A failing eigenfunction expansion associated with an indefinite Sturm-Liouville problem (Q764939)

Results concerning eigenfunction expansions are proved for the indefinite Sturm-Liouville problem \[ - f'' = \lambda r f, \;\;\;\;f' (- 1) = f' (1) = 0, \] where the weight function \(r \in L^{1} [- 1, 1].\) First, conditions are given in order that the eigenfunctions \(f_{n}\) of the problem form a Riesz basis of the weighted space \(L_{| r |}^{2} [- 1, 1].\) Second, failing expansion results for a sufficiently large class of the weights are proved. It is shown, in particular, a connection between these problems and the regularity properties of the associated non-semibounded symmetric sesquilinear form.