Ultrarelativistic (Cauchy) spectral problem in the infinite well

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 $D\subset R$.

11 pp, 2 figures

References in corpus (2)

• Ten equivalent definitions of the fractional Laplace operator
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well

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