A failing eigenfunction expansion associated with an indefinite Sturm–Liouville problem

We consider the indefinite Sturm–Liouville problem − f ″ = λ r f , f ′ ( − 1 ) = f ′ ( 1 ) = 0 where r ∈ L 1 [ − 1 , 1 ] satisfies x r ( x ) > 0 . Conditions are presented such that the (normed) eigenfunctions f n form a Riesz basis of the Hilbert space L | r | 2 [ − 1 , 1 ] (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function f ∈ L | r | 2 [ − 1 , 1 ] having no eigenfunction expansion f = ∑ β n f n . Furthermore, a sequence ( α n ) ∈ l 2 is constructed such that the “Fourier series” ∑ α n f n does not converge in L | r | 2 [ − 1 , 1 ] . These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form t [ u , v ] = ∫ u ′ v ¯ ′ p d x with Dirichlet boundary conditions in L 2 [ − 1 , 1 ] where p = 1 / r . For the associated operator T t we construct elements in the difference between dom t and the domain of the associated regular closed form, i.e. dom | T t | 1 / 2 .