Spectral Representations for Schrdinger Operators with by Yoshimi Saito

By Yoshimi Saito

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Extra resources for Spectral Representations for Schrdinger Operators with Long-Range Potentials by Yoshimi Saito

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Z(y) = Z ( y , k ) of y on of t h i s chapter. function). 8) r ~ (y,k) with r : lYl, = y/lyl if 0 < c. 4. Let function for being as in ( 5 . 9 ) . u = ei~v - {0}. 9) ative f Z(tc~,k)dt 0 1. be proved in ~7 by making use o f the next. 1 be s a t i s f i e d . {L,k,~} with Then we have k • IR - {0} u' - i k u , i s given by ( 5 . 8 ) . Then, there e x i s t s and let be the r a d i - L • FI+S(I,X), K B where be a compact set in such t h a t i. 11) f o r any r a d i a t i v e where ~ function v for {L,k,~} with k • K and L • FI+~(I,X), i s as in ( 5 .

2) as m-~. 3) v m = 1,2 . . > (c><_ C~:(I,X)). ,X)Ioc m as m ÷ ~. 9) relation of ( 2 . 1 3 ) , we see t h a t #Iv' - i k v l l ~ _ l , ( O , R ) ~ CO, and hence, because of the a r b i t r a r i n e s s of Therefore for v i s the r a d i a t i v e function R > O, v'-ikv c L2,6_I(I,X). {L,k,~}. As is e a s i l y seen from the d i s c u s s i o n above, any subsequence of {v n} contains a subsequence which converges in the r a d i a t i v e {v n} itself function v converges to for v {L,k,Ll. 10) follows that L2 _,~(I,X) " H ~ ' B ( I , X ) I o c .

K,~[f]) ]) - v ( " k n '~[fn I)" -3 + 0 m m m which contradicts is complete. fn (n = 1,2 .... 46) L2,5(I,X) k e ~+. 46). 3. P. 5. 49) Let kn - Ln, n = 1,2 . . . C(r). This w i l l be useful in if6. be the operators of the form d2 d ~ + B(r) + Cn(r), Cn(r ) = Con(r) + Col(r) (r c I) with Cjn(r) = O~jn(rm)x for j = 0,I. 1 with tively. QO and The constants c,~ and independent of n = 1,2 . . . 1. {Ln,kn,g n} such that be the r a d i a t i v e function f o r Let v n, kn ~ ~+' n = 1,2 . . . 52) k ~ ~+ and vn in where v stant C such that -~ v L c F6(I,X ).

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