## 1 Introduction
### Motivation in Physics  
In the context of physics the above-mentioned NES constructions were motivated by the needs of molecular physics so that the non-analyticity of \(V(x)\) in the origin appeared inessential. Nevertheless, even in this area of physics the analyticity of the benchmark models \(V(x)\) re-emerged as essential, mainly due to the related possibility of a qualitative analysis of _quantum_ dynamics using the mathematical tools of the _classical_ theory of catastrophes [14].  
The simplest forms of the theory (viz., the so called fold and cusp catastrophes) proved mostly needed [15]. Let us just remind the readers briefly that in this language the equilibrium of a classical dynamical system is identified with a minimum of a suitable \"Lyapunov\" potential \(V(x)\)[16]. This enables one to classify all of the qualitatively different patterns of the bifurcation of the equilibria in terms of the parameter-dependence of the minima.  
By far the most interesting and useful example of such a bifurcation pattern is called cusp catastrophe. In this case one conventionally chooses  
\[V(x)=V^{(cusp)}(x,a,b)=x^{4}+ax^{2}+bx\,. \tag{3}\]  
It is then easy to verify that such a Lyapunov function possesses a single minimum at \(a>0\) but two minima (i.e., a double-well shape) at \(a<0\) and at a not too large \(b\). It is also worth noting that in the \(a\)-\(b\) plane the boundary of transition between the two regimes is a spiked, cusp-shaped curve [17]. Thus, one can use the formalism for a qualitative description of dynamics, say, of the Josephson junction between the two Bose-Einstein condensates [18].  
After one turns attention to the genuine quantum forms of dynamics, the work with the classical \(a\)-\(b\) plane geometry of model (3) ceases to be satisfactory. One can expect that the classical bifurcation phenomena become smeared out by the quantization. In particular, in the double-well case, due to the tunneling, one would only have to move far away from the fine-tuned cusp singularity in order to be able to distinguish, experimentally, between the low-lying states which remain localized, in general, either near the left minimum \(R_{-}\) or near the right minimum \(R_{+}\) of \(V^{(cusp)}(x,a,b)\).  
In the latter, deep-double-well dynamical regime the tunneling becomes suppressed since the growth of the distance \(R_{+}\)-\(R_{-}\) between the minima becomes accompanied by their increasing separation mediated by a growing and thickening barrier [19]. The approximate calculation of the bound states may be then facilitated by the use of a fairly efficient local harmonic-oscillator approximation,  
\[V^{(cusp)}(R_{\pm}+\xi,a,b)=V^{(cusp)}(R_{\pm},a,b)+\omega_{\pm}^{2}\,\xi^{2}+ \mathcal{O}(\xi^{3})\,.\]  
More or less equivalently, one may replace \(V^{(cusp)}(x,a,b)\) by a global double-well harmonic oscillator,  
\[V(x,a,b)=\min\left[\omega_{-}^{2}\,(x-R_{-})^{2}+D_{-}\,,\,\omega_{+}^{2}\,(x- R_{+})^{2}+D_{+}\right]. \tag{4}\]  
Whenever the difference \(\left|\omega_{-}-\omega_{+}\right|\) or \(\left|V^{(cusp)}(R_{-},a,b)-V^{(cusp)}(R_{+},a,b)\right|\) becomes large, the low-lying states remain localized just near one of the minima. The realization of a genuine quantum effect of bifurcation may only be expected to occur when \(\omega_{-}^{2}\approx\omega_{+}^{2}\) and \(D_{-}\approx D_{+}\). Thus, we arrive at the spatially symmetric toy-model potential  
\[V(x,d,\omega)=\omega^{2}\,\min\left[(x+d)^{2},(x-d)^{2}\right] \tag{5}\]  
in which we may scale out \(\omega\to 1\) and obtain the ultimate double-well version of the one-parametric potential (2) of our present interest.  
It is worth adding that our asymptotically quadratic one-dimensional potential (2), viz., \(V(x,d,1)=x^{2}-2d\left|x\right|+d^{2}\) with \(d\
eq 0\) represents in fact just a single-well interaction at a negative displacement \(d<0\), and it only acquires the more interesting double-well shape at a positive \(d>0\) (see its sample in Figure 1 where \(d=2\)).