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Eigenvalue problems for the complex PT-symmetric potential V(x)= igx

Zafar Ahmed
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Abstract

The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as $E_n \sim n^2.$ The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if $|g|<(1.22330447)$. The soft-box potential turns out to be a scattering potential not possessing reflectionless states.

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Zafar Ahmed. 2006-11-21. Eigenvalue problems for the complex PT-symmetric potential V(x)= igx. https://doi.org/10.1016/j.physleta.2006.11.057

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