By S. Albeverio, P. Combe, R. Hscogh-Krohn, G. Rideau, M. Sirugue-Collin, M. Sirugue, R. Stora
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Extra resources for Feynman Path Integrals: Proceedings of the International Colloquium Held in Marseille, May 1978
V ( i . e . ) * × , A~R ( t , . ) * Xe This formula is the analogue of the formula (3~)in Sect. e. 15) is the Banach space norm of the Fresnel space II~fll , where II~fll is the total variation of the bounded complex measure uf whose Fourier transform is f ) . Hence we see that utA'~ ~ is a bounded map of F ( % , ~ s ) ) i n t o i t s e l f , the bound being uniform in We shall now see t h a t the l i m i t as t i e s of Z. 1,J A R ~ ~ 0 e x i s t s . We have from the support proper- that - ~X =i z ~Xj)~j i~A i*j ~i cam (Ti - ~~j, ~ (Ti - Tj, ~i - ~ ) aj , since ~A~ (0,~) = 0 .
14(f) = l(f) when B = is s t r i c t l y e f ( y ) d~. H p o s i t i v e and with range dense in D then i Izl ( f ) = e f (x) d~ . HB In p a r t i c u l a r f o r B ~ 1 i 14(f) = I (f) = e f(y) dy. H Thus the normalized i n t e g r a l w i t h respect to a quadratic form reduces to the normalized i n t e g r a l when the quadratic form is the u n i t one on P 3) I f B-z and is bounded and everywhere defined then A (~, @) = (~, B-z B) v a, BE H. a d i r e c t sum decomposition B+ strictly D = H with e q u i v a l e n t norms Moreover H = H+ @ H_ p o s i t i v e , where of H .
G. [~2]- [~9]). 11) one has, for some sequence of f i n i t e dimensional projections V2 ~ Is - Pn ~I d Ip W' (~) ~ f d IPWI (~) < ~2, f e H- I PnH hence Thus in Pn converging strongly to 1, that e H - PnH H = Hz ¢ H2 , l~-Pn~l ~2 ~ d I~W I (~) %2 < for n sufficiently large. one can apply Theor. 12) big. 14) and where Y2 ) = Y2 , d. ~ d a Yj Consider now the phase ~(~y2) as a f u n c t i o n on the H i l b e r t space Hz, of f i n i t e codimension, f o r f i x e d Y2E H2 . 13). Y2 Now PW is the measure whose Fourier transform is W(.
Feynman Path Integrals: Proceedings of the International Colloquium Held in Marseille, May 1978 by S. Albeverio, P. Combe, R. Hscogh-Krohn, G. Rideau, M. Sirugue-Collin, M. Sirugue, R. Stora