MONOTONICITY OF EIGENVALUES


LEON GREENBERG

lng@math.umd.edu

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






For $ 0 < y \le 1$, consider the family of Sturm-Liouville problems:

  $\displaystyle -(p(x)u')' + q(x)u = \lambda r(x)u, \ \ 0 < x < y, \hspace{1.5in} \ \ (1a)$    
  $\displaystyle \alpha _0 u(0) + \beta _0 p(0)u'(0) = 0, \hspace{2.3in} \ \ (1b)$    
  $\displaystyle \alpha _1 u(y) + \beta _1 p(y)u'(y) = 0, \hspace{2.3in} \ \ (1c)$    

where $ p\in C^1[0,1], \ q,\ r\in C[0,1]$, and $ p,\ r > 0$. The Sturm-Liouville problem (1) has a sequence of eigenvalues: $ \lambda _0(y) < \lambda _1(y) < \lambda _2(y) < \cdots$. A well known Sturmian property states that if the second boundary condition (1c) is a Dirichlet condition (i.e. if $ \beta _1 = 0$) then the eigenvalues $ \lambda _n(y)$ are strictly decreasing functions of $ y$. We ask the question: Does any vestige of this property remain true if $ \beta _1 \ne 0$? The answer is yes.

Theorem 1   There exists an integer $ n_0 \ge 0$ such that for $ n \ge n_0$, the eigenvalues $ \lambda _n(y)$ are strictly decreasing functions of $ y$.

Theorem 2   For given functions $ p,\ r$ and a given integer $ n_0 \ge 0$, there exists a continuous function $ q$ such that the eigenvalues $ \lambda _n(y), \ 0 \le n \le n_0,$ are not decreasing. In fact, for any given $ y_0 \in (0,1)$, $ q$ can be found so that $ \lambda _n(y)$ is increasing in a neighborhood of $ y_0, \ 0 \le n \le n_0$.



EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts