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Theoretical and Mathematical Physics

Approximate analytic solution of the Logunov–Tavkhelidze equation for a one-dimensional oscillator potential in the relativistic configuration representation

Yu. A. Grishechkin, V. N. Kapshaǐ

Published Jun 1, 2022• 2 citations
Publisher

Abstract

We construct approximate analytic solutions of the Logunov–Tavkhelidze equation in the case of a potential that, in the one-dimensional relativistic configuration representation, has the form analogous to the potential of the nonrelativistic harmonic oscillator in the coordinate representation. The wave functions are obtained in both the momentum and relativistic configuration representations. The approximate values of the energy of the relativistic harmonic oscillator are the roots of transcendental equations. The wave functions in the relativistic configuration representation have additional zeros in comparison with the wave functions of the corresponding states of the nonrelativistic harmonic oscillator in the coordinate representation.

DOI
10.1134/s0040577922060058

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We investigate the dynamics of the spin-less relativistic particle subject to an external eld of a harmonic oscillator potential.The KleinGordon equation with one-and three-dimensional vector and scalar parabolic potentials is solved using the expansion of the wavefunction in properly selected basis-sets.The resonance states are determined using the complex coordinate rotation method.The analytic expressions for the rst-and secondorder relativistic energy corrections are derived perturbatively.The relativistic model of the harmonic oscillator in the momentum representation, originally proposed by Znojil, is also discussed.
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Abstract We determine explicitly the exact transcendental bound states energies equation for a one-dimensional harmonic oscillator perturbed by a single and a double point interactions via Green’s function techniques using both momentum and position space representations. The even and odd solutions of the problem are discussed. The corresponding limiting cases are recovered. For the harmonic oscillator with a point interaction in more than one dimension, divergent series appear. We use to remove this divergence an exponential regulator and we obtain a transcendental equation for the energy bound states. The results obtained here are consistent with other investigations using different methods.
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