Article

# 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.

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**ABSTRACT:**We study the spectral gaps of the Schrödinger operatorMathematical Proceedings of the Cambridge Philosophical Society 06/2007; 143(01):185 - 199. DOI:10.1017/S030500410700031X · 0.83 Impact Factor - [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). DOI:10.3836/tjm/1244208857 · 0.33 Impact Factor - [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 12/2009; DOI:10.1017/S1446788709000135 · 0.33 Impact Factor

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