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Ultrarelativistic (Cauchy) spectral problem in the infinite well. (English) Zbl 1371.42027
Summary: We analyze spectral properties of the ultrarelativistic (Cauchy) operator $$|\Delta |^{1/2}$$, provided its action is constrained exclusively to the interior of the interval $$[-1, 1]\subset \mathbb R$$. 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 not the eigenfunctions of $$|\Delta |^{1/2}_D$$, $$D = (-1, 1)$$. This clearly demonstrates that the traditional Fourier multiplier representation of $$|\Delta |^{1/2}$$ becomes defective, while passing from $$\mathbb R$$ to a bounded spatial domain $$D \subset \mathbb R$$.
##### MSC:
 42B35 Function spaces arising in harmonic analysis 47A10 Spectrum, resolvent
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