EIGENVALUE ESTIMATES FOR THE WEIGHTED LAPLACIAN ON METRIC TREES

Abstract

The Laplacian on a metric tree is $\Delta u = u''$ on its edges, with the appropriate compatibility conditions at the vertices. We study the eigenvalue problem on a rooted tree $\Gamma$: $$-\lambda \Delta u = V u \quad \text{on } \Gamma, \qquad u(o)=0.$$ Here $V \ge 0$ is a given ‘weight function’ on $\Gamma$, and $o$ is the root of $\Gamma$. The eigenvalues for such a problem decay no faster than $\lambda_n = O(n^{-2})$, this last case being typical for one-dimensional problems. We obtain estimates for the eigenvalues in the classes $l_p$, with $p > \frac12$, and their weak analogues $l_{p,\infty}$ with $p \ge \frac12$. The results for $p<1$ and $p>1$ are of different character. The case $p<1$ is studied in more detail. To analyse this case, we use two methods; in both of them the problem is reduced to a family of one-dimensional problems. One of the methods is based upon a useful orthogonal decomposition of the Dirichlet space on $\Gamma$. For a particular class of trees and weights, this method leads to the complete spectral analysis of the problem. We illustrate this in several examples, where we were able to obtain the asymptotics of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05; secondary 34L15, 34L20.