By B. S. Pavlov (auth.), Pavel Exner, Petr à eba (eds.)

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Additional info for Applications of Self-Adjoint Extensions in Quantum Physics: Proceedings of a Conference Held at the Laboratory of Theoretical Physics, JINR Dubna, USSR, September 29 – October 1, 1987

Sample text

V ) >L2(L) -< f~(u), Im Gi#o(V)>L2(L) . (37) 38 So we have the natural variables (ccnjugated in the sense cf Scbclev classes) cf the symplectic form: u-_~ Im Gigo(U) , 9;(u) • Hew ene can calculate ~ ~, u- fcr a given u ~ ven by Lemma 4 The integral ~ ~ ( u ) S~(u) ~l dy n ~ : {X~R6: dy = lim ~ n ? fA is given by u ds dist(X,n) : #}, (38) and u- = Im Gi~ O = lim (u(X) - S Re G i ~ o÷( U ) dy) On the base cf Lemma $ we can write the bcundary ccnditicn (;I (39) for the thin manifold L in R 6.

To formulate the correspondence rule, we treat the singular perturbations in a non-additive way. The information about a perturbatlou is contained in boundary conditions of some new (different from the free one) selfadJoint operator. 5y 2. ar perturbation Let A map G be a linear dense subset in a complex Hilbert space b: G x G - * C such that conjugate linear in the domain to G vg G Q(b) = G. is linear in (',') gives an example of a billnear form on b(u,u) ~ o for all H sequence We write Let {Un)CG, tition of G b I s) H..

Here ~ means ~-function with support at the sphere Z 2" The R form of the potential V(}~) allows us to obtain an explicit solution for the Lippmann-Schwinger equation for the resolvent 34 (25) G(A) = GO(,~) - GO(,X) v(~) G(,x) This solution locks like G(X) = QO(,~) + ,)- , Y l m > G o ( A ) Ylm x (26) , - 1 -1 We denote here GO(A) the free resclvent of the Laplace operator in R3: GO(A) = ( - 4 - ~)-I . Now we can obtain the expression for the external component of the scattered wave as the coefficient cf the spherical wave in the asymptctics of the Green function G(x,x I, A) when [x11-~ ~ [11].

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