Halliday-Suranyi Approach to the Anharmonic Oscillator

In Ref. 22, Halliday and Suranyi consider demanding that

to fix $Z$ at each order $k$. This corresponds to the method of Fastest Apparent Convergence (FAC) used in pQCD.[23] For $k=1$, we need to solve

which, in general, has one real and two complex solutions. For the $m=0$ (i.e. $X=0$) case, the three solutions overlap and we have

The value of $E_{n}^{(0)}(Z)$ at $Z=Z_{n,\text{FAC}}^{(1)}$ is

Note that this expression scales as $\sim n^{4/3}$ for large $n$ as it should.

Graphically, \erefFAC is equivalent to searching for values of $Z$ at which the graphs for the 0th, and $k$th order approximations cross:

Another method for choosing $Z$ would be to require

This corresponds to the Principle of Minimum Sensitivity (PMS) used in pQCD.[23] For the $k=1$ case, PMS is equivalent to minimizing the expectation value of $\hat{H}=\hat{H}_{0}+\hat{V}$ using the harmonic oscillator eigenfunctions as the variational trial functions.[24] Indeed, $E_{n}^{(0)}(Z)$ and $E_{n}^{(1)}(Z)$ are respectively the expectation values of $\hat{H}_{0}$ and $\hat{V}$ for the $n$th eigenstate of $\hat{H}_{0}$. Imposing \erefPMScondition for $k=1$, we find

which, in general, has one real and two complex solutions. For the $m=0$ (i.e. $X=0$) case, we find

and the approximate value at $Z=Z_{n,\text{PMS}}^{(1)}$ is [24]

which is numerically similar to \erefZPapprox. For $k\geq 2$, PMS does not correspond to any variational calculation.

Graphically, \erefPMScondition looks for the values of $Z$ for which the slope of the $k$th order approximation is flat. This is illustrated for the $X=0$ case in \frefPMSfig, in which the six graphs shown are for the $n=0$ to $n=5$ states. For $k=1$, there is a unique flat location as we saw above. For $k=2$, there are two flat locations in which the one on the left is a local minimum while the one of the right is a local maximum. Though we cannot tell which one should be choosen beforehand, comparison with the exact results suggests we should choose the local minimum point on the left.

For $k=3$, there are three flat locations, where the two outer points are local minima, while the one in the middle is a local maximum. For all the cases considered, the left local minimum is closer to the exact result than the central local maximum. For the ground state, $n=0$, the right local minimum is closer to the exact result than the left local minimum, but undershoots it. For $n\geq 2$, though we cannot tell from \frefPMSfig, the right local minimum dips into the negative. Thus, we choose the left local minimum as our approximation for $E_{n}(0,M)$.

The values of $Z$ chosen in this way are plotted in \frefPMS1, and the resulting approximate values normalized to the exact value are shown in \frefPMS2. At $k=3$, the approximate values are within 1% of the exact value for all $n$ considered.

The problem with \erefZPcondition and \erefPMScondition is that they do not uniquely determine $Z$ for $k\geq 2$, and choosing the smallest real solution was somewhat arbitrary and there is no guarantee that this choice would be optimal. To remedy this problem, let us consider the following. Denote the perturbative expansion of the eigenstates of $\hat{H}$ in powers of $\hat{V}$ as

where $|n^{(k)}\rangle$ includes all terms proportional to $k$ powers of $\hat{V}$. We have commented in the previous subsection that

so the PMS condition, \erefPMScondition, applied to the $k=1$ case minimizes $\langle H\rangle_{n}^{(0)}$ using $|n^{(0)}\rangle$ as the trial function. Now, consider the expectation value of $\hat{H}$ for the state $|n^{0}\rangle+|n^{(1)}\rangle$ :

Minimizing $\langle H\rangle_{n}^{(1)}$ by varying $|n^{0}\rangle+|n^{(1)}\rangle$ should improve the approximations for the $n=0$ (ground state, even parity) and the $n=1$ (1st excited state, odd parity) cases without ever going below the exact values.

The numerator and denominator of \erefH1def are

Therefore,

We see that $\langle H\rangle_{n}^{(1)}$ can be considered the 3rd-order perturbative energy $\sum_{j=0}^{3}E_{n}^{(j)}$ corrected by a class of higher order terms which have been resummed into the factor $(1+\langle n^{(1)}|n^{(1)}\rangle)^{-1}$. This is similar in spirit to Renormalization Group resummation, or the Brodsky-Lepage-Mackenzie method used in pQCD.[23] The correction renders $\langle H\rangle_{n}^{(1)}$ positive definite, and avoids the problem $\sum_{j=0}^{3}E_{n}^{(j)}$ has of becoming negative around the right local minimum for $n\geq 2$.

To calculate $\langle H\rangle_{n}^{(1)}$ we need, in addition to \erefHSterms,

The graphs of $\langle H\rangle_{n}^{(0)}$ and $\langle H\rangle_{n}^{(1)}$ as well as their stationary points are shown for the $X=0$ case in \frefVARfig for $n=0$ through $n=5$. Comparing the $Z$-dependence of $\langle H\rangle_{n}^{(1)}$ (dashed line) and $\sum_{j=0}^{3}E_{n}^{(j)}$ (dotted line), we can see that they both have three stationary points: two local minima and one local maximum.

For the $n=0$ (ground state, even parity) and $n=1$ (1st exited state, odd parity) cases, the global minimum of $\langle H\rangle_{n}^{(1)}$ will provide the best approximation without undershooting the exact values. Indeed, comparing the graphs of $\langle H\rangle_{n}^{(1)}$ (dashed line) and $\sum_{j=0}^{3}E_{n}^{(j)}$ (dotted line) for these cases in \frefVARfig, we can see that the right local minimum has been lifted to just above the exact value. These global minima of $\langle H\rangle_{0}^{(1)}$ and $\langle H\rangle_{1}^{(1)}$ compared to the exact values are

which are amazingly accurate (better than 0.1%) for a 3rd order perturbative calculation without any small expansion parameter.

For the $n\geq 2$ cases, the global minimum of $\langle H\rangle_{n}^{(1)}$ does not have any special meaning. Indeed, for the $n=2$ case we can see from \frefVARfig that the global minimum of $\langle H\rangle_{2}^{(1)}$ is a poor approximation. Nevertheless, for all $n\geq 2$ cases the right local minimum is now positive, and the spread of $\langle H\rangle_{n}^{(1)}$ in the range between the two local minima are greatly reduced compared to that of $\sum_{j=0}^{3}E_{n}^{(j)}$. This is shown in \frefEHspread. For $n\geq 3$, the function $\langle H\rangle_{n}^{(1)}$ is flat enough between the two local minima, so it does not matter what value of $Z$ is chosen in that range.

As we continue this procedure to higher orders, we conjecture that $\langle H\rangle_{n}^{(k)}$ will become flatter for a wider range of $Z$ with increasing $k$. For instance, the next function to consider in the sequence is

which we expect to have three local minima and two local maxima. The “wiggles” in $\langle H\rangle_{n}^{(2)}$ between these extrema should be smaller than those of $\langle H\rangle_{n}^{(1)}$.