Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices
The paper deals with the boundary value problems for the Sturm-Liouville equations \N\[\N-y_j'' + q_j(x_j) y_j = \lambda y_j, \quad j = 1, 2, \dots, g, \N\]\Non a tree graph (i.e. graph without cycles) with equal edge lengths. The potentials are assumed to be all equal and symmetric. The Dirichlet boundary conditions are imposed in the pendant vertices and the standard matching conditions (continuity and Kirchhoff's ones), in the interior vertices except for the root vertex. The authors consider the recovery of the graph structure from the two spectra, corresponding to the Dirichlet boundary condition and the Neumann one in the root. It is proved that, for caterpillar trees, the two spectra uniquely determine the shape of a graph. However, in general, the uniqueness does not hold. The authors present an example of co-spectral snowflake trees. The technique is based on the structure of characteristic functions and eigenvalue asymptotics.
scientific article; zbMATH DE number 7906392