Resolvent formulas of general type and its application to point interactions

Journal of Evolution Equations (Impact Factor: 0.64). 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.

  • Source
    [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.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Let ℋ be a Hilbert space, H 0 a self-adjoint operator in ℋ and R 0 (z), Imz≠0 its resolvent; H 0 is assumed to have only absolutely continuous spectrum with σ ac (H 0 )=ℝ· Let ℋ s be the H 0 -scale of Hilbert spaces.The operator R 0 (z) is regarded as an element of ℒ(ℋ,ℋ)∩ℒ(ℋ s ,ℋ s+2 )· For T∈ℒ(ℋ 2 ,ℋ -2 ) and Imz≠0, define R T (z)=R 0 (z)-R 0 (z)(1+TW(z)) -1 TR 0 (z), where W(z)=(z-i)R 0 (z)R 0 (i)· Assumptions on T are such that there exists a unique self-adjoint operator H T whose resolvent is R T (z) [see S. T. Kuroda and H. Nagatani, Oper. Theory Adv. Appl. 108, 99–105 (1999; Zbl 0976.47009)]. Let ℛ=rangeT· The author considers the case dim ℛ=N, and obtains conditions which imply that λ∈ℝ is also an eigenvalue for H T · He investigates the wave operators W ± (H 0 ,H T ) and shows that they exist and are asymptotically complete. The particular case N=1 is considered at the end of the paper and is illustrated by two concrete examples.
    Tokyo Journal of Mathematics 12/2002; 25(2002). · 0.33 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.