## 5 Conclusions  
In conclusion, let us re-emphasize that in the current literature there exists a large number of papers and studies devoted to the experiments and realistic models of quantum physical systems described by a simplified, one-dimensional (i.e., ordinary differential) Schrodinger equation (i.e., often, by our Equation (1) which is defined on the whole real line of the coordinate). Still, in contrast to the dominance of the scene by the papers describing the bound states lying in a single-well potential, one finds perceivably less papers dealing with the potentials having a double-well shape. Naturally, one of the reasons is purely formal. One often finds that many methods of solving Schrodinger equations (and, among them, the perturbation-expansion methods) happen to fail when applied to the double-well systems. This leads to a remarkable conflict between the prevailing, purely mathematically motivated preference of the single-well models in various methodical considerations, and the frequently encountered lack of some sufficiently efficient construction recipes, especially in the context of the double-well scenarios which are much more interesting, in the phenomenological context, mainly due to the emerging manifestations of the influence of the tunneling effect. In such a context we sought, in our present paper, for a nontrivial support of the existing tendencies towards an extension of the class of the \"exactly solvable\" 1D potentials \(V(x)\) from their completely analytic subset to its non-analytic generalizations. We are persuaded that such a tendency is natural and well motivated. We also believe that in the nearest future our approach might find some immediate and successful applications, e.g., during some of the popular and, usually, partially non-numerical descriptions of the physical systems (as studied, e.g., in [39]) in which, in the language of mathematics, the double-well structure of the potential \(V(x)\) is necessary, but in which the phenomenon of the avoided level crossing is expected to occur while being not so easily detected by the mere finite-precision experimental measurements of the spectra. Last but not least let us mention that among the physical systems to which our present methodical innovation could also appear, in the nearest future, to be immediately applicable are also, potentially, all of the various more-dimensional realistic multi-well models as encountered, e.g., in quantum chemistry [40] and/or in the quantum-computation-motivated mesoscopic physics of the systems of quantum dots [41]. Indeed, under the widespread interaction-analyticity constraint, just the approximate solutions of the partial differential Schrodinger equations are usually available in these systems. In the methodical context, a strengthening and/or independent complement of our present arguments may be also found in Ref. [11] where we sampled the methodical as well as practical gains of the approach by introducing a spiked, centrally symmetrized 1-D potential of the Morse-oscillator type. In spite of its non-analyticity in the origin the model was still shown to exhibit several features of the more conventional complete exact solvability. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Shift** & \multicolumn{2}{c}{**Energy Estimates**} \\ \hline \(d\) & **Lower** & **Upper** \\ \hline
0 & 5 & 5 \\
0.25 & 4.34600 & 4.34700 \\
0.50 & 3.79410 & 3.79420 \\
0.75 & 3.34470 & 3.34471 \\
1 & 3 & 3 \\
1.50 & 2.64860 & 2.64870 \\
2.00 & 2.73500 & 2.73510 \\ \(\infty\) & 3 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Shift-dependence of the 2 nd-excited-state energy (at \(d=1\) the state is quasi-exact). For the similar and generic 1-D interactions characterized by the various incomplete forms of solvability (such as QES) the situation appears mathematically more complicated because the presence of the non-analyticities at some coordinates may restrict, severely, the practical as well as methodical applicability of the incomplete constructions in quantum physics.