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1. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai (November 2019, International Mathematics Research Notices)

   Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $$H$$ as a perturbation of the free operator $$H_0$$, where $$(H_0u)(n)= u({n+1})+u({n-1})$$. For $$H_0$$ (no perturbation), $$\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$$ and $$H_0$$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $$H_0+V$$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $$\limsup _{n\to \infty } n|V(n)|=a<\infty .$$ We obtain a lower/upper bound of $$a$$ such that $$H_0+V$$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $$\{ E_j\}_{j=1}^N$$ in $(-2,2)$ with $$0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$$, we construct the explicit potential $$V(n)=\frac{O(1)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}_{j=1}^N$$.5: Given any countable set of points $$\{ E_j\}$$ in $(-2,2)$ with $$0\notin \{ E_j\}+\{ E_j\}$$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $$|V(n)|\leq \frac{h(n)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}$$. 

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2. A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

   https://doi.org/10.1093/imrn/rnz156  

   T-R McLaughlin, Kenneth; Nabelek, Patrik V. (November 2019, International Mathematics Research Notices)

   Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation. 

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3. Generalized Bounded Distortion Property

   https://doi.org/10.1007/s10884-024-10376-5  

   Borissov, Gregory; Monakov, Grigorii (July 2024, Journal of Dynamics and Differential Equations)

   Abstract We prove the nonstationary bounded distortion property for$$C^{1 + \varepsilon }$$ $C^{1+\epsilon }$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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4. 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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5. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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