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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)

   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. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai ( November 2019 , International Mathematics Research Notices)

   In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

    

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4. 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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5. Ohm’s law lost and regained: observation and impact of transmission and velocity zeros

   https://doi.org/10.1038/s41467-024-54012-8  

   Joshi, Krishna ; Kurtz, Israel ; Shi, Zhou ; Genack, Azriel_Z ( December 2024 , Nature Communications)

   The quantum conductance and its classical wave analogue, the transmittance, are given by the sum of the eigenvalues of the transmission matrix. However, neither measurements nor theoretical analysis of the transmission eigenchannels have been carried out to explain the dips in conductance found in simulations as new channels are introduced. Here, we measure the microwave transmission matrices of random waveguides and find the spectra of all transmission eigenvalues, even at dips in the lowest transmission eigenchannel that are orders of magnitude below the noise in the transmission matrix. Transmission vanishes both at topological transmission zeros, where the energy density at the sample output vanishes, and at crossovers to new channels, where the longitudinal velocity vanishes. Zeros of transmission pull down all the transmission eigenvalues and thereby produce dips in the transmittance. These dips and the ability to probe the characteristics of even the lowest transmission eigenchannel are due to correlation among the eigenvalues. The precise tracking of dips in the conductance by peaks in the density of states points to a further correlation between zeros and poles of the transmission matrix. The conductance approaches Ohm’s law as the sample width increases in accord with the correspondence principle.

    

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