Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval

TL;DR

This paper establishes a precise asymptotic law for the eigenvalues of certain pseudo-differential operators in an interval, extending previous results to a broader class of functions with detailed error estimates.

Contribution

The authors derive a two-term Weyl-type asymptotic formula with explicit error bounds for eigenvalues of psi(-Delta) operators, generalizing prior specific cases.

Findings

Eigenvalues follow a specific asymptotic formula with O(1/n) error

Extension of previous results to a broader class of Bernstein functions

Explicit relation between eigenvalues and solutions of a transcendental equation

Abstract

We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to infinity as x goes to infinity. This extends previous results obtained by the authors for the fractional Laplace operator (psi(x) = x^{alpha/2}) and for the Klein-Gordon square root operator (psi(x) = (1+x^2)^{1/2} - 1). The formula for the eigenvalues in (-a,a) is of the form lambda_n = psi(mu_n^2) + O(1/n), where mu_n is the solution of mu_n = (n pi)/(2 a) - theta(mu_n)/a, and theta(mu) is given as an integral involving psi.