January 5, 2004
Michael Solomyak
Title: On the spectrum of the Laplacian on regular metric trees
A metric tree is a tree whose edges are viewed as non-degenerate line segments. The Laplacian 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 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 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.