## 1 Introduction  
In the traditional realistic applications of quantum mechanics one picks up a tentative one-dimensional potential \(V(x)\) in a way inspired, say, by the principle of correspondence. Subsequently, one solves the corresponding \"user-friendly\" (i.e., ordinary differential) bound-state Schrodinger equation  
\[-\ \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\,\psi_{n}(x)+V(x)\,\psi_{n}(x)=E_{n} \,\psi_{n}(x)\,,\quad\psi_{n}(x)\in L^{2}(\mathbb{R})\,,\quad n=0,1,\ldots \tag{1}\]  
numerically, arriving at the acceptance or rejection of the model in dependence on the agreement or disagreement between the predictions and the measurements [1].  
The task becomes simplified when the needs of the quantum phenomenology are met by a function \(V(x)\) which is analytic. Then it makes sense to restrict attention to the potentials which are exactly solvable (ES, [2]) or at least partially exactly _alias_ \"quasi-exactly\" solvable (QES, [3]). Indeed, the \"redundant\" analyticity requirement may prove useful, e.g., when one treats the elementary Schrodinger Equation (1) as a mere methodical laboratory needed, say, during the study of supersymmetric systems [4].  
Whenever a suitable ES or QES model is not available, people feel motivated to assign the solvability status also to some non-analytic models \(V(x)\). Typically, these models are of the piecewise constant square-well type [5]. In applications, their decisive formal merit is that their bound-state wave functions \(\psi_{n}(x)\) can be defined, using the matching method, in terms of elementary trigonometric functions.  
In between the analytic and non-analytic extremes one finds a few parity-symmetric models \(V(x)=V(-x)\) in which the analyticity is only broken in the origin. Under this assumption it was possible to assign the QES solvability status to the spatially symmetrized quartic anharmonic oscillator [6] or to the symmetrized sextic anharmonic oscillator [7].  
In the non-polynomial exact solvability (NES) context the non-QES exact spectra and bound states of the parity-symmetric models proved obtainable, for a few parity-symmetric potentials, via the \(x=0\) matching of their exact special-function left and right parts \(\psi_{n}^{(\pm)}(x)\) of the wave functions. This enabled one to assign the NES status to the symmetrized exponential potential \(V(x)\sim\exp|x|\) (with \(\psi_{n}^{(\pm)}(x)\) expressed in terms of Bessel functions [8,9]), to its asymptotically decreasing alternative [10] and/or to the symmetrized Morse potential \(V(x)\) (i.e., to the sum \(V(x)\) of the two exponentials for which \(\psi_{n}^{(\pm)}(x)\) appeared proportional to a confluent hypergeometric function [11,12]).  
In our present paper we intend to complement these results by showing that in some of the parity-symmetric quantum models and, in particular, for  
\[V(x)=V_{[d]}(x)=\left\{\begin{array}{cc}(x-d)^{2},&x>0,\\ (x+d)^{2},&x<0\end{array}\right. \tag{2}\]  
in Equation (1), the QES and NES regimes can coexist. One might also add that disproportionately, the ES quantum systems with the single-well harmonic-oscillator potential \(V(x)=V^{(HO)}(x)=x^{2}\) is discussed in virtually any textbook on quantum mechanics (cf., e.g., [13]). In contrast, the most straightforward twinned, \(d\)-displaced generalization (2) of the harmonic-oscillator model with \(d\in\mathbb{R}\setminus\{0\}\) is never mentioned. In what follows, we intend to fill the gap.