the Stationary Wave Operators for 'Oscillating' Long-Range Potentials

“…As a corollary of Lemma 3.2 we can now prove the following Based on Propositions 2.1, 3.1 and 4.1, we can now follow the idea employed in Kitada [13], Ikebe-Isozaki [10] and Kako [11], where is treated the case of "non-oscillating" long-range potentials, to prove the above theorem. …”

“…Since the original paper of Bollard [7], the long-range scattering theory for the Schrodinger operators -A + V(x) has been studied by many authors (e.g., Buslaev-Matveev [5], Amrein-Martin-Misra [2], Alsholm-Kato [1], Hormander [9], Kitada [13], Ikebe-Isozaki [10] and Kako [11]). These works treat the case that the potential V(x) approaches zero without too much oscillation at infinity: (0.1) P«7(x) = (Kr-l«l-*)(|a|=0, 1, 2,...) for some 5>Q…”

“…(F=F* is the gradient in R", r=|x| and a = (a 1 ,..., a n ) are multi-indices with |a|=a 1 + ---+a n ), and prove the existence ( [1], [2], [5], [9], [11]) and the completeness ( [10], [13]) of the modified wave operators As we see in [10] and [13], Y+(x, A) can be obtained from X + (£, f) by a kind of Legendre transformation in classical mechanics and are used to establish the completeness of W%. On the other hand, Y ± (x 9 A) are directly used in [11] to obtain another formulation of the modified wave operators.…”

Asymptotic completeness in long range scattering. II

“…Various spectral properties have been obtained by Lavine [7], Ikebe [8] and Saito [9]. The completeness problem for a large class of long range potentials was studied by Weidmann [10] and Georgescu [11] for the spherically symmetric case and by Ikebe and Isozaki [12] and Kitada [13] for more general situation. Also Agmon [14] has given a proof of completeness using eigenfunction expansion method.…”