Bessel-type operators with an inner singularity (Q1946548)

The authors study the Bessel-type differential equation \[ -y''+\frac{\frac{\alpha}{2}\left(\frac{\alpha}{2}+1\right)}{(x-1)^2}y=\lambda\, y,\quad 0\leq x\leq a, \] where \(a>1\). Note that this equation has an inner singularity at \(x=1\). If \(\alpha\geq 1\), then this singularity is in the limit point case at \(x=1\) from both sides, and hence treating it in a Hilbert space in \(L^2[0,a]\), e.g., for Dirichlet boundary conditions at \(x=0\) and \(x=a\), a unique self-adjoint operator is associated with this differential expression. However, it is possible to construct self-adjoint extensions in some (exit) Pontryagin space. The authors give an explicit construction of this Pontryagin space and provide a description of the self-adjoint operators which can be associated with this differential equation.