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 L 2(0, a), e.g. for Dirichlet boundary conditions at x = 0 and xa, 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.

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Correspondence to Matthias Langer.

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Heinz Langer thanks the Leverhulme Trust for awarding him a Leverhulme Visiting Professorship. Without this substantial support the paper could never have been written. He also thanks the School of Mathematics of the University of Wales at Cardiff, especially Professors Malcolm Brown and Marco Marletta, for their hospitality. All three authors are grateful to the Isaac Newton Institute for Mathematical Sciences in Cambridge for offering them excellent working conditions. Last but not least, Malcolm Brown and Heinz Langer thank the University of Strathclyde for their hospitality.

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Brown, B.M., Langer, H. & Langer, M. Bessel-Type Operators with an Inner Singularity. Integr. Equ. Oper. Theory 75, 257–300 (2013).

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Mathematics Subject Classification(2010)

  • Primary 34B30
  • 34B20
  • Secondary 47E05
  • 34L40
  • 47B50
  • 46C20


  • Bessel equation
  • singular potential
  • symmetric operators in Pontryagin spaces
  • self-adjoint extensions
  • Weyl function
  • generalized Nevanlinna function