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1. Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

   https://doi.org/10.1090/btran/117  

   Eichinger, Benjamin; Lukić, Milivoje; Young, Giorgio (January 2023, Transactions of the American Mathematical Society, Series B)

   There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets. 

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2. On the number of closed gaps of discrete periodic one-dimensional operators

   https://doi.org/10.1063/5.0175428  

   Arroyo, Andrew; Castro, Faye; Fillman, Jake (May 2024, Journal of Mathematical Physics)

   From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices. 

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3. PARITY OF k -DIFFERENTIALS IN GENUS ZERO AND ONE

   https://doi.org/10.1093/qmath/haag030  

   Chen, Dawei; Chen, Evan; Lau, Kenny; Ono, Ken; Zhang, Jujian (September 2026, The Quarterly Journal of Mathematics)

   Abstract Here we completely determine the spin parity of k-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis [Conjecture A.10]. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix). We emphasize that it is this combinatorial identity, and not the geometric results on k-differentials, that was formalized. The formalized statement is displayed and explained in the Appendix. 

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4. A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices

   https://doi.org/10.1007/s11785-025-01804-5  

   Eichinger, Benjamin; Lukić, Milivoje; Young, Giorgio (October 2025, Complex Analysis and Operator Theory)

   A new way of encoding a non-self-adjoint Jacobi matrix J by a spectral measure of |J| together with a phase function was described by Pushnitski–Štampach in the bounded case. We present another perspective on this correspondence, based on Weyl functions instead of moments, which simplifies some proofs and generalizes the correspondence to the unbounded case. In particular, we find a bijection between proper Jacobi matrices with positive off-diagonal elements, and a class of spectral data. We prove that this mapping is continuous in a suitable sense. To prove injectivity of the map, we prove a local Borg–Marchenko theorem for unbounded non-self-adjoint Jacobi matrices in this class that may be of independent interest. 

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5. Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

   https://doi.org/10.1002/mana.202100561  

   Damanik, David; Fillman, Jake; Wang, Chunyi (September 2023, Mathematische Nachrichten)

   This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum. 

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