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1. Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems

   https://doi.org/10.3389/fams.2023.1165371  

   Cockburn, Bernardo; Du, Shukai; Sánchez, Manuel A. (April 2023, Frontiers in Applied Mathematics and Statistics)

   We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work. 

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2. Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators

   https://doi.org/10.1007/s11118-019-09776-w  

   Guillen, Nestor; Kitagawa, Jun; Schwab, Russell W. (April 2019, Potential analysis)

   Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings. 

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3. Etudes for the inverse spectral problem

   https://doi.org/10.1112/jlms.12772  

   Makarov, Nikolai; Poltoratski, Alexei (May 2023, Journal of the London Mathematical Society)

   Abstract In this note, we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second‐order differential equations on a half‐line. Our goal is to extend the classical results developed in the work of Marchenko, Gelfand–Levitan, and Krein to broader classes of canonical systems and to illustrate the solution algorithms and formulae with a variety of examples. One of the main ingredients of our approach is the use of truncated Toeplitz operators, which complement the standard toolbox of the Krein–de Branges theory of canonical systems. 

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4. Some binomial formulas for non-commuting operators

   https://doi.org/10.1090/conm/733/14743  

   Kuchment, Peter; Lvin, Sergey (January 2019, Contemporary mathematics - American Mathematical Society)

   Let D and U be linear operators in a vector space (or more generally, elements of an associative algebra with a unit). We establish binomialtype identities for D and U assuming that either their commutator [D,U] or the second commutator [D, [D,U]] is proportional to U. Operators D = d/dx (differentiation) and U- multiplication by eλx or by sin λx are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors’ medical imaging research [2–5]. 

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5. Photonic interlacing architectures for learning discrete linear unitaries

   https://doi.org/10.1103/4vth-6n7q  

   Markowitz, M; Miri, M-A; Ovchinnikov, A; Zelaya, K (September 2025, Physical Review Research)

   Recent investigations suggest that the discrete linear unitary group $U\left(N\right)$can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of $U\left(N\right)$can be decomposed into a sequence of $N$-parameter phases alternating with one-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent $U\left(N\right)$for a finite number of layers. This is numerically verified by using Haar random matrices as targets, showing a convergence for exactly $N$layers. As a specific application, we propose an integrated all-optical logic gate device that performs OR, NAND, XOR, and XAND tasks within a lossless and passive optical circuit design. 

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