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51) A"= A+ a(cp, ·)cp, cp E 1l-2(A) \ 1i-1(A). We have shown that any self-adjoint operator corresponding to this formal expression is an extension of the symmetric operator A0 . Considering rank one form bounded perturbations we have determined the unique self-adjoint extension of the operator A 0 which coincides with the linear operator A" defined in the generalized sense. In the case under consideration the linear operator Aa is not in general defined on the space H"'(A) = Dom (A 0*). The reason is that the linear functional cp is not defined on this domain.

From the condition that the range of the function ~ belongs to the Hilbert space L2 (R) we get -~' (+0) + ~, (-0) + a~(O) = If the function~ satisfies the latter condition, then the restriction of the operator Ba to the domain Dom (Ba) = 0. Ba~ E L 2 (R). Consider { ~ E Wi(R \ {0}) : ~( -0) = ~( +0) =~(0) ~'(+0)- ~'(-0) = a~(O)}. The restrictions of the operators Ba and B 0* to this domain coincide. This operator is a self-adjoint extension of the operator B 0 and this operator can be considered as a natural definition for the operator Ba in the framework of the theory of self-adjoint operators.

The imaginary part of f(t) is always equal to 1 if the parameter cis real. Suppose conversely that the function f(t) is equal to a given constant. 58) ( 'Pc, A 2- : 1cp = C= f(t) - i. 58) is always real. It is necessary to show that the extension of the functional is homogeneous in this case. In fact it is enough to prove this property only for the elements (1/(A- i)) cp and (1/{A + i)) cp. B ( ( 'Pc, = r8 ( 'Pc, A ~ i cp J. pc, A+ icp) = t -8 1 (cpc, A+ icp), and the lemma is pron'n. D It has been shown during the proof of the latter theorem that every homogeneous extension of the functional cp is defined by the real parameter c.

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