Singular point-like perturbations of the Laguerre operator in a Pontryagin space

The spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna function

Q(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).

If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.