January 5, 2004

Michael Solomyak

Title: On the spectrum of the Laplacian on regular metric trees

Abstract:

A metric tree $\Gamma$ is a tree whose edges are viewed as non-degenerate line segments. The Laplacian $\Delta$ on such tree is the operator of second order differentiation on each edge, complemented by the Kirchhoff matching conditions at the vertices. The spectrum of $\Delta$ can be quite varied, reflecting the geometry of a tree.

We consider a special class of trees, namely the so-called regular metric trees. Any such tree possesses a rich group of symmetries. As a result, the space $L^2(\Gamma)$ decomposes into the orthogonal sum of subspaces reducing the Laplacian. This leads to detailed spectral analysis of the operator. We survey recent results on this subject.