Transmutation Theory and Applications (Notas De Matematica, by Robert Carroll

By Robert Carroll

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J < m. 8. e. J) Then t h e measure 28 ROBERT CARROLL - REMARK 5-10. One can transmute P = D2 t r a n s m u t a t i o n s B1: D2 + Q and B2 = Bil: q1 i n t o Q = D2 P -+ - q2 by composing D2 (one can a l s o c o n s t r u c t t h e t r a n s m u t a t i o n d i r e c t l y which w i l l be discussed l a t e r f o r more general op1 2 e r a t o r s ) . g. i f B, % I + Kh and B2 % I + Lk one o b t a i n s f o r B = B1B2: P -+ Q L a t e r i n Chapter 2 we w i l l study t h e f o r m a t i o n o f s p e c t r a l p a i r i n g s t o repr e s e n t such t r a n s m u t a t i o n k e r n e l s K(y,<).

20) i s v e r y general, a l t h o u g h f i s r e s t r i c t e d . We The po- t e n t i a l q may be complex valued and R 4 i s n o t n e c e s s a r i l y a measure ( c f . The d i s t r i b u t i o n a c t i o n o f RQ i n v o l v e s X r e a l [C40; Mc4] f o r examples). g e n e r a l l y and one p i c k s up complex eigenvalues f o r example i n t h e sense o f c o n t o u r i n t e g r a t i o n ( c f . [C40]). 20) can be ~il ( RQ,F(i)lpQ ( x ) ) A,h I f now q and h a r e r e a l one has a s e l f a d j o i n t s i t u a t i o n w i t h $ ‘ PQ ~ , ~ ( X ) Hence .

G i n o r countable u n i o n t o p o l o g i e s ( c f . [C19; G f l ] ) . 2 Z (resp. CK ) i f t h e e x p o n e n t i a l t y p e o f a l l gn i s bounded by some u and gn 2 g i n L1 ( r e s p . L ); such s e q u e n t i a l convergence i s a l l we need c o n s i d e r 2 2 (as for U - c f . 3). E v i d e n t l y Z C CK and i f g1,g2 E CK t h e n g = -f g 1g 2 E Z ( i n f a c t t h e v e c t o r space o f such p r o d u c t s forms a dense s e t i n Z ) . 2, The dual Z ' o f Z i s a space o f g e n e r a l i z e d f u n c t i o n s ( i n which t h e so c a l l e d g e n e r a l i z e d s p e c t r a l f u n c t i o n R w i l l be found) w i t h act i o n on Z denoted by ( R , v ) o r ( R,V)~.

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