Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

A D Alhaidari

Table 3. Partial list of exactly solvable potential functions in the polynomial class associated with ${H}_{n}^{(\mu ,\nu )}(z;\alpha ,\theta ).$ The coordinate transformation y(x) enters in the basis (44) that supports a tridiagonal matrix representation for the corresponding wave operator. The presence of the ${V}_{1}$ term in all of these potentials inhibits exact solvability of the Schrödinger wave equation in the standard formulation of quantum mechanics.

V(x) x y(x) ${V}_{0}+\displaystyle \frac{{V}_{+}-{V}_{-}\,\sin (\pi x/L)}{{\cos }^{2}(\pi x/L)}+{V}_{1}\,\sin \left(\pi x/L\right)$ $-\tfrac{1}{2}L\leqslant x\leqslant +\tfrac{1}{2}L$ $\sin \left(\pi x/L\right)$ $\displaystyle \frac{1}{1-{(x/L)}^{2}}\left\{{V}_{0}+\displaystyle \frac{{V}_{+}}{{(x/L)}^{2}}+\displaystyle \frac{{V}_{-}}{1-{(x/L)}^{2}}+{V}_{1}\left[2{(x/L)}^{2}-1\right]\right\}$ $0\leqslant x\leqslant L$ $2{\left(x/L\right)}^{2}-1$ $\displaystyle \frac{1}{{{\rm{e}}}^{\lambda x}-1}\left[{V}_{0}+{V}_{-}{{\rm{e}}}^{\lambda x}+\displaystyle \frac{{V}_{+}}{1-{{\rm{e}}}^{-\lambda x}}+{V}_{1}\left(1-2{{\rm{e}}}^{-\lambda x}\right)\right]$ $x\geqslant 0$ $1-2{{\rm{e}}}^{-\lambda x}$ ${V}_{-}+\displaystyle \frac{{V}_{+}}{{\sinh }^{2}(\lambda x)}+\displaystyle \frac{{V}_{0}+{V}_{1}[2{\tanh }^{2}(\lambda x)-1]}{{\cosh }^{2}(\lambda x)}$ $x\geqslant 0$ $2{\tanh }^{2}(\lambda x)-1$ ${V}_{+}-{V}_{-}\,\tanh (\lambda x)+\displaystyle \frac{{V}_{0}+{V}_{1}\,\tanh (\lambda x)}{{\cosh }^{2}(\lambda x)}$ $-\infty \lt x\lt +\infty $ $\tanh (\lambda x)$ ${V}_{0}+\displaystyle \frac{{V}_{+}}{{\sin }^{2}(\pi x/L)}+\displaystyle \frac{{V}_{-}}{{\cos }^{2}(\pi x/L)}-{V}_{1}\,\cos (2\pi x/L)$ $0\leqslant x\leqslant \tfrac{1}{2}L$ $2{\sin }^{2}(\pi x/L)-1$ ${V}_{0}+\displaystyle \frac{{V}_{+}-{V}_{-}\,\cosh (\lambda x)}{{\sinh }^{2}(\lambda x)}+{V}_{1}\,\cosh \left(\lambda x\right)$ $x\geqslant 0$ $\cosh \left(\lambda x\right)$