Boundary value problems for singular Sturm-Liouville equation (Q2761888)

The author of this interesting paper considers a construction of maximal dissipative, maximal accumulative and selfadjoint extensions of the minimal operator generated by the differential operator \(-d^2/dx^2+l(l-1)x^{-2}+\delta x^{-\alpha }+q(x)\). Here, \(\delta ,\alpha\in [1,2)\), \(l\) is a real number that satisfies \(|l|<1\), \(q(x)\) is a real-valued bounded function. NEWLINENEWLINENEWLINEThe main result is that there exist unitary matrices and contraction transformations \(E\) and \(K\), respectively, such that the extensions of the minimal operator \(L_0\) generated by the above stated differential operator and the boundary conditions \((K-E)Y+i(K-E)Y'=0\), \((K-E)Y-i(K-E)Y'=0\), are maximal dissipative and maximal accumulative, respectively. Conversely, each maximal dissipative and maximal accumulative extension of \(L_0\) is generated by the above stated differential operator and the boundary conditions. Moreover, the contraction transformation \(K\) defined in the complex space \(\mathbb{C}^2\) is determined uniquely by the extension of the operator.