Completeness of the Wave Operators for Scattering Problems of Classical Physics by John R. Schulenberger and Calvin

Many wave propagation phenomena of classical physics are governed by equations of the Schrödinger form iDtu *=Aw where (1) A iEtà-i't* AêDj. (E(x) and the Aj are Hermitian matrices, E(x) is positive definite and the Aj are constant.) The time-evolution of such phenomena is described by a group of unitary operators exp(-itA) on the Hilbert space 3C with the energy norm (2) \\u\\l f u(x)*E(x)u(x) dx. J R If E(x) is replaced by a constant Eo the corresponding space and operator are denoted by Ho and Ao. In this paper it is shown that the wave operators (3) W±(A, Ao, J) s-lim exp(*7A)7 exp(-#A0)Pΰ «-•±«0 exist and are complete if A.o(p) ^-E^ HCj-i Ajpj satisfies (4) rank AQ(p) =m-k for all p £ R n {0}, E(x) and DjE(x) are continuous and bounded (j = l, 2, • • • , w), E(x) is uniformly positive definite, \im\x\+„E(x)=EQ uniformly in x/\x\ and (5) f (1 + | x |)" | E(x) Eo \*dx < oo for some M > »/2. J R (In (3), J:Ho—>H is the identification map; Ju — u, and PQ is the orthogonal projection onto the absolutely continuous subspace for Ao. W± are complete if their ranges equal H*°, the absolutely continuous subspace for A.) 1. Wave operators. An abstract theory of scattering with two AMS 1970 subject classifications. Primary 35P25, 47A40; Secondary 73D99, 76Q05, 78A45.