
Requiring that they satisfy the Schr\"odinger equation, we obtain a recursion
relation for $a_{2l+P}^{[2k]}$ and $c^{[2m]}$:
\begin{multline}
\label{eq:recur1}
\qquad (4l+1+2P)a_{2l+P}^{[2k]}-(2l+2+P)(2l+1+P)a_{2(l+1)+P}^{[2k]}\\
 -2a_{2(l-2)+P}^{[2(k-1)]}+a_{2(l-3)+P}^{[2(k-2)]}
 =2\sum_{m=0}^{k}c^{[2m]}a_{2l+P}^{[2(k-m)]}.\qquad
\end{multline}
Second, the perturbation theory around the harmonic oscillator
states of the side potential wells is defined by shifting the origin of the
coordinate to one of the minimum of the side potentials $q\to q\pm 1/g$ and
then decomposing the Hamiltonian:
\begin{align}
H^{\mathrm{ST}}(q\pm 1/g;g,0)=-\frac{1}{2}\frac{\mathrm{d}^2}{\mathrm{d} q^2}+\frac{4}{2}q^{2}
 \pm 6g q^{3}+\frac{13}{2}g^{2}q^{4}\pm 3g^{3}q^{5}+\frac{1}{2}g^{4}q^{6}.
\end{align}
The perturbative corrections are similarly introduced by\footnote{Here
we note that all the perturbative coefficients of odd powers in $g$ for
the spectrum vanish due to the parity symmetry of the original Hamiltonian
\eqref{eq:HamST0}.}
\begin{align}
E(g)=\sum_{m=0}^{\infty}g^{2m}c^{[2m]},\qquad
 \psi(q;g)=\mathrm{e}^{-q^{2}}\sum_{k=0}^{\infty}g^{k}
 \sum_{l=0}^{\infty}a_{l}^{[k]}q^{l}.
\end{align}
The recursion relation for $c^{[m]}$ and $a_{l}^{[k]}$ in this case is then
given by
\begin{multline}
\label{eq:recur2}
\qquad (4l+2)a_{l}^{[k]}-(l+2)(l+1)a_{l+2}^{[k]}\pm 12 a_{l-3}^{[k-1]}\\
 +13 a_{l-4}^{[k-2]}\pm 6 a_{l-5}^{[k-3]}+a_{l-6}^{[k-4]}
 =2\sum_{m=0}^{[k/2]}c^{[2m]}a_{l}^{[k-2m]}.\qquad
\end{multline}
For the lowest state, $c^{[0]}=1$ and $a_{l}^{[0]}=0$ for all $l>0$.

On the other hand, as we have mentioned previously, the imaginary parts
of the nonperturbative contributions are to be canceled with those of
the perturbative ones in the framework of the valley method. This leads to
the following dispersion relation~\cite{ST02,AKOSW99}:
\begin{align}
\label{eq:dispe}
c^{[2m]}=-\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d} g^2
 \frac{\operatorname{Im} E_{\mathrm{NP}}(g)}{g^{2m+2}}\,.
\end{align}
This relation enables us to predict the large order behavior of
the perturbation series for the eigenvalues. For the lowest three states
($n_{0}=n_{\pm}=0$) in the present case ($\epsilon=0$), we obtain from
Eqs.~\eqref{eqs:npcSTs1}--\eqref{eqs:npcSTs2} and \eqref{eq:dispe}
\begin{subequations}
\label{eqs:lobps}
\begin{align}
\label{eq:lobps0}
c_{0}^{[2m]}&\sim -\frac{2^{5/4}}{\pi\Gamma(3/4)}\,2^{m}\Gamma\biggl(
 m+\frac{3}{4}\biggr)\equiv\bar{c}_{0}^{[2m]},\\
c_{1(2)}^{[2m]}&\sim -\frac{8\sqrt{2}}{\pi^{3/2}}\,
 2^{m}\Gamma\biggl(m+\frac{3}{2}\biggr)\equiv\bar{c}_{1(2)}^{[2m]}.
\end{align}
\end{subequations}
Therefore, we can check the validity of the results \eqref{eqs:npcSTs1}--%
\eqref{eqs:npcSTs2} by comparing the predicted asymptotic forms
$\bar{c}^{[2m]}$ in Eqs.~\eqref{eqs:lobps} with the exact perturbative
coefficients $c^{[2m]}$ calculated using the recursion relations
\eqref{eq:recur1} and \eqref{eq:recur2}.

\begin{table}[h]
\caption{The ratios of the exact values of the perturbative coefficients
$c^{[2m]}$ to the predicted asymptotic values $\bar{c}^{[2m]}$.}
\label{tb:large}
\begin{center}
\[
\arraycolsep = 5pt
\begin{array}{rll}
\hline
m & c_{0}^{[2m]}/\bar{c}_{0}^{[2m]} & c_{1(2)}^{[2m]}/\bar{c}_{1(2)}^{[2m]}
 \rule{0pt}{15pt}\rule[-7pt]{0pt}{0pt}\\
\hline
 20 & 0.8946472445 & 0.7797002850\\
 40 & 0.9493285320 & 0.8904365552\\
 60 & 0.9665686152 & 0.9268736279\\
 80 & 0.9750492671 & 0.9451085558\\
100 & 0.9800964967 & 0.9560611732\\
120 & 0.9834448543 & 0.9633690036\\
140 & 0.9858286759 & 0.9685922139\\
160 & 0.9876123158 & 0.9725115700\\
180 & 0.9889971178 & 0.9755611660\\
200 & 0.9901034160 & 0.9780016290\\
220 & 0.9910075563 & 0.9799989041\\
240 & 0.9917603143 & 0.9816636733\\
260 & 0.9923967744 & 0.9830725934\\
280 & 0.9929419564 & 0.9842804381\\
300 & 0.9934141831 & 0.9853273870\\
\hline
\end{array}
\]
\end{center}
\end{table}

In Table~\ref{tb:large}, we show the ratios $c^{[2m]}/\bar{c}^{[2m]}$ up to
the order $m=300$. We easily see that the exact values indeed tend to the
predicted asymptotic values for both the ground and excited states and thus
confirm the correctness of our formulas, at least, for their imaginary parts.


\section{Interplay between Nonperturbative and Perturbative Corrections}
\label{sec:inter}

In Section~\ref{sec:nonpe}, we have investigated the nonperturbative
corrections for the excited states and confirmed that our formulas
obtained from the valley method calculation are more accurate than the ones
obtained from the instanton calculation with the dilute gas approximation.
Although the analysis of the large order behavior
in the previous section ensures the correctness of the imaginary parts
of our formulas for both the ground and excited states, it does not
necessarily mean that the real part of our formula for the ground state
is also correct. In order to check the accuracy of the nonperturbative
spectral shift for the ground state we must resort to other means.

As we have mentioned previously, there exist perturbative corrections
to the harmonic oscillator spectrum in addition to nonperturbative ones.
Hence, each of the spectrum is formally expressed as
\begin{align}
\label{eq:decom}
E(g)=E^{(0)}+E_{\mathrm{NP}}(g)+E_{\mathrm{P}}(g),
\end{align}
where $E^{(0)}$ denotes the harmonic oscillator spectrum when $g=0$,
$E_{\mathrm{NP}}$ the purely nonperturbative part which cannot be represented by
a formal power series in $g^{2}=\omega^{-1}$, and $E_{\mathrm{P}}$ the remaining
perturbative part. The decomposition \eqref{