By Richard B Melrose
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Extra resources for Geometric scattering theory
21 e i t AH ( ! ) (t) ! 9) H z R = supp(BH ) t 2R : 0 R ; 0 ; ; ; C fj j g 0 ) ;1 ; f ; ; g The scattering operator is the operator on R Sn 1 de ned by BH or (t) ! ( ) + BH acting as a convolution operators in the t variable. The most immediate invariants of such an operator are the singularities of the kernel. 10) sing supp(BV ) t = 0 = ! : In fact more is true. Not only is the singular support of BV contained ; 0 0 f g 0 cuto as in Footnote 8 it follows that v1 ; v2 has 2compact support. By unique continuation for elements of the null space of ; it follows that v1 ; v2 must vanish on the complement of O1 O2 and hence on its boundary.
Fig. 6. 4, I have already mentioned that the Lax-Phillips transform, for odd dimensions, gives an isomorphism of the nite energy space HFE (Rn) to L2 (R Sn 1) and intertwines the free wave group and the translation group. 33). 29. 2 ; ; 2 ; ; 8 That is, restriction back to ;R R] Sn;1: 9 Existence of which is guaranteed by the Hille-Yosida theorem. 19) uses this result. For t > 2R the Lax-Phillips semigroup Z(t) is smoothing, and hence trace class as an operator on L2 ( R R] Sn 1): Its trace is V (t) and its non-zero eigenvalues are exp(i t) for D(V ): In fact, for T > 0 a subspace of L2 ( R R] n 1 S ) can be constructed on which the operator ZV (t) has, for t > T the same eigenvalues as it has on L2 ( R R] Sn 1) and on which it is of trace class.
This shows that there must be at least one pole of the analytic continuation of the resolvent, so D(V ) is not empty. Suppose D(V ) were nite. 19) would be a nite sum. e. e. 12) the coe cient of t0 vanishes in odd dimensions. Thus D(V ) cannot be both nite and non-empty. 4 Lax-Phillips semigroup t . .. . ;R . . . . . ............. R s .. ..... . ! Fig. 6. 4, I have already mentioned that the Lax-Phillips transform, for odd dimensions, gives an isomorphism of the nite energy space HFE (Rn) to L2 (R Sn 1) and intertwines the free wave group and the translation group.
Geometric scattering theory by Richard B Melrose
Categories: Algebraic Geometry