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1. 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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2. Quantum Ergodicity for Periodic Graphs

   https://doi.org/10.1007/s00220-023-04826-2  

   McKenzie, Theo; Sabri, Mostafa (September 2023, Communications in Mathematical Physics)

   This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger operators H on periodic graphs. This means that most eigenfunctions of H on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on Z^d, the triangular lattice, the honeycomb lattice, Cartesian products, and periodic Schrödinger operators on Z^d. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues. 

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3. Ambarzumian-type Problems for Discrete Schrödinger Operators

   https://doi.org/10.1007/s11785-021-01169-5  

   Hatinoğlu, Burak; Eakins, Jerik; Frendreiss, William; Lamb, Lucille; Manage, Sithija; Puente, Alejandra (November 2021, Complex Analysis and Operator Theory)

   Abstract We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator. 

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4. Density of States and Delocalization for Discrete Magnetic Random Schrödinger Operators

   https://doi.org/10.1093/imrn/rnab017  

   Becker, Simon; Han, Rui (May 2021, International Mathematics Research Notices)

   Abstract We study discrete magnetic random Schrödinger operators on the square and honeycomb lattice. For the non-random magnetic operator on the hexagonal lattice with any rational magnetic flux, we show that the middle two dispersion surfaces exhibit Dirac cones. We then derive an asymptotic expansion for the density of states on the honeycomb lattice for oscillations of arbitrary rational magnetic flux. This allows us, as a corollary, to rigorously study the quantum Hall effect and conclude dynamical delocalization close to the conical point under disorder. We obtain similar results for the discrete random Schrödinger operator on the $$\mathbb Z^2$$-lattice with weak magnetic fields, close to the bottom and top of its spectrum. 

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5. On the spectrum of the periodic Anderson–Bernoulli model

   https://doi.org/10.1063/5.0096118  

   Wood, William (October 2022, Journal of Mathematical Physics)

   We analyze the spectrum of a discrete Schrödinger operator with a potential given by a periodic variant of the Anderson model. In order to do so, we study the uniform hyperbolicity of a Schrödinger cocycle generated by the SL(2,R) transfer matrices. In the specific case of the potential generated by an alternating sequence of random values, we show that the almost sure spectrum consists of at most 4 intervals. 

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