Eigenfunction expansions and scattering theory for perturbed elliptic partial differential operators

Abstract

1. A number of papers discussing the spectral decomposition and eigenfunction expansion for partial differential operators appeared in the last few years. Browder [l], [2], [3], [4], Gârding [5] and Mautner [12 ] proved the existence of an abstract eigenfunction expansion for elliptic partial differential operators. In 1953 A. Ya. Povzner [13] considered the detailed spectral decomposition of — A-\-q(x). This was completed by T. Ikebe [6] who used the theory of wave operators as developed by Kato [8] and Kuroda [10], [ll]. In this note we investigate an eigenfunction expansion for the operator P(D)+q(x) where P(D) is a linear homogeneous elliptic partial differential operator with constant coefficients. Detailed proofs of the results will appear elsewhere. 2. The Euclidean w-space will be denoted by Rn or Mn with ele­ ments x=(xu • - • , xn) or k=(ki> • • • , kn) respectively. ff(x)dx de­ notes integration with respect to Lebesgue measure. We set d — Dj = for 1 ^ j S nidxj Let P(x) be a homogeneous elliptic polynomial, i.e. P(x)jâqx| 2p where 2p is the order of P(x). Then P(D) =P(Dh • • • , Dn) is a linear homogeneous elliptic partial differential operator. All through this note we assume that 4p>n. It is well known that P(D) can be ex­ tended to a selfadjoint operator P(D) in Lz(Rn). Let g(x)GC2[„/2] with q{x) — 0{\x\~n~h) for some h>0. Then by Theorem 1 of [ll], P(D)+q(x) is a selfadjoint operator in L<i(Rn). Let {Et} andjP*}, — oo<£< + oo,be the resolutions of the identity for P(D) and P(D) +q(x) respectively. Define