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Exact solutions of the sine hyperbolic type potential

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Abstract

We study quantum system with a symmetric sine hyperbolic type potential \(V(x)=V_{0}[\sinh ^4(x)-k\sinh ^2(x)]\), which becomes single or double well depending on whether the potential parameter k is taken as negative or positive. We find that its exact solutions can be written as the confluent Heun functions \(H_{c}(\alpha , \beta , \gamma , \delta , \eta ; z)\), in which the energy level E is involved inside the parameter \(\eta \). The properties of the wave functions, which is strongly relevant for the potential parameter k, are illustrated for a given potential parameter \(V_{0}\). It is shown that the wave functions are shrunk to the origin when the negative potential parameter |k| increases, while for a positive k which corresponding to a double well, the wave functions with a certain parity are changed sensitively.

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Acknowledgements

This work is supported by project 20180677-SIP-IPN, COFAA-IPN, Mexico and partially by the CONACYT project under grant No. 288856-CB-2016.

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Correspondence to Shi-Hai Dong.

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Dong, Q., Torres-Arenas, A.J., Sun, GH. et al. Exact solutions of the sine hyperbolic type potential. J Math Chem 57, 1924–1931 (2019). https://doi.org/10.1007/s10910-019-01045-w

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  • DOI: https://doi.org/10.1007/s10910-019-01045-w

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