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

**Read Online or Download 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 PDF**

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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(,~) +