Arithmetic Spectral Transitions for the Maryland Model

Abstract

We give a precise description of spectra of the Maryland model $(h_{\lambda ,\alpha ,\theta }u{)}_n=u_{n+1}+u_{n-1}+\lambda \tan \pi (\theta +n\alpha )u_n$ for all values of parameters. We introduce an arithmetically defined index $\delta (\alpha ,\theta )$ and show that for $\alpha \notin \mathbb{Q},$ ${\sigma }_{sc}(h_{\lambda ,\alpha ,\theta })=\overline{\{e:{\gamma }_{\lambda }(e)<\delta (\alpha ,\theta )\}}$ and ${\sigma }_{pp}(h_{\lambda ,\alpha ,\theta })=\{e:{\gamma }_{\lambda }(e)\ge \delta (\alpha ,\theta )\}$. Since ${\sigma }_{ac}(h_{\lambda ,\alpha ,\theta })=\varnothing ,$ this gives complete description of the spectral decomposition for {\it all} values of parameters $\lambda ,\alpha ,\theta$, making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion. The set of eigenvalues can be explicitly identified for all parameters, using the {\it quantization condition}. We also establish, for the first time for this or any other model, a quantization condition for singular continuous spectrum (an arithmetically defined measure zero set that supports singular continuous measures) for all parameters.

CPAM to appear

Cited by in corpus (17)

• Anomalous mobility edges in one-dimensional quasiperiodic models
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Floquet engineering of topological localization transitions and mobility edges in one-dimensional non-Hermitian quasicrystals
• Non-Hermitian Maryland Model
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Pure point spectrum for the Maryland model: a constructive proof
• Almost Everything About the Unitary Almost Mathieu Operator
• Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach
• Critical States Generators from Perturbed Flatbands
• Singular continuous spectrum and generic full spectral/packing dimension for unbounded quasiperiodic Schrödinger operators
• Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces
• Arithmetic Phase Transitions For Mosaic Maryland Model
• Stability of the gapless pure point spectrum of self-adjoint operators
• Almost Mathieu operators with completely resonant phases
• Continuous quasiperiodic Schrödinger operators with Gordon type potentials
• Anderson Localization for Long-Range operators with Singular Potentials
• Perturbative diagonalization and spectral gaps of quasiperiodic operators on ${\ell }^2({\mathbb{Z}}^d)$ with monotone potentials