I am looking for the exact data on $E_0(\lambda)$ for the anharmonicity $\lambda x^4$. The perturbative expansion is the following: $E_0(\lambda)\approx 0.5(1+1.5\lambda -5.25\lambda ^2+41.625\lambda^3-...)$, but I need a curve or a table for the exact values of the ground state energy for different $\lambda$ including a "strong coupling" regime $\lambda\gt 1$. Quickly accessible data in any form will be appreciated. Thanks.


1 Answer 1


The problem is not exactly solvable. But the recent paper http://journals.jps.jp/doi/abs/10.7566/JPSJ.83.034003 gives exact Pade-approximants of various orders, which are probably quite accurate.

  • $\begingroup$ Thanks to Vladimir and you for this interesting contribution! I'm surprised you didn't import the question to PO with your usual comment: "There's an answer here on PO" ;) $\endgroup$ Mar 31, 2015 at 20:51
  • $\begingroup$ @LarryHarson: The paper contains the exact values of $E_0$ for $\lambda = 0.05,\, 0.1,\, 0.5,\, 5,\, 50$. The paper notations are somewhat different, and “my” $E_0(\lambda)$ is expressed via "their" $E_0(1,\beta)$ in the following way: $E_0(\lambda)=0.5E_0(1,2\lambda)$. There is also an exact curve for $0\le \lambda\le 1$ in another paper: physics.ucsc.edu/~peter/115/anharmonic.pdf. I used the curve data and $E_0(\lambda=5)$ to plot Fig. 8 on PO (physicsoverflow.org/27226/…) $\endgroup$ Apr 1, 2015 at 9:50
  • $\begingroup$ @LarryHarson: I link to an answer on PO only if it is longer than a few lines. $\endgroup$ Apr 1, 2015 at 13:27
  • $\begingroup$ An exact curve for $\lambda < 5$ can be found here arxiv.org/pdf/quant-ph/0305128.pdf, Figure 2. Here again, one has to divide the energy and the g-axis by 2 to be coherent with my notations. $\endgroup$ Apr 3, 2015 at 10:28

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