Article
Resolvent formulas of general type and its application to point interactions
Journal of Evolution Equations (Impact Factor: 0.78). 11/2001; 1(4):421440. DOI: 10.1007/PL00001381
ABSTRACT
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 onedimension will be discussed systematically, recapturing selfadjoint boundary conditions associated to the problem.
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 onedimension will be discussed systematically, recapturing selfadjoint boundary conditions associated to the problem.

 "The theory of H −2 perturbations is closely related to that of quantum Hamiltonians with singular interactions. Indeed, the onedimensional 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 rankone perturbations of a Hill operator. We give a necessary and sufficient condition for a real sequence to be the spectrum of a singular rankone perturbation of the Hill operator.  [Show abstract] [Hide abstract]
ABSTRACT: We study, in general, infinite rank singular perturbations of selfadjoint operators. For the given unbounded selfadjoint operator A acting on a separable Hilbert space H and, generally speaking, unbounded selfadjoint 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 selfadjoint in H realizations of this expression and describe their domains and resolvents. This new approach, based on extension theory of symmetric operators with the exit into rigged Hilbert spaces, allows to develop infinitedimensional version of singular perturbations of selfadjoint operators and in the case of finite rank perturbations to get results close to the corresponding ones obtained by S. Albeverio and P. Kurasov. For a nonnegative selfadjoint operator A we establish for the first time necessary and sufficient conditions on G that guarantee the existence of nonnegative selfadjoint operators among the realizations of A + G, as well as when those nonnegative realizations contain the Kreinvon Neumann nonnegative extremal selfadjoint extension. Singular perturbations of the Laplace operator in R 3 by delta potentials are considered.  [Show abstract] [Hide abstract]
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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