The index problem for eigenvalues for coupled boundary conditions and Fulton’s conjecture

Abstract

This paper deals with regular self-adjoint Sturm–Liouville problems with coupled boundary conditions. The spectrum of the problems can be either bounded from below, or unbounded from both below and above. We consider the following question: given an eigenvalue, how to determine its index or indices efficiently? Using results on the level surfaces of the nth eigenvalue and certain inequalities among eigenvalues, the determination is converted into that of the index of the same eigenvalue for an appropriate separated boundary condition and hence can be achieved in terms of the Prufer angle. As an application of this complete solution of the index problem, we show that Fulton’s conjecture about such an index is true.