Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

Abstract

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized.

arXiv comment64 pages

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Cited by in corpus (3)

• Quantum Unique Ergodicity for Cayley graphs of quasirandom groups
• Pointwise Weyl law for graphs from quantized interval maps
• Scattering resonances of large weakly open quantum graphs