Resolvent formulas of general type and its application to point interactions

Journal of Evolution Equations (Impact Factor: 0.78). 11/2001; 1(4):421-440. DOI: 10.1007/PL00001381


Kiselev and Simon ([13]) considered rank one singular perturbations of general type and formulate such perturbation in terms
of a resolvent formula. In an attempt to generalize it beyond the rank one case, it is found that this expression in its generality
describes resolvents of a rather wide class of closed operators (or all selfadjoint operators). Bounded operators from the
domain of unperturbed operator with the graph norm to its dual space will serve as a parameter. As an application point interactions
in one-dimension will be discussed systematically, recapturing selfadjoint boundary conditions associated to the problem.

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    • "The theory of H −2 -perturbations is closely related to that of quantum Hamiltonians with singular interactions. Indeed, the one-dimensional Schrödinger operators with point interactions—especially δand δ -interactions—have been vigorously studied by means of the H −2 -perturbation theory; see[2,5,16]and the references therein. Our work here is motivated by this background. "
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    ABSTRACT: We investigate an inverse spectral problem for the singular rank-one perturbations of a Hill operator. We give a necessary and sufficient condition for a real sequence to be the spectrum of a singular rank-one perturbation of the Hill operator.
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    ABSTRACT: We study, in general, infinite rank singular pertur-bations of self-adjoint operators. For the given unbounded self-adjoint operator A acting on a separable Hilbert space H and, gen-erally speaking, unbounded self-adjoint operator G from H +2 (A) into H −2 (A) with the property R(G) ∩ H = {0} we construct the family of regularizations of the formal expression A + G in terms of skew unbounded projections onto H +2 , define corresponding self-adjoint in H realizations of this expression and describe their do-mains and resolvents. This new approach, based on extension the-ory of symmetric operators with the exit into rigged Hilbert spaces, allows to develop infinite-dimensional version of singular perturba-tions of self-adjoint operators and in the case of finite rank pertur-bations to get results close to the corresponding ones obtained by S. Albeverio and P. Kurasov. For a nonnegative self-adjoint opera-tor A we establish for the first time necessary and sufficient condi-tions on G that guarantee the existence of nonnegative self-adjoint operators among the realizations of A + G, as well as when those nonnegative realizations contain the Krein-von Neumann nonnega-tive extremal self-adjoint extension. Singular perturbations of the Laplace operator in R 3 by delta potentials are considered.
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    ABSTRACT: Krein’s formula and its modication are discussed from the view- point that they describe all selfadjoint operators in relation to a given unper- turbed operator. A direct proof of Krein’s formula is also given for the case when,the restricted operator is not necessarily densely dened,and possibly has innite deciency,indices.
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