Bessel-Type Operators with an Inner Singularity

We consider a Bessel-type differential expression on [0, a], a > 1, with the singularity at the inner point x = 1, see (1.2) below. This singularity is in the limit point case from both sides. Therefore in a Hilbert space treatment in L2(0, a), e.g. for Dirichlet boundary conditions at x = 0 and x = a, a unique self-adjoint operator is associated with this differential expression. However, in papers by J. F. van Diejen and A. Tip, Yu. Shondin, A. Dijksma, P. Kurasov and others, in more general situations, self-adjoint operators in some Pontryagin space were connected with this kind of singular equations; for (1.2) this connection appeared also in the study of a continuation problem for a hermitian function by H. Langer, M. Langer and Z. Sasvári. In the present paper we give an explicit construction of this Pontryagin space for the Bessel-type equation (1.2) and a description of the self-adjoint operators which can be associated with it.