Fourth order canonical forms of singular self-adjoint boundary conditions

Canonical forms of self-adjoint boundary conditions are well known in the second order (Sturm–Liouville) case for both regular and singular problems. These are critical for the theoretical investigation of the eigenvalues as well as their numerical computation. Recently canonical forms have been found for fourth order regular problems. These are much more complicated than the second order ones. Here we find canonical forms for fourth order singular problems with one or both endpoints of the domain interval singular and with arbitrary deficiency index d. In the regular fourth order case d=4, in the singular case d can assume any value between 0 and 4 and, moreover, depends on the nature of the singularities at the two endpoints. These different values of d and their dependence on the endpoints introduces serious additional complications.