The Left Set The Oscillation Of The Sturm-liouville Operator Characteristic Function

In this paper, we study the oscillations of the eigenfunction of Sturm-Liouville problem with indefinite coefficients and general separated boundary conditions on (0,l).Using Prufer transform, we give a description of the signature of an eigenvalue, and give the relation between the signature of an eigenvalue and the signs of the corresponding leading coefficients of Weyl function and the Priifer angle at this eigenvalue. Finally we obtain a formula which can be used to calculate the numbers of oscillation points in [0,l} of the eigenfunction corresponding to the nth eigenvalue. The results in this paper are the extendability of the results obtained by P. Binding about Sturm-Liouville problem with indefinite coefficients and some special separated boundary condition.