Singular Perturbations of Differential Operators: Solvable - download pdf or read online

By S. Albeverio, P. Kurasov

Differential and extra common self-adjoint operators regarding singular interactions come up obviously in a variety of matters resembling classical and quantum physics, chemistry, and electronics. This publication is a scientific mathematical learn of those operators, with specific emphasis on spectral and scattering difficulties. The equipment mentioned are in keeping with a brand new proposal of symplectic constitution of the "boundary form." appropriate for researchers in research or mathematical physics, this quantity may be used as a textual content for a sophisticated direction at the purposes of research.

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Consider an arbitrary vector 'TJ from the domain Dom (A 0 ) C H. p, 'l/;)('TJ, tp) = (Ary,'lj;). p) = 0 (as an element from Dom (A0 )) and the operator A is defined in the generalized sense on the vectors from 1i1(A). Let 'ljJ E Dom (A 0*) = Hcp(A). 47) {A~- b('l/;) A2 ~ 1 cp} + [a(cp, ~)b( 'ljJ) + ab( 'l/;) (cp, A2~ 1cp)) cp. The expression in the braces { } belongs to the original Hilbert space H. e. if the following equality holds (cp, ~) =- (~ + ( cp, A2~ 1 cp)) b('lj;). 48) is self-adjoint.

To define the corresponding rank one perturbations we fix the real parameter c = ('Pc, (Aj(A 2 + 1)) cp). p coincides with B if the coupling constant is chosen equal to -1 0:= - - . ')'+C The theorem is proven. 0 The same result holds in the case when the symmetric operator A 0 is not densely defined. 0 are defined on the same domain. The exceptional extension is not an operator but an operator relation. 2). The latter theorem implies that the self-adjoint extensions of any symmetric operator \vith unit deficiency indices can be considered as rank one perturbations of a self-adjoint operator and can therefore be parametrized by the additive parameter a instead of the nonadditive parameters 'Y and v appearing in the boundary conditions.

2). 3). For

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