Ultrarelativistic (Cauchy) spectral problem in the infinite well

TL;DR

This paper investigates the spectral characteristics of the ultrarelativistic (Cauchy) operator confined to a finite interval, revealing new spectral data and demonstrating the failure of traditional Fourier methods in this bounded domain setting.

Contribution

It provides new high-accuracy spectral data and analytically proves that common trigonometric eigenfunctions are not valid for the operator on a bounded interval.

Findings

New high-accuracy spectral data obtained

Trigonometric functions are not eigenfunctions of the operator on the interval

Fourier multiplier representation becomes defective in bounded domains

Abstract

We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|\Delta |^{1/2}$, provided its action is constrained exclusively to the interior of the interval $[-1,1] \subset R$. To this end both analytic and numerical methods are employed. New high-accuracy spectral data are obtained. A direct analytic proof is given that trigonometric functions $\cos(n\pi x/2)$ and $\sin(n\pi x)$, for integer $n$ are {\it not} the eigenfunctions of $|\Delta |_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|\Delta |^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$.