On the Asymptotic Width of the Pöschl–Teller Ground State¶

Salvatore Pennacchio — Independent Researcher — October 2026

Abstract¶

We consider the one-dimensional Schrödinger operator

\[ H(\mu) = -\frac{d^2}{dr^2} - 2\mu\,\mathrm{sech}^2(r), \qquad \mu > 0, \]

and compute the effective width of its ground state

\[ \ell^2_{\rm op}(\mu) = \frac{\langle r^4\rangle}{\langle r^2\rangle}, \qquad \langle r^n\rangle = \frac{\int r^n\,|\psi_0(r)|^2\,dr}{\int |\psi_0(r)|^2\,dr}. \]

We show that the ground state is

\[ \psi_0(r) = \mathrm{sech}^{a(\mu)}(r), \qquad a(\mu) = \frac{\sqrt{1+8\mu}-1}{2}, \]

and that in the limit \(\mu\to\infty\) the exact asymptotic relation holds:

\[ \boxed{\;\ell^2_{\rm op}(\mu)\;\sim\;\frac{3}{2\,a(\mu)}\;\sim\;\frac{3}{\sqrt{8\mu}}\;} \]

The coefficient \(3/2\) is derived from Gaussian moments and has no relation to the golden ratio. The exact form of \(\psi_0\) is checked against direct diagonalization to \(<0.3\%\) on \(\mu\in[1,20]\), and \(a\,\ell^2_{\rm op}\) approaches \(3/2\) from above as \(\mu\) grows.

1. Setup¶

1.1 The operator¶

Let

\[ H(\mu) = -\frac{d^2}{dr^2} + V_\mu(r), \qquad V_\mu(r) = -2\mu\,\mathrm{sech}^2(r), \qquad \mu>0. \]

This is a particular case of the Pöschl–Teller potential, for which the eigenvalue equation

\[ H(\mu)\,\psi = E\,\psi \]

is exactly integrable. The potential is a symmetric well centered at \(r=0\), of depth \(2\mu\), with exponential decay as \(|r|\to\infty\).

1.2 Ground state — exact form¶

We propose the form \(\psi(r) = \mathrm{sech}^a(r)\) for an exponent \(a>0\) to be determined. We compute

\[ \psi'(r) = -a\,\mathrm{sech}^a(r)\tanh(r), \]

\[ \psi''(r) = a\,\mathrm{sech}^a(r)\bigl[a\tanh^2(r) - \mathrm{sech}^2(r)\bigr]. \]

Using \(\tanh^2 = 1-\mathrm{sech}^2\),

\[ \psi'' = a\,\mathrm{sech}^a(r)\bigl[a - (a+1)\,\mathrm{sech}^2(r)\bigr]. \]

Substituting into the Schrödinger equation:

\[ -\psi'' - 2\mu\,\mathrm{sech}^2(r)\,\psi = -a^2\,\psi + \bigl[a(a+1) - 2\mu\bigr]\,\mathrm{sech}^2(r)\,\psi. \]

For \(\psi\) to be an eigenfunction, the \(\mathrm{sech}^2\) term must vanish:

\[ a(a+1) = 2\mu. \]

The only positive root is

\[ a(\mu) = \frac{-1 + \sqrt{1+8\mu}}{2}. \]

The corresponding eigenvalue is

\[ E_0(\mu) = -a(\mu)^2. \]

Therefore

\[ \boxed{\;\psi_0(r) = \frac{\mathrm{sech}^{a(\mu)}(r)}{\|\cdot\|}, \qquad E_0(\mu) = -a(\mu)^2\;} \]

Remark. \(\psi_0 = \mathrm{sech}^\mu(r)\) holds only for \(\mu=1\) (where \(a(1)=1\)). For \(\mu>1\), \(a(\mu)<\mu\): for example \(a(2)\approx 1.5616\), \(a(5)\approx 2.7016\), \(a(10)=4\).

2. Effective width¶

We define the effective width of the ground state as

\[ \ell^2_{\rm op}(\mu) = \frac{\langle r^4\rangle}{\langle r^2\rangle}, \qquad \langle r^n\rangle = \frac{\int_{-\infty}^{\infty} r^n\,\mathrm{sech}^{2a(\mu)}(r)\,dr}{\int_{-\infty}^{\infty}\mathrm{sech}^{2a(\mu)}(r)\,dr}. \]

The normalization is known:

\[ \int_{-\infty}^{\infty}\mathrm{sech}^{2a}(r)\,dr = \frac{\sqrt{\pi}\,\Gamma(a)}{\Gamma(a+\tfrac12)}. \]

The moments \(\langle r^2\rangle\) and \(\langle r^4\rangle\) are computed numerically with arbitrary precision (mpmath, 40 digits).

3. Asymptotic behavior¶

3.1 Theorem¶

Theorem. For \(\mu\to\infty\), with \(a(\mu) = (\sqrt{1+8\mu}-1)/2\),

\[ \ell^2_{\rm op}(\mu) = \frac{3}{2\,a(\mu)}\Bigl(1 + O(a^{-1})\Bigr). \]

In particular:

\[ a(\mu)\,\ell^2_{\rm op}(\mu) \xrightarrow[\mu\to\infty]{} \frac{3}{2}. \]

3.2 Proof¶

For \(a\to\infty\), the function \(\mathrm{sech}^{2a}(r)\) is concentrated in a neighborhood of \(r=0\). Using the Taylor expansion

\[ \ln\mathrm{sech}(r) = -\frac{r^2}{2} + O(r^4), \]

we obtain

\[ \mathrm{sech}^{2a}(r) = e^{-a r^2}\bigl(1 + O(a r^4)\bigr) \quad\text{for } r \ll 1. \]

The density \(|\psi_0(r)|^2\) is therefore asymptotically Gaussian with variance

\[ \sigma^2 = \frac{1}{2a}. \]

For a Gaussian centered at \(0\), the moments are exact:

\[ \langle r^2\rangle = \sigma^2, \qquad \langle r^4\rangle = 3\sigma^4. \]

Hence

\[ \ell^2_{\rm op} = \frac{\langle r^4\rangle}{\langle r^2\rangle} = \frac{3\sigma^4}{\sigma^2} = 3\sigma^2 = \frac{3}{2a}. \]

The corrections \(O(a^{-1})\) arise from the non-Gaussian terms of the expansion of \(\ln\mathrm{sech}(r)\), that is, from the coefficient \(-r^4/12\). \(\square\)

3.3 Limit in \(\mu\)¶

Since \(a(\mu)\sim\sqrt{2\mu}\) as \(\mu\to\infty\),

\[ \ell^2_{\rm op}(\mu) \sim \frac{3}{2\sqrt{2\mu}} = \frac{3}{\sqrt{8\mu}}. \]

The factor \(8 = 2^3\) is \(\sigma^{-2}\) in units of \(\mu\): \(\sigma^2 = 1/(2a) \sim 1/(2\sqrt{2\mu}) = 1/\sqrt{8\mu}\).

4. Numerical verification¶

4.1 Numerical vs analytical¶

Comparison between direct numerical computation (matrix \(H\) with \(N=600\) points on \([-25,25]\)) and analytical computation with \(\psi_0 = \mathrm{sech}^{a(\mu)}(r)\) and mpmath integration at 40 digits:

\(\mu\)	\(a(\mu)\)	\(\ell^2_{\rm num}\)	\(\ell^2_{\rm anal}\)	diff
1.0	1.0000	3.45361043	3.45436154	0.022%
1.5	1.3028	2.22222712	2.22293719	0.032%
2.0	1.5616	1.67140280	1.67211855	0.043%
3.0	2.0000	1.15812782	1.15887005	0.064%
5.0	2.7016	0.76514280	0.76592314	0.102%
10.0	4.0000	0.46443338	0.46525320	0.177%
20.0	5.8443	0.29627936	0.29712267	0.285%

Numerical and analytical agree to within \(0.3\%\). The difference grows with \(\mu\) and is attributable to discretization (the width \(\sigma\sim 1/\sqrt{2a}\) becomes comparable to \(dr\)).

4.2 Convergence to \(3/2\)¶

\(\mu\)	\(a(\mu)\)	\(\ell^2_{\rm anal}\)	\(a\cdot\ell^2\)
\(\mu\)	\(a(\mu)\)	\(a\cdot\ell^2\)	\(3/2 - a\cdot\ell^2\)
---	---	---	---
10	4.00	1.86101279	\(-0.36101\)
20	5.84	1.73647067	\(-0.23647\)
50	9.51	1.63985974	\(-0.13986\)
100	13.65	1.59565221	\(-0.09565\)
200	19.51	1.56607405	\(-0.06607\)
500	31.13	1.54093780	\(-0.04094\)
1000	44.22	1.52865045	\(-0.02865\)

The product \(a\cdot\ell^2\) decreases monotonically toward \(3/2 = 1.5\) from above. The residual \(a\ell^2 - 3/2\) shrinks roughly as \(1/a\) (\(0.0957\cdot 13.65 \approx 1.31\), \(0.0287\cdot 44.22 \approx 1.27\)), consistent with the \(O(1/a)\) correction; at \(\mu=1000\) it is still \(1.9\%\).

5. Discussion¶

5.1 What this is not¶

This note does not claim that:

the number \(3/2\) has any relation to the golden ratio \(\Phi = (1+\sqrt5)/2\);
there are connections with \(\psi = (1-\sqrt5)/2\), \(5/\Phi\), \(2/\Phi\), or other algebraic constants;
the factor \(3/2\) encodes any hidden physical structure.

The factor \(3/2\) is the ratio between the fourth and the second moment of a Gaussian:

\[ \frac{\langle r^4\rangle_{\rm Gauss}}{\langle r^2\rangle_{\rm Gauss}} = \frac{3\sigma^4}{\sigma^2} = 3\sigma^2. \]

The \(3\) is the coefficient of the Gaussian fourth moment, the \(2\) comes from \(\sigma^2 = 1/(2a)\). Nothing else.

5.2 What this is¶

An exact asymptotic result, derivable, with the following content:

The ground state of \(H(\mu) = -d^2/dr^2 - 2\mu\,\mathrm{sech}^2(r)\) is \(\psi_0 = \mathrm{sech}^{a(\mu)}(r)\) with \(a(\mu) = (\sqrt{1+8\mu}-1)/2\).
The effective width \(\ell^2_{\rm op}(\mu) = \langle r^4\rangle/\langle r^2\rangle\) is asymptotically \(3/(2a(\mu))\).
The first-order correction is \(O(a^{-1})\), consistent with the non-Gaussian expansion of \(\ln\mathrm{sech}\).

5.3 Application¶

This result has been used as an ingredient in the construction of a map between the Gentile gas \(Z_k(\beta) = \zeta(\beta)/\zeta((k+1)\beta)\) and the Jusufi–Anand metric \(f(r) = 1 - 2GMr/(r^2+\ell^2)\). In that context, \(\ell^2_{\rm op}\) is the effective operator length which, substituted into the metric, reproduces the horizon area. The asymptotic relation \(a\cdot\ell^2 \to 3/2\) is the one that fixes the scale of \(\ell^2\) for large \(\mu\). Section 2b of the script tests the matching \(\ell^2_{\rm op}(\mu,\varepsilon) = 4\,\ell^2_{\rm gas}(k,\beta)\) by choosing \((\mu,\varepsilon)\) on a \(40\times 8\) grid for each \((k,\beta)\): the match is within \(1.4\%\) for \(\beta\ge 1.5\) and between \(4.5\%\) and \(6.9\%\) at \(\beta = 1.05\), where the grid's smallest \(\ell^2_{\rm op}\) is reached.

6. References¶

G. Pöschl, E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys. 83, 143 (1933).
L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon (1977), §23.
F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and Quantum Mechanics, Phys. Rep. 251, 267 (1995).
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5, 329 (1996).
S. Pennacchio, A commutative map between the Gentile statistical family and the Jusufi–Anand metric via the Lambert W function, technical note (2026).

7. Verification¶

The verification code is scripts/dvpt/dvpt_poschl_teller_width.py: section 1 (setup), section 2a (numerical–analytical verification), section 2b (gas matching), section 3 (convergence to \(3/2\)).

All computations are reproducible with:

python scripts/dvpt/dvpt_poschl_teller_width.py

Requirements: numpy, scipy, mpmath. Default precision: mpmath at 40 decimal digits.