We study quantum system with a symmetric sine hyperbolic type potential V(x)=V0[sinh4(x)-ksinh2(x)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x)=V_{0}[\sinh ^4(x)-k\sinh ^2(x)]$$\end{document}, 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 Hc(α,β,γ,δ,η;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{c}(\alpha , \beta , \gamma , \delta , \eta ; z)$$\end{document}, in which the energy level E is involved inside the parameter η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}. The properties of the wave functions, which is strongly relevant for the potential parameter k, are illustrated for a given potential parameter V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{0}$$\end{document}. 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.

中文翻译：

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正弦双曲型势的精确解

我们研究具有对称正弦双曲型势的量子系统 V(x)=V0[sinh4(x)-ksinh2(x)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts } \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x)=V_{0}[\sinh ^4(x)-k\sinh ^2(x)]$$\end{document}，根据势参数 k 取负值还是正值而变成单井或双井。我们发现它的精确解可以写成汇合 Heun 函数 Hc(α,β,γ,δ,η;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts } \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{c}(\alpha , \beta , \ γ , δ , ε ; z)$$\end{document}，其中能级 E 涉及参数 η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}。对于给定的势参数 V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage 说明了与势参数 k 密切相关的波函数的属性{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{0}$$\end{document}。结果表明，当电位参数|k|为负时，波函数向原点收缩。增加，而对于对应于双阱的正 k，具有一定奇偶性的波函数变化敏感。