Rank One Perturbations in a Pontryagin Space with One Negative Square

✍ Scribed by Vladimir Derkach; Seppo Hassi; Henk de Snoo

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 428 KB
• Volume: 188
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3837

✦ Synopsis

Let N 1 denote the class of generalized Nevanlinna functions with one negative square and let N 1, 0 be the subclass of functions Q(z) ¥ N 1 with the additional properties lim y Q . Q(iy)/y=0 and lim sup y Q . y |Im Q(iy)| < .. These classes form an analytic framework for studying (generalized) rank one perturbations A(y)= A+y [ • , w] w in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N 1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(y) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(y), are based on a recent factorization result for generalized Nevanlinna functions.