Dissipative eigenvalue problems for a Sturm–Liouville operator with a singular potential

Abstract

In this paper we consider the Sturm–Liouville operator d²/dx² − 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L²(a, 0) and L²(0, b) of the space L²(a, b) = L²(a, 0) ⊕ L²(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L²(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition lim_x→0+(f′(x)−f′(−x)) = γf(0) with γ∈C⁺∪R or by the Dirichlet condition f(0+) = f(0−) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L²(a, b).