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1. Exact and approximate energy sums in potential wells

   https://doi.org/10.1088/1751-8121/ab69a6  

   Berry, M. V.; Burke, Kieron (February 2020, Journal of Physics A: Mathematical and Theoretical)

   Abstract Sums of theNlowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regimeN  ≫  1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levelsN  +  1,N  +  2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10⁻⁷. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on howNand the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits. 

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2. Faster Kernel Matrix Algebra via Density Estimation.

   Backurs, Arturs; Indyk, Piotr; Musco, Cameron; Wagner, Tal. (January 2021, International Conference on Machine Learning (ICML))

   We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix K∈ R^{n*n} corresponding to n points x1,…,xn∈R^d. In particular, we consider estimating the sum of kernel matrix entries, along with its top eigenvalue and eigenvector. We show that the sum of matrix entries can be estimated to 1+ϵ relative error in time sublinear in n and linear in d for many popular kernels, including the Gaussian, exponential, and rational quadratic kernels. For these kernels, we also show that the top eigenvalue (and an approximate eigenvector) can be approximated to 1+ϵ relative error in time subquadratic in n and linear in d. Our algorithms represent significant advances in the best known runtimes for these problems. They leverage the positive definiteness of the kernel matrix, along with a recent line of work on efficient kernel density estimation. 

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3. A Chebyshev Locally Optimal Block Preconditioned Conjugate Gradient Method for Product and Standard Symmetric Eigenvalue Problems

   https://doi.org/10.1137/23M1566017  

   Zhang, Tianqi; Xue, Fei (November 2024, SIAM Journal on Matrix Analysis and Applications)

   The discretized Bethe-Salpeter eigenvalue (BSE) problem arises in many body physics and quantum chemistry. Discretization leads to an {\color{black}algebraic} eigenvalue problem involving a matrix $$H\in \mathbb{C}^{2n\times 2n}$$ with a Hamiltonian-like structure. With proper transformations, {\color{black}the real BSE eigenproblem of form I and the complex BSE eigenproblem of form II} can be transformed into real product eigenvalue problems of order $$n$$ and $2n$, respectively. We propose a new variant of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) {\color{black}enhanced with polynomial filters} to improve the robustness and effectiveness of a few well-known algorithms for computing the lowest eigenvalues of the product eigenproblems. {\color{black}Furthermore, our proposed method can be easily employed to solve large sparse standard symmetric eigenvalue problems.} We show that our ideal locally optimal algorithm delivers Rayleigh quotient approximation to the desired lowest eigenvalue that satisfies a global quasi-optimality, which is similar to the global optimality of the preconditioned conjugate gradient method for the iterative solution of a symmetric positive definite linear system. The robustness and efficiency of {\color{black}our proposed method} is illustrated by numerical experiments. 

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4. Behavior of absorbing and generating $p$ -Robin eigenvalues in bounded and exterior domains

   https://doi.org/10.1016/j.na.2025.113943  

   Bundrock, Lukas; Giorgi, Tiziana; Smits, Robert (January 2026, Nonlinear Analysis)

   We establish rigorous quantitative inequalities for the first eigenvalue of the generalized 𝑝-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter 𝛼 is positive, and the superconducting generation regime (𝛼 < 0), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all 𝑝 and all small real 𝛼, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as 𝛼 → 0 for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. 

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5. Eigenvalue spacing for 1D singular Schrödinger operators

   https://doi.org/10.3233/ASY-221814  

   Hillairet, Luc; Marzuola, Jeremy L. (May 2023, Asymptotic Analysis)

   The aim of this paper is to provide uniform estimates for the eigenvalue spacing of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain. 

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