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1. Semiclassical estimates for measure potentials on the real line

   https://doi.org/10.4171/JST/500  

   Larraín-Hubach, Andrés; Shapiro, Jacob (August 2024, Journal of Spectral Theory)

   We prove an explicit weighted estimate for the semiclassical Schrödinger operatorP = - h^{2} \partial^{2}_{x} + V(x;h)onL^{2}(\R), withV(x;h)a finite signed measure, and whereh >0is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues forP, and a limiting absorption resolvent estimate with sharph-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support. 

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2. Stability of a point charge for the repulsive Vlasov–Poisson system

   https://doi.org/10.4171/JEMS/1518  

   Pausader, Benoît; Widmayer, Klaus; Yang, Jiaqi (August 2024, Journal of the European Mathematical Society)

   We consider solutions of the repulsive Vlasov–Poisson system which are a combination of a point charge and a small gas, i.e., measures of the form\delta_{(\mathcal{X}(t),\mathcal{V}(t))}+\mu^{2}d\mathbf{x}d\mathbf{v}for some(\mathcal{X}, \mathcal{V})\colon \mathbb{R}\to\mathbb{R}^{6}and a small gas distribution\mu\colon \mathbb{R}\to L^{2}_{\mathbf{x},\mathbf{v}}, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on\mu_{0}=\mu(t=0)are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution \mudecays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of \mu_{0}, the electric field decays at an optimal rate, and we derive modified scattering dynamics for the motion of the point charge and the gas distribution. Our proof makes crucial use of the Hamiltonian structure. The linearized system is transport by the Kepler ODE, which we integrate exactly through an asymptotic action-angle transformation. Thanks to a precise understanding of the associated kinematics, moment and derivative control is achieved via a bootstrap analysis that relies on the decay of the electric field associated to\mu. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates. 

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3. Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

   https://doi.org/10.1088/1361-6544/adc653  

   Liu, Wencai; Matos, Rodrigo; Treuer, John N (April 2025, Nonlinearity)

   Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on ${\mathbb{Z}}^d$. Specifically, we consider operators of the form $H=-\mathrm{\Delta }+V+v$, where Δ is the discrete Laplacian, $V:{\mathbb{Z}}^d\to \mathbb{R}$is a periodic potential, and $v:{\mathbb{Z}}^d\to \mathbb{C}$represents a decaying impurity. We establish quantitative conditions under which the equation $-\mathrm{\Delta }u+Vu+vu=\lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u\equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes ofλ. 

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4. Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

   https://doi.org/10.1017/etds.2024.76  

   ARANA-HERRERA, FRANCISCO (February 2025, Ergodic Theory and Dynamical Systems)

   Abstract In her thesis, Mirzakhani showed that the number of simple closed geodesics of length$$\leq L$$on a closed, connected, oriented hyperbolic surfaceXof genusgis asymptotic to$$L^{6g-6}$$times a constant depending on the geometry ofX. In this survey, we give a detailed account of Mirzakhani’s proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces. 

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5. Universal coarse geometry of spin systems

   https://doi.org/10.1007/s11005-025-01949-6  

   Elokl, Ali; Jones, Corey (June 2025, Letters in Mathematical Physics)

   Abstract The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associatedcoarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state$$\phi $$ $\varphi$on an (abstract) spin system with an infinite collection of sitesX, we define a universal coarse structure$$\mathcal {E}_{\phi }$$ $E_{\varphi }$on the setXwith the property that a state has decay of correlations with respect to a coarse structure$$\mathcal {E}$$ $E$onXif and only if$$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ $E_{\varphi }\subseteq E$. We show that under mild assumptions, the coarsely connected completion$$(\mathcal {E}_{\phi })_{con}$$ ${\left(E_{\varphi }\right)}_{\mathrm{con}}$is stable under quasi-local perturbations of the state$$\phi $$ $\varphi$. We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit$$\alpha $$ $\alpha$and the coarse structure of the state$$\psi \circ \alpha $$ $\psi \circ \alpha$where$$\psi $$ $\psi$is any product state. 

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