Resolvent formulas of general type and its application to point interactions

Journal of Evolution Equations (Impact Factor: 0.79). 11/2001; 1(4):421-440. 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 one-dimension will be discussed systematically, recapturing selfadjoint boundary conditions associated to the problem.

  • [Show abstract] [Hide abstract]
    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.
    Journal of the Australian Mathematical Society 01/2009; · 0.45 Impact Factor
  • Source
    Tokyo Journal of Mathematics 01/2002; 25(2002). · 0.26 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    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.