Ultrarelativistic (Cauchy) spectral problem in the infinite well

Abstract

We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|Δ|^{1/2}$, 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 $\cos(nπx/2)$ and $\sin(nπx)$, for integer $n$ are {\it not} the eigenfunctions of $|Δ|_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|Δ|^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$.

11 pp, 2 figures