Pöschl–Teller potential

Definition

In its symmetric form is explicitly given by

${\displaystyle V(x)=-\frac{\lambda (\lambda +1)}{2}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2(x)}$

and the solutions of the time-independent Schrödinger equation

${\displaystyle -\frac{1}{2}{\psi }^{″}(x)+V(x)\psi (x)=E\psi (x)}$

with this potential can be found by virtue of the substitution ${\displaystyle u=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(\mathrm{x})}$, which yields

${\displaystyle {\left[(1-u^2){\psi }^{\prime }(u)\right]}^{\prime }+\lambda (\lambda +1)\psi (u)+\frac{2E}{1-u^2}\psi (u)=0}$.

Thus the solutions ${\displaystyle \psi (u)}$ are just the Legendre functions ${\displaystyle P_{\lambda }^{\mu }(\tanh (x))}$ with ${\displaystyle E=-\frac{{\mu }^2}{2}}$, and ${\displaystyle \lambda =1,2,3\cdots }$, ${\displaystyle \mu =1,2,\cdots ,\lambda -1,\lambda }$. Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer ${\displaystyle \lambda }$, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg–De Vries equation.

The more general form of the potential is given by

${\displaystyle V(x)=-\frac{\lambda (\lambda +1)}{2}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2(x)-\frac{\nu (\nu +1)}{2}{\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}}^2(x).}$