Completeness Theorems for a Non-Standard Two-Parameter Eigenvalue Problem

We consider two simultaneous Sturm-Liouville systems coupled by two spectral parameters. However, unlike the standard multiparameter problem, we now suppose that the principal part of each of the differential operators is multiplied by a different parameter. In a recent paper, Faierman and Mennicken derived various results concerning the eigenvalues and eigenfunctions, and in particular, they established the oscillation theory for this system. Here we continue this investigation focusing on the completeness of the set of eigenfunctions in a suitable function space. If either one of the potentials is identically zero, the completeness of the eigenfunctions is established, whereas, if this condition fails, then we show the existence of an essential spectrum having non-zero points. The completeness problem for this latter case will be left for a later work. M?ller and Watson supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory.