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arXiv · 1505.01277

Ultrarelativistic (Cauchy) spectral problem in the infinite well

Elena V. Kirichenko·Piotr Garbaczewski·Vladimir Stephanovich·Mariusz Żaba
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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 $D\subset R$.

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Elena V. Kirichenko, Piotr Garbaczewski, Vladimir Stephanovich, Mariusz Żaba. 2016-04-09. Ultrarelativistic (Cauchy) spectral problem in the infinite well. https://doi.org/10.5506/aphyspolb.47.1273

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