Quasi-isospectrality on quantum graphs

TL;DR

This paper demonstrates that quantum graphs with nearly identical spectra, differing only on sparse sets, actually share the same eigenvalue and length spectra, revealing a form of spectral rigidity.

Contribution

It establishes conditions under which near-spectral agreement implies full spectral and length-spectrum equivalence for quantum graphs.

Findings

Eigenvalue spectra agree except on sparse sets imply identical spectra and length spectra.

Length spectra agreement outside sparse sets leads to identical eigenvalue and length spectra.

Results highlight spectral rigidity in quantum graphs with near-matching spectra.

Abstract

Consider two quantum graphs with the standard Laplace operator and non-Robin type boundary conditions at all vertices. We show that if their eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then the eigenvalue-spectra and the length-spectra of the two quantum graphs are identical, with the possible exception of the multiplicity of the eigenvalue zero. Similarly if their length-spectra agree everywhere aside from a sufficiently sparse set, then the quantum graphs have the same eigenvalue-spectrum and length-spectrum, again with the possible exception of the eigenvalue zero.