Singular spectrum for radial trees

Abstract

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that generically (in the sense of Baire) radial trees have purely singular continuous spectrum.

15 pages

References in corpus (3)

• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• The absolutely continuous spectrum of one-dimensional Schr"odinger operators
• Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

Cited by in corpus (10)

• Spectral Theory of Infinite Quantum Graphs
• Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures
• On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs
• Absolutely continuous spectrum for multi-type Galton Watson trees
• On the Decomposition of the Laplacian on Metric Graphs
• Zero Measure and Singular Continuous Spectra for Quantum Graphs
• Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees
• Non-compact quantum graphs with summable matrix potentials
• Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition
• The curious spectra and dynamics of non-locally finite crystals