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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 \(D \subset \mathbb R\).
42B35 Function spaces arising in harmonic analysis
47A10 Spectrum, resolvent
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