Spectral Properties and Oscillation Theorems for Mixed Boundary-Value Problems of Sturm-Liouville Type

This paper presents analogs, for certain mixed boundary-value problems, of the spectral and oscillatory properties exhibited by classical Sturm-Liouville systems. ‘l’he usual analysis of the Sturm-Liouville eigenvalue problem is based on special ad hoc methods. In contrast, Gantmacher and Krein [3; see also references therein] showed that these fundamental spectral properties are direct consequences of the total positivity of the Green’s function for the problem. They further showed that the total positivity of the Green’s function is itself the mathematical espression of certain basic physical properties of vibrating mechanical systems, which are typically modeled by SturnPLiouville systems. Subsequent to the work in [3], extensive studies have revealed se\-cl-al important classes of boundary-value problems with separated boundaryconditions whose Green’s functions are totally positive or sign regular. As in the classical Sturm-I,iouville problem, these boundary-value problems exhibit a rich oscillation theorv. Some principal contributors in this area are Gantmacher and Krein [3], Karlin [6-81, Karen [12], Krein [13], and Krein and 1:inkrlstein