## 4 The Displacement-Dependence of the Spectrum  
The numerical results of the above-outlined algorithm are sampled in Figure 3 and by Tables 3-5. The key advantage of such a special-function-based version of the well known shooting algorithm is that in our model the trial-and-error functions \(\psi(x)\) are known in the exact (albeit rather clumsy, computer-stored) special-function confluent-hypergeometric form (which is closed and analytic but, unfortunately, still too long for a printed display). This enables us to keep the numerical errors under active control and to construct, systematically, the pairs of the lower and upper estimates of the bound-state energies.  
In Table 3 sampling the ground-state results, the non-monotonicity of the \(d-\)dependence of the energy is slightly counterintuitive but still easy to understand. It is obvious that the growth of the positive shift \(d>0\) leads, at the beginning, to the broadening of the well without a significant change of its size near the origin. This implies the possibility of an approximation of the potential by a broader effective well, i.e., in a solvable approximation, by \(V^{(eff)}=\omega^{2}\chi^{2}\) with \(\omega<1\) yielding the smaller ground state energy. Later, i.e., at the larger shifts \(d\) the central repulsive spike starts growing more quickly, separating the two HO wells and making them only weakly coupled. This must lead, approximately, to the double degeneracy of the low-lying spectrum and, in particular, to the unavoidable return of the ground state energy to its initial value at the sufficiently large \(d\)s.  
Table 4 illustrates that as long as the wave function of the first excited state vanishes in the origin, it is much less sensitive to the influence of the central spike. According to our tests, the energy just decreases with the growth of \(d\) and, in contrast to the behavior of the ground state energy, it never drops below \(E=1\) at the large shifts \(d\gg 1\). Marginally we may add that the test is facilitated by the observation that at the relevant spectral bound \(E=1\), the wave-function solutions with the correct odd-parity behavior in the origin may be further simplified and expressed in terms of error functions. This simplification of the general solution also facilitates the demonstration of the complex-value nature of the asymptotic nodal zero at any \(E<1\).  
Table 5 shows that the \(d\)-dependence of the second excited level (i.e., its decrease followed by the increase) is similar to its ground-state predecessor. Figure 3 complements these results by a typical sample of the first three wave functions at \(d=3/2\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Shift** & \multicolumn{2}{c}{**Energy Estimates**} \\ \hline \(d\) & **Lower** & **Upper** \\ \hline
0 & 3 & 3 \\
0.25 & 2.483910 & 2.483920 \\
0.50 & 2.060760 & 2.060770 \\
0.75 & 1.724710 & 1.724720 \\
1.00 & 1.468460 & 1.468470 \\
1.50 & 1.157479 & 1.157480 \\
2.00 & 1.035760 & 1.035770 \\ \(\infty\) & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The shift-dependence of the 1 st-excited-state energy.  
Figure 3: The first three double-well wave functions at \(d=3/2\).