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1. Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

   https://doi.org/10.1093/imrn/rnaa362  

   Jitomirskaya, Svetlana; Kocić, Saša (February 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s $${{\mathcal{H}}}$$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $$C^{1+BV}$$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $$\mu $$, and for $$\mu $$-almost all $$x\in{{\mathbb{T}}}^1$$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $$V$$. 

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2. On the Spectra of Separable 2D Almost Mathieu Operators

   https://doi.org/10.1007/s00023-021-01080-x  

   Takase, Alberto (June 2021, Annales Henri Poincaré)

   null (Ed.)

   Abstract We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets. 

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3. Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

   https://doi.org/10.1093/imrn/rnz319  

   Han, Rui; Zhang, Shiwen (May 2020, International Mathematics Research Notices)

   Abstract We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies. 

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4. Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent

   https://doi.org/10.1007/s00222-022-01157-2  

   Avila, Artur; Damanik, David; Zhang, Zhenghe (February 2023, Inventiones mathematicae)

   Abstract We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains. 

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5. Essential Self-Adjointness of the Laplacian on Weighted Graphs: Harmonic Functions, Stability, Characterizations and Capacity

   https://doi.org/10.1007/s11040-025-09498-z  

   Inoue, Atsushi; Ku, Sean; Masamune, Jun; Wojciechowski, Radosław K (June 2025, Mathematical Physics, Analysis and Geometry)

   We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth–death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schrödinger operators, use the characterizations for birth–death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the $$\ell^2$$-Liouville property. 

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