Pöschl–Teller — the full P-symbol (hypergeometric ₂F₁)

Pöschl–Teller — the full P-symbol

The Pöschl–Teller \(\mathrm{sech}^2\) well is the representative of the non-confluent class: the equation reduces to the Gauss \(_2F_1\) with its three Fuchsian singular points \(0,1,\infty\) — Riemann's P-symbol in full. Finite bound spectrum \(E_n=-\tfrac{\alpha^2}{2}(\lambda-n)^2\); for integer \(\lambda\) the potential is transparent. Finite-difference computation in the Python backend (via gw2py).

While hydrogen, Morse and the harmonic oscillator use \(_1F_1\) (Kummer, two singularities merged), Pöschl–Teller reduces to the Gauss \(_2F_1\) with the three regular singular points \(0,1,\infty\) intact: Riemann's P-symbol used in full.

\(\mathrm{sech}^2\) well

\[ V(x)=-\frac{\lambda(\lambda+1)\,\alpha^2}{2}\,\mathrm{sech}^2(\alpha x). \]

Reduction to Gauss \(_2F_1\)

With the variable \(\xi=\tfrac12\big(1-\tanh(\alpha x)\big)\in(0,1)\), the Schrödinger equation becomes the Gauss hypergeometric equation, with the three regular singular points \(0,1,\infty\): Riemann's P-symbol in its full form, without confluence. The eigenfunctions are \(_2F_1\) (associated Legendre / Jacobi functions).

Bound spectrum (finite): \[ E_n=-\frac{\alpha^2}{2}\,(\lambda-n)^2,\qquad n=0,1,\dots,\lfloor\lambda\rfloor. \]

Transparency and solitons

For integer \(\lambda\) the potential is reflectionless (transparent at every energy): this is the link with the solitons of the Korteweg–de Vries equation and the Bargmann potentials.

State to represent

State type Level \(n_1\)

Level \(n_2\)

Potential parameters

Depth \(\lambda\) = 4

Bound states: \(n=0,\dots,\lfloor\lambda\rfloor\). Integer \(\lambda\) ⇒ transparent potential.

Inverse width \(\alpha\) = 1

The \(_2F_1\) branch: three Fuchsian points, no confluence

While hydrogen, Morse and the harmonic oscillator use \(_1F_1\) (Kummer, where two singularities have merged), Pöschl–Teller reduces to the Gauss \(_2F_1\) with the three regular singular points \(0,1,\infty\) intact. It is Riemann's P-symbol used in full.

The full-P-symbol family

All these potentials reduce to the Gauss \(_2F_1\) (hypergeometric branch of the Natanzon class): Pöschl–Teller (\(\mathrm{sech}^2\)) and trigonometric, Eckart, Rosen–Morse (hyperbolic and trigonometric), Hulthén, Manning–Rosen, Scarf I and II. All shape-invariant (SUSY QM), with a spectrum given by a closed form descending from the termination of the \(_2F_1\).

Computational evidence

With \(\lambda=4,\ \alpha=1\) the engine reproduces \(E_n=-(\lambda-n)^2/2 = -8,\,-4.5,\,-2,\,-0.5\) to \(\sim1.5\times10^{-3}\); the number of bound states matches \(\lfloor\lambda\rfloor+1\).

References

G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933).
NIST DLMF, ch. 15 (Gauss hypergeometric). dlmf.nist.gov/15.
G. A. Natanzon, Theor. Math. Phys. 38, 146 (1979).
F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995). doi.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: Pöschl-Teller, sech squared, Gauss hypergeometric, 2F1, Riemann P-symbol, transparent potential, solitons, SUSY, quantum mechanics