MONOTONICITY OF EIGENVALUES

LEON GREENBERG

lng@math.umd.edu

Department of Mathematics
University of Maryland
College Park
Maryland MD 20742 , USA

For , consider the family of Sturm-Liouville problems:

where , and . The Sturm-Liouville problem (1) has a sequence of eigenvalues: . A well known Sturmian property states that if the second boundary condition (1c) is a Dirichlet condition (i.e. if ) then the eigenvalues are strictly decreasing functions of . We ask the question: Does any vestige of this property remain true if ? The answer is yes.

Theorem 1   There exists an integer such that for , the eigenvalues are strictly decreasing functions of .

Theorem 2   For given functions and a given integer , there exists a continuous function such that the eigenvalues are not decreasing. In fact, for any given , can be found so that is increasing in a neighborhood of .

EPSRC Gregynog Workshop, 21-26 July 2002