[Paper Review] Measure-geometric Laplacians on the real line

Marc Keßeböhmer, Tony Samuel+1 more|arXiv (Cornell University)|Feb 13, 2018

Spectral Theory in Mathematical Physics21 references3 citations

TL;DR

This paper generalizes measure-geometric Laplacians to measures $η = \nu + \delta$ on the real line, where $\nu$ is continuous and $\delta$ is a finite sum of Dirac masses. It establishes that the resulting Laplacian $\Delta_\eta$ is a densely defined, self-advoint, unbounded operator with compact resolvent, and provides a complete spectral characterization via explicit eigenvalue and eigenfunction formulas, including asymptotic growth rates of the eigenvalue counting function.

ABSTRACT

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\uın, given an atomless Borel probability measure $η$ supported on a compact subset of $\mathbb{R}$, Freiberg and Zähle introduced a measure-geometric approach to define a first order differential operator $ abla_η$ and a second order differential operator $Δ_η$, with respect to $η$. We generalise this approach to measures of the form $η= ν+ δ$, where $ν$ is continuous and $δ$ is finitely supported. We determine analytic properties of $ abla_η$ and $Δ_η$ and show that $Δ_η$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $Δ_η$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

Motivation & Objective

• To extend the measure-geometric Laplacian framework of Freiberg and Zähle to measures with both continuous and finitely many atomic components.
• To analyze the analytic and spectral properties of the first-order differential operator $\nabla_\eta$ and the second-order operator $\Delta_\eta = -\nabla_\eta^* \circ \nabla_\eta$.
• To determine the eigenvalues and eigenfunctions of $\Delta_\eta$ for such mixed measures, and to establish the asymptotic behavior of the eigenvalue counting function.

Proposed method

• Define $\nabla_\eta$ and $\Delta_\eta$ using a measure-geometric approach based on the cumulative distribution function of $\eta = \nu + \delta$, generalizing the classical derivative and Laplacian.
• Prove that $\Delta_\eta$ is densely defined, linear, self-adjoint, non-positive, and has compact resolvent in $L^2_\eta$.
• Derive a system of transcendental equations that eigenfunctions must satisfy, involving sine functions of the transformed cumulative distribution $F_\nu(x)$, with jump conditions at atomic points.
• Use functional analytic techniques to show that eigenfunctions are piecewise sine functions with phase shifts determined by the atomic weights and positions.
• Establish orthonormality and completeness of eigenfunctions via a transformation to a periodic setting, reducing the problem to a cycle graph framework.
• Apply spectral asymptotics techniques to derive the eigenvalue counting function's growth rate, showing $\pi N_\eta(x)/\sqrt{x} \to 1$ as $x \to \infty$.

Experimental results

Research questions

• RQ1 How can the measure-geometric Laplacian be extended to measures with both continuous and discrete components?
• RQ2 What are the spectral properties of $\Delta_\eta$ when $\eta = \nu + \delta$ with $\nu$ continuous and $\delta$ finitely supported?
• RQ3 Can the eigenvalues and eigenfunctions of $\Delta_\eta$ be explicitly computed for such mixed measures?
• RQ4 How does the eigenvalue counting function $N_\eta(x)$ grow asymptotically for $\Delta_\eta$?
• RQ5 Do the eigenfunctions exhibit symmetry or continuity at atomic points, and how does this affect spectral multiplicity?

Key findings

• The operator $\Delta_\eta$ is densely defined, self-adjoint, non-positive, and has compact resolvent in $L^2_\eta$.
• Eigenfunctions of $\Delta_\eta$ are piecewise sine functions of the form $f(x) = a_j \sin(bF_\nu(x) + \gamma_j)$ on intervals between atoms.
• The eigenvalues are given by $\lambda = -b^2$, where $b$ satisfies a system of nonlinear equations involving the atomic weights and positions.
• For the example $\eta = \Lambda + \pi^{-1}\delta_{1/2} + \pi^{-1}\delta_1$, the eigenvalues are $\lambda^{(-2,2)} = 4\pi^2$, $\lambda^{(-1,2)} = \pi^2$, $\lambda^{(1,2)} \approx 21.8$, $\lambda^{(2,2)} \approx 106.9$, $\lambda^{(3,2)} \approx 267.2$, and $\lambda^{(4,2)} \approx 505.3$.
• The eigenvalue counting function satisfies $\lim_{x \to \infty} \pi N_\eta(x)/\sqrt{x} = 1$, indicating square-root-type spectral growth.