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1. On the convergence of Fourier spectral methods involving noncompact operators

   https://doi.org/10.1093/imanum/drae044  

   Trogdon, Thomas (July 2024, IMA Journal of Numerical Analysis)

   Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $$\mathcal L u = f$$. The framework posits the existence of a left-Fredholm regulator for $$\mathcal L$$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients. 

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2. Dyson gas on a curved contour

   https://doi.org/10.1088/1751-8121/ac5a8f  

   Wiegmann, P; Zabrodin, A (March 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β -ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy ( N is the number of particles) by iterating the ‘loop equation’ that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour. 

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3. Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

   https://doi.org/10.21468/SciPostPhys.16.6.154  

   Seiberg, Nathan; Seifnashri, Sahand; Shao, Shu-Heng (January 2024, SciPost Physics)

   We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry. 

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4. On realization of isometries for higher rank quadratic lattices over number fields

   https://doi.org/10.1090/tran/8670  

   Chan, Wai Kiu; Li, Han (July 2022, Transactions of the American Mathematical Society)

   Let F F be a number field, and n ≥ 3 n\geq 3 be an integer. In this paper we give an effective procedure which (1) determines whether two given quadratic lattices on F n F^n are isometric or not, and (2) produces an invertible linear transformation realizing the isometry provided the two given lattices are isometric. A key ingredient in our approach is a search bound for the equivalence of two given quadratic forms over number fields which we prove using methods from algebraic groups, homogeneous dynamics and spectral theory of automorphic forms. 

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5. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems II: Linear Theory

   https://doi.org/10.1007/s10884-021-09942-y  

   Hupkes, H. J.; Van Vleck, E. S. (September 2022, Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Equ 28:955, 2016). For small spatial grid-sizes, we establish some useful Fredholm properties for the operator that arises after linearizing our system around the travelling wave solutions to the original Nagumo PDE. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is a spatially stretched and twisted version of the standard second order differential operator that is associated to the PDE waves. 

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