Rank One Perturbations in a Pontryagin Space with One Negative Square

• Published in: Journal of functional analysis Vol. 188; no. 2; pp. 317 - 349
• Main Authors: Derkach, Vladimir, Hassi, Seppo, de Snoo, Henk
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2002
• Subjects: Friedrichs extension; generalized Nevanlinna function; Pontryagin space; rank one perturbation; selfadjoint extension; symmetric operator; rank one perturbation; Friedrichs extension; Pontryagin space; symmetric operator; generalized Nevanlinna function; selfadjoint extension
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1006/jfan.2001.3837

Let N1 denote the class of generalized Nevanlinna functions with one negative square and let N1, 0 be the subclass of functions Q(z)∈N1 with the additional properties limy→∞Q(iy)/y=0 and limsupy→∞y|ImQ(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω]ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions.