A short note on decay rates of odd partitions: an application of spectral asymptotics of the Neumann–Poincaré operators

Here, we introduce a theorem currently proved uniquely by the asymptotic behaviors of eigenvalues of a compact operator. Specifically, a problem of partitions is considered, and the Neumann–Poincaré operator is employed as the compact linear operator. Then a theorem is proved by the spectrum of the Neumann–Poincaré operator. Although the following proposed problem looks artificial, our result in the partitions seems to be proven uniquely by the spectral theory of the Neumann–Poincaré operators: Odd partitions of the unit interval [0, 1] are considered, that is, we divide the unit interval [0, 1] into 2N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2N+1$$\end{document} disjoint non-zero intervals LN,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{N, k}$$\end{document} (k=1,…,2N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1, \ldots , 2N+1$$\end{document}), and the sum of corresponding lengths ∑k=12N+1|LN,k|=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^{2N+1} |L_{N, k}| =1$$\end{document} for each N∈N≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N}_{\ge 0}$$\end{document}. Thus we obtain a countable set of real numbers P={|LN,k|k=1,2,…,2N+1,N∈N≥0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=\{ |L_{N, k}| \;\ k=1, 2, \ldots , 2N+1, \ N\in \mathbb {N}_{\ge 0} \}$$\end{document} by odd partitions of the unit interval. One can enumerate the set P in decreasing order to obtain the non-increasing sequence a1=|L0,1|=1>a2≥a3≥⋯>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a_1=|L_{0, 1}|=1> a_2 \ge a_3 \ge \cdots >0. \end{aligned}$$\end{document}We show that for any C≥1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C \ge 1/2$$\end{document}, there exist odd partitions of the unit interval such that aj∼Cj-1/2asj→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a_j \sim C j^{-1/2} \quad \text{ as }\ j \rightarrow \infty . \end{aligned}$$\end{document}Here, the coefficient C=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=1/2$$\end{document} corresponds to the optimal decay. We prove this fact by a fundamental property of the Riemann zeta function and by eigenvalue asymptotics for some compact linear operators known as the Neumann–Poincaré operators.