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

We consider the indefinite Sturm-Liouville problem −f″=λrf, f′(−1)=f′(1)=0 where r∈L1[−1,1] satisfies xr(x)>0. Conditions are presented such that the (normed) eigenfunctions fn 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=∑βnfn. Furthermore, a sequence (αn)∈l2 is constructed such that the “Fourier series” ∑αnfn 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¯′pdx with Dirichlet boundary conditions in L2[−1,1] where p=1/r. For the associated operator Tt we construct elements in the difference between domt and the domain of the associated regular closed form, i.e. dom|Tt|1/2.