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1. 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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2. On the Spectrum of Sturmian Hamiltonians of Bounded Type in a Small Coupling Regime

   https://doi.org/10.1007/s00023-025-01581-z  

   Luna, Alexandro (May 2025, Annales Henri Poincaré)

   We prove the nonstationary bounded distortion property for smooth dynamical systems on multidimensional spaces. The results we obtain are motivated by potential application to study of spectral properties of discrete Schrödinger operators with potentials generated by Sturmian sequences. 

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3. Breather gas fission from elliptic potentials in self-focusing media

   https://doi.org/10.1103/PhysRevE.111.014204  

   Biondini, Gino; El, Gennady A; Luo, Xu-Dan; Oregero, Jeffrey; Tovbis, Alexander (January 2025, Physical Review E)

   We present an analytical model of integrable turbulence in the focusing nonlinear Schrödinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter 𝑚 of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter 𝑚 , and we show that it is greater than 2 for all nonzero 𝑚 , implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases. 

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4. Anderson localization for Schrödinger operators with monotone potentials over circle homeomorphisms

   https://doi.org/10.4171/JST/531  

   Kerdboon, Jiranan; Zhu, Xiaowen (October 2024, Journal of Spectral Theory)

   In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough. 

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5. Electronic absorption spectra from off-diagonal quantum master equations

   https://doi.org/10.1063/5.0106888  

   Lai, Yifan; Geva, Eitan (September 2022, The Journal of Chemical Physics)

   Quantum master equations (QMEs) provide a general framework for describing electronic dynamics within a complex molecular system. Off-diagonal QMEs (OD-QMEs) correspond to a family of QMEs that describe the electronic dynamics in the interaction picture based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. The fact that OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrödinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrödinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from the electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained this way is studied in the context of a biexciton benchmark model, by comparing spectra calculated based on time-local and time-nonlocal OD-QMEs to spectra calculated based on a Redfield-type QME and the non-perturbative and quantum-mechanically exact hierarchical equations of motion method. 

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