The authors study the spectrum of indefinite Sturm-Liouville problems associated with the differential equation \(-(py')'+qy=\lambda wy\) on \(J\), where \(J=(a,b)\), \(-\infty\leq a<b\leq\infty\), and the coefficients satisfy the basic conditions \(1/p,q,w\in L_{\text{loc}}(J,\mathbb R)\), \(p>0\), \(| w| >0\) a.e. on \(J\), \(w\) changes sign on \(J\), and suitable boundary conditions. The form of these boundary conditions depends on the classification of the endpoint \(a,b\) of the interval \(J\) as regular, or singular and, when singular, whether limit-points or limit-circle in the space \(H=L^ 2(J,| w| )\).
Sufficient conditions are found for the cases: (i) The spectrum is real and unbounded below as well as above. (ii) The essential spectrum is empty. Also an upper bound for the number of nonreal eigenvalues is found. These results are obtained by studying the interplay between the indefinite problems and the corresponding definite problems (with weight function \(| w| \)) and by using the operator theory in a Krein space.