ISSN:
0170-4214
Keywords:
Mathematics and Statistics
;
Applied Mathematics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
Notes:
Given self-adjoint operators Hj, on Hilbert spaces Hj, j = 0,l, and J ∊ B (H0, H1) (where B (H0 H1) denotes the set of bounded linear operators from H0to H1), define the wave operators where P0 is the projection onto the subspace for absolute continuity for H0. We use (i) to study the scattering problem associated with a pair of equations each of the form where L is a positive, self-adjoint operator on a Hilbert space X, m is a positive integer and the αj are distinct positive constants. Methods patterned after those of Kato are used to study two equations (that is for L = L0 and L = Ll) each of the form (ii). We show that they are equivalent to equations of the form where each Ĥk is a self-adjoint operator on an associated Hilbert space Hk. Now suppose∼he-wave operators W±,(L1 L0) exist and are complete. Then we can find a J ∊ B(H1 H0) such that W+(Ĥl, Ĥ0,J) exists. In the case where Lo and L1 have the same domain, H1 and H0 are equal as vector spaces, and under certain conditions (on Li, i = 0, 1) H0 and H1 have equivalent norms. Assuming these conditions, let J'∊ B(H1' H0) be the identity map. We show that (with an additional assumption on L0 and L1) W+(Ĥ1Ĥ0,J) exists andisequal to W+(Ĥl,Ĥ0, J).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/mma.1670150106
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