Some Remarks on the Eigenvalue Multiplicities of the Laplacian on Infinite Locally Finite Trees

Abstract

We consider the continuous Laplacian on an infinite uniformly locally finite network under natural transition conditions as continuity at the ramification nodes and the classical Kirchhoff flow condition at all vertices in a L ∞-setting. The characterization of eigenvalues of infinite multiplicity for trees with finitely many boundary vertices (von Below and Lubary, Results Math 47:199–225, 2005, 8.6) is generalized to the case of infinitely many boundary vertices. Moreover, it is shown that on a tree, any eigenspace of infinite dimension contains a subspace isomorphic to $${\ell^\infty({\mathbb N})}$$ . As for the zero eigenvalue, it is shown that a locally finite tree either is a Liouville space or has infinitely many linearly independent bounded harmonic functions if the edge lengths do not shrink to zero anywhere. This alternative is shown to be false on graphs containing circuits.