SUSY formalism for the symmetric double well potential

Abstract

Using first- and second-order supersymmetric Darboüx formalism and starting with symmetric double well potential barrier we have obtained a class of exactly solvable potentials subject to moving boundary condition. The eigenstates are also obtained by the same technique.

Appendix

In general, the form of γ(β) are:

$$ \begin{array}{rll} \gamma(\beta)&=&\frac{1}{4}\cot^2p -\frac{j\sin p\cot(j+1)p}{\sin jp}\left[1-\frac{j\sin p}{\sin jp\sin 2(j+1p)}\right] \\ &&+\,\frac{1}{[j\sin p-\sin jp\cos(j+1)p]^2}\\ &&\times\,\left[j\sin jp+\frac{j}{2}\sin^2jp-\frac{j^2}{4}(1+12j)\sin p\sin(2j+1)p \right.\\ &&\;\;\;\quad +\,\left(j^2+j+\frac{1}{4}\right)\frac{\sin^2jp}{\sin^2(j+1)p}\\ &&\;\;\;\quad+\left(j^2+\frac{1}{4}\right)\sin^2jp\sin^2(j+1)p +j\sin^2jp\sin^2(2j+1)p\\ &&\;\;\;\quad +\,\frac{j^2\cos jp}{\sin(j+1)p}(\sin p-\sin jp) +\frac{j^4\sin^p\text{cos}(2j+1)p}{\sin jp\sin(j+1)p}\\ &&\;\;\;\quad\left.\times\left[1+ \frac{\tan(2j+1)p\sin(2j+1)p}{4\sin jp\sin(j+1)p}\right]\right] \end{array} $$

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for regions (1) and (2) and

$$ \begin{array}{rll} \gamma(\beta)&=&\frac{1}{\Theta_\text{s}}\left[\frac{3}{2}\alpha^2_\text{s}\alpha^2_{\text{s}+1}+2\alpha_\text{s}\alpha_{\text{s}+1}\,{\rm tanh}\,A_{\rm s}\,{\rm tanh}\,C_{\rm s}(\alpha^2_{\text{s}+1}\,{\rm sinh} \,A_{\rm s}\,{\rm cosh}\,A_{\rm s} \right. \\ &&\;\;\phantom{\frac{1}{\Theta_\text{s}}} -\,\alpha^2_{\rm s}\,{\rm sinh}\,C_{\rm s}\,{\rm cosh}\,C_{\rm s})-(\alpha^{4}_{{\rm s}+1}\,{\rm cosh}^{2}\,A_{\rm s}+\alpha^{4}_{\rm s}\,{\rm cosh}^{2}\,{C_{\rm s}})\\ &&\;\;\phantom{\frac{1}{\Theta_s}}+\,\frac{1}{4}\,{\rm cosh}^2\,C_{\rm s}\,{\rm sech}^2\,A_s+\frac{1}{4}\,{\rm cosh}^2\,A_{\rm s}\,{\rm sech}^2\,C_{\rm s} \\ &&\;\;\phantom{\frac{1}{\Theta_s}}\times\left(j_s+\frac{1}{2}\right)(\alpha^2_{\rm s}\,{\rm cosh}^2\,C_{\rm s}-\alpha^2_{{\rm s}+1}\,{\rm cosh}^2\,A_{\rm s})\\ &&\;\;\phantom{\frac{1}{\Theta_s}}\left.+\,\frac{(2j_{\rm s}+1)^2}{4}\,{\rm cosh}^2\,A_{\rm s}\,{\rm cosh}^2\,C_{\rm s}\right] \\ &&-\,\frac{1}{4}(\alpha^2_{{\rm s}+1}\,{\rm tanh}^2\,C_{\rm s}+\alpha^2_{\rm s}\,{\rm tanh}^2\,A_{\rm s}-2\alpha_{\rm s}\alpha_{{\rm s}+1}\,{\rm tanh}\,A_{\rm s}\,{\rm tanh}\,C_{\rm s}) \end{array} $$

(23)

for region (3) and symmetric case. Here Θ_s = [α _s + 1 cosh A _s sinh C _s − α _s cosh C _s sinh A _s].

For region (3) and antisymmetric case.

$$ \begin{array}{rll} \gamma(\beta)&=&\frac{1}{\Theta_\text{a}}\left[2\alpha^2_\text{a}\alpha^2_{\text{a}+1}\vphantom{\left(j_\text{a}+\frac{1}{2}\right)^2}\right.\\ &&\;\;\phantom{\frac{1}{\Theta_\text{a}}}-\,\alpha_\text{a}\alpha_{\text{a}+1}\,{\rm coth}\,A_{\rm a}\,{\rm coth}\,C_{\rm a}(\alpha^2_{{\rm a}+1}\,{\rm sinh}^2\,A_{\rm a} +\alpha^2_{\rm a}\,{\rm sinh}^2\,C_{\rm a})\\ &&\;\;\phantom{\frac{1}{\Theta_a}}+\,\alpha^{4}\,{\rm sinh}^2\,A_{\rm a}+\alpha^{4}\,{\rm sinh}^2\,A_{\rm a} \times\frac{\alpha^2_{a+1}}{4}\,{\rm sinh}\,A_{\rm a}\,{\rm cosech}\,C_{\rm a}\\ &&\;\;\phantom{\frac{1}{\Theta_a}}-\, \frac{\alpha^2_{\rm a}}{4}\,{\rm sinh}\,C_{\rm a}\,{\rm cosech}\,A_{\rm a}+\left(j_{\rm a}+\frac{1}{2}\right)(\alpha^2_{{\rm a}+1}\,{\rm sinh}^2\,A_{\rm a}\\ &&\;\;\phantom{\frac{1}{\Theta_a}}-\left.\alpha^2_{\rm a}\,{\rm sinh}^2\,C_{\rm a})\left(j_\text{a}+\frac{1}{2}\right)^2\,{\rm sinh}^2\,A_{\rm a}\,{\rm sinh}^2\,C_{\rm a}\right]\\ &&-\,\left(\frac{\alpha^2{\text{a}+1}}{4}\,{\rm coth}^2\,C_{\rm a}+ \frac{\alpha^2_{\rm a}}{4}{\rm coth}^2\,A_{\rm a} -2\alpha_{\rm a}\alpha_{{\rm a}+1}\,{\rm coth}\,A_{\rm a}\,{\rm coth}\,C_{\rm a}\right),\\ \end{array} $$

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where $\Theta_{\rm a}=(\alpha_{{\rm a}+1}\,{\rm sinh}\,A_{\rm a}\,{\rm cosh}\,C_{\rm a}-\alpha_{\rm a}\,{\rm cosh}\,A_{\rm a}\,{\rm sinh}\,C_{\rm a})^2\alpha$.