Abstract. Spectral and scattering theory is discussed for the Schrodinger operators H = -A + V and H0 = -A when the potential V is central and may be rapidly oscillating and unbounded. A spectral representation for H is obtained along with the spectral properties of H. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on V is derived which is both necessary and sufficient for the M/aller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.1. Introduction. We shall study the Schrodinger operator H --A + V on L2(Rd ) (d > 2) when the potential V is central, oscillating or perhaps rapidly oscillating, in which case V may be unbounded. A spectral representation (or "eigenfunction expansion") for H will be derived along with the "expected" spectral properties. We shall further prove the existence and completeness of the modified wave operators for the selfadjoint operators, H and H0 = -A and derive a necessary and sufficient condition on V to assure the existence and completeness of the Mailer (i.e. " usual") wave operators.To place this present work in perspective, we shall briefly review some of the relevant results from scattering theory. For potentials V which are short range, i.e., roughly V(x) -0{\ x |~'~E) as | x |-> oo, for some e > 0, Agmon [1] [23, 44,48,53]. On the other hand, Dollard [17, 18] has shown that these wave operators do not exist when V is the Coulomb

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An example of a bounded self adjoint operator A is constructed so that A⊗I + α(I⊗A) is purely singularly continuous but A⊗1 + β(I⊗A) is purely absolutely continuous, for some real α and β. In fact α - β can be chosen arbitrarily small.

ABSTRACT. Existence and completeness of wave operators is established by a straightforward transposition of the original short range result of Enss into an appropriate two-Hilbert space setting. Applied to long range quantum mechanical potential scattering, this result in conjunction with recent work of Isozaki and Kitada reduces the problem of proving existence and completeness of wave operators to that of approximating solutions of certain partial differential equations on cones in phase space. As an application existence and completeness of wave operators is established for Schrödinger operators with a long range multiplicative and possibly rapidly oscillating potential.1. Introduction. The geometric time-dependent methods of Enss now play a fundamental role in the study of quantum mechanical potential scattering theory. However, until recently the success of this method has been most dramatic in the treatment of short range interactions. All this has been changed by the recent work of Isozaki and Kitada [20]. Their approach, using "time-independent modifiers" allows them to apply Enss's arguments to prove existence and completeness of wave operators for Schrödinger operators with long range potentials.In the present paper we transpose the usual short range result of Enss into an appropriate two-Hilbert space setting. Applied to the study of long range scattering, in conjunction with (our version of) Isozaki and Kitada's "modifiers", this result reduces the problem of proving existence and completeness of wave operators to that of approximating solutions of certain partial differential equations on cones in phase space. In the context of Isozaki and Kitada's result [20] this equation is an eikonal equation and we use their solution to give an alternate proof of their result. Then we extend this result to allow more general potentials.The following elementary argument will elucidate the problems to be encountered below as well as the relationship between this work and [20]. Let Ho and H be selfadjoint operators on L2(R"), Ho = -^A and H is a long range perturbation of Ho'-H = Ho + V. The usual approach to scattering theory for H and Ho is to introduce the (Dollard) modified wave operators