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One-dimensional quasi-relativistic particle in the box. (English) Zbl 1278.81068
Asymptotics of the eigenvalues \(E_{n}\) of the one dimensional quasi-relativistic Hamiltonian or the Klein-Gordon square-root Hamiltonian with electrostatic potential, i.e., the Friedrichs extension on \(L^{2}(-a,a)\) of the operator \(\tilde H\) defined by \(\tilde Hf = (- \hbar^{2}c^{2} \frac{d^{2}}{dx^{2}}+m^{2}c^{4})^{1/2}f_{|(-a,a)}\), \(\forall f \in C^{\infty}_{0}(-a,a)\) are given, uniformly in \(n,\hbar, m, c \) and \(a\). The simplified version of the main theorem states that all the energy levels are non-degenerate and \(E_{n}=(\frac{k\pi}{2}-\frac{\pi}{2})\frac{\hbar c}{a} +O(\frac{1}{n})\) as \(n \rightarrow \infty\). In fact a more refined result where the error of the approximation is less than \(C_{1} \max (mc^{2}, \hbar ca^{-1})\exp(-C_{2}\hbar ^{-1}mca)n^{-1}\) is proved. As a byproduct, \(L^{2}\) approximations and \(L^{\infty}\) estimations for the eigenfunctions are obtained.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35P15 Estimates of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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