Oscillation of Eigenfunctions of Weighted Regular Sturm-Liouville Problems

Abstract

We investigate the zeros of eigenfunctions of regular Sturm–Liouville boundary value problems with general weight functions w. In particular we are interested in the case when the set of zeros of w has positive measure. We find that in this case the first eigenfunction may have one or more zeros in the interval, in contrast to the classical case when w is positive. Necessary and sufficient conditions on w and the other coefficients are found such that the first eigenfunction has no zero.

Second order elliptic equations with degenerate weight

“…For convenience, we assume throughout that dfl' G C°° . Much of the motivation for our work comes from two papers of Everitt, Kwong and Zettl, [5], [6] where the n = 1 case of this problem was considered (under much more general conditions on the coefficients and the boundary conditions), and where more references may be found. It seems unreasonable to expect that our methods will lead to new results when specialized to ordinary differential equations.…”

Singular Sturm?Liouville problems with nonnegative and indefinite weights

“…In fact, it is shown in [4] that, given any nonnegative integer k, one can construct a regular boundary value problem of the type (2.10) such that the eigenfunction associated with the smallest eigenvahie has exactly k zeros in (a, b). If P, Q and W are the coefficients of the differential equation for this boundary value problem, then we can define functions p, q, and w, p(t) = P(t)(~b(t)) 2, q(t) = Q(t)(q~(t)) 2, (2.12)…”

Right Semidefinite Eigenvalue Problems

“…We also show that completeness may fail for such problems in H a . For further discussion of semidefinite weight Sturm-Liouville problems we refer to [10]. In § 7 we consider certain elliptic partial differential equations with L ∞ coefficients, and again completeness in H b is automatic.…”