On multiplicity of eigenvalues in quantum graph theory

Abstract

Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at pendant vertices and continuity and Kirchhoff's conditions at interior vertices. It is shown that the eigenvalues of such problems approach asymptotically the eigenvalues of the problem on the same tree with zero potentials on the edges. It is shown that between any two eigenvalues of maximal multiplicity (ppen−1) where ppen is the number of pendant vertices there are pin eigenvalues (with account of multiplicity, where pin is the number of interior vertices in the tree).