Approximaths

Anharmonic oscillator II

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Remember that the perturbative series for the anharmonic oscillator is

Anharmonic oscillator I

\displaystyle E_0(\epsilon) = \frac{1}{2} + \frac{3}{4}\epsilon - \frac{21}{8}\epsilon^2 + \frac{333}{16}\epsilon^3 + \mathcal{O}(\epsilon^4)

We used Padé approximants to compute the ground state energy E_0(\epsilon). Now we aim to calculate E_0(\epsilon) using Borel summation. The (formal) Borel sum is given by

\displaystyle \int_{0}^{\infty} e^{-t} \sum_{n=0}^\infty \frac{a_n}{n!} (\epsilon t)^n \, dt.

Using the first three coefficients of the perturbative series, the truncated Borel transform is approximated by

\displaystyle \mathcal{B}E_0(t) \approx \frac{1}{2} + \frac{3}{4} t - \frac{21}{16} t^2,

so the truncated Borel sum writes

\displaystyle E_0(\epsilon) \approx \int_0^\infty e^{-t} \left(\frac{1}{2} + \frac{3}{4} \epsilon t - \frac{21}{16} \epsilon^2 t^2 \right) dt = \frac{1}{2} + \frac{3}{4} \epsilon - \frac{21}{8} \epsilon^2.

since

\displaystyle \int_0^\infty \frac{1}{2} e^{-t} \, dt = \frac{1}{2}
\displaystyle \int_0^\infty e^{-t} \cdot \frac{3}{4} \epsilon \, t \, dt = \frac{3}{4} \epsilon
\displaystyle \int_0^\infty e^{-t} \cdot \left( -\frac{21}{16} \epsilon^2 t^2 \right) dt = -\frac{21}{8} \epsilon^2

This doesn’t really seem like progress, since the first terms of the Borel sum are identical to those of the perturbative expansion. The perturbative expansion E_0(\epsilon) = \sum_{n=0}^\infty a_n \epsilon^n above has coefficients that grow factorially. Bender and Wu showed that for large n,

\displaystyle a_n \sim -r\, (-1)^n \left(\frac{3}{2}\right)^n \Gamma\!\left(n+\tfrac12\right), \qquad r>0.

Since

\displaystyle \Gamma\!\left(n+\tfrac12\right) \sim n!\, n^{-1/2},
formule avec n! en rouge et gras

thus the series diverges for all \epsilon \neq 0.

Borel summation improves convergence by dividing out this factorial growth.

In summary, even with just a few terms, Borel summation correctly recovers the perturbative results for the anharmonic oscillator then It turns the divergent series into a well-defined and useful result.

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