Oscillation Properties of Singular Quantum Trees

Abstract

We discuss the possibility of generalizing the Sturm comparison and oscillation theorems to the case of singular quantum trees, that is, to Sturm-Liouville differential expressions with singular coefficients acting on metric trees and subject to some boundary and interface conditions. As there may exist non-trivial solutions of differential equations on metric trees that vanish identically on some edges, the classical Sturm theory cannot hold globally for quantum trees. However, we show that the comparison theorem holds under minimal assumptions and that the oscillation theorem holds generically, that is, for operators with simple spectra. We also introduce a special Prüfer angle, establish some properties of solutions in the non-generic case, and then extend the oscillation results to simple eigenvalues.

Keywords

• quantum tree
• distributional potential
• Sturm comparison and oscillation theorem
• nodal count
• Prüfer angle