## 3 Special-Function Solvability
### Single-Well Dynamical Regime with \(D<0\)  
Very recently we imagined that any quantization of a classical catastrophe must replace the usual schematic time-dependent trajectories \(x=x(t)\) in a classical space of states by certain selfadjoint operators \(\hat{x}(t)\) in a suitable infinite-dimensional physical Hilbert space \(\mathcal{H}^{(P)}\)[31; 32; 33]. Thus, after quantization the concepts such as bifurcation (etc.) must necessarily be transferred from the domain of algebraic geometry to the area of functional analysis and spectral theory. Naturally, this implies that the non-analyticity of \(V_{[d]}(x)\) becomes inessential while its solvability may start playing the role of a significant merit in comparison with a purely numerical nature of AHO Schrodinger equations [34; 35; 36; 37]. In other words, the twinned harmonic oscillator (2) with a variable real displacement \(d\) and with two minima if \(d>0\) seems suitable and acceptable as a benchmark quantum model with catastrophic features. Besides the serendipitous and rather formal QES property as discussed above, a turn of attention to the fully general twinned harmonic-oscillator model (2) is recommended from a phenomenological perspective. Another independent reason would be again mathematical. After an elementary change of variables one reveals that Equation (1) coincides with the two branches of the confluent hypergeometric equation which are matched together in the origin. In this sense our model (1) + (2) resembles square wells and remains solvable via the matching of logarithmic derivatives of \(\psi(x)\) at \(x=0\). In the literature the conceptual appeal of the complete and, in principle, arbitrarily precise solvability based on the use of confluent hypergeometric special functions may be found advocated by multiple authors [1]. In what follows we will defend and accept such an NES point of view, demonstrating its applicability to the one-parametric family of potentials (2) in both of their one-centered and two-centered realizations. Once we recall the left-right symmetry (i.e., parity-symmetry, \(\mathcal{P}\)-symmetry) of our Hamiltonian, we may temporarily restrict our attention to the differential Equation (1) on one of the half-lines (i.e., say, with \(x\in(0,\infty)\)). Then we may change variables as follows,  
\[x-d=\sqrt{z}\in(-d,\infty)\,,\quad\psi(x)=\exp(-z/2)\,w(z)\,.\]  
Whenever we choose a negative \(d=-\mu^{2}<0\), this converts Equation (1) into [PERSON]'s equation  
\[z\frac{d^{2}w(z)}{dz^{2}}+(b-z)\frac{dw(z)}{dz}-aw(z)=0\,,\quad z\in(\mu^{4}, \infty) \tag{25}\]  
(cf. [29], SS13.2(i)) with parameters \(a=(1-E)/4\) and \(b=1/2\). As long as \(z\geq\mu^{2}>0\) we may ignore the branch-point singularity of Equation (25) at \(z=0\) and we may write down the general solution as a superposition of two independent confluent hypergeometric series, viz.,  
\[w_{(even)}(z)=M(a,b,z)=\ _{\rm 1}F_{1}(a,b,z)=\]  
\[=\sum_{s=0}^{\infty}\frac{(a)_{s}}{(b)_{s}s!}z^{s}=1+\frac{a}{b}z+\frac{a(a+1)}{ b(b+1)2!}z^{2}+\cdots \tag{26}\]  
and  
\[w_{(odd)}(z)=z^{1-b}\,M(a-b+1,2-b,z)\,. \tag{27}\]  
Subsequently, we have to satisfy the asymptotic physical boundary condition. This forces us to combine the two independent solutions \(w_{(even)}(z)\) and \(w_{(odd)}(z)\) of Equation (25) in the following unique manner,  
\[w_{(physical)}(z)=U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}\ M(a,b,z)+\]  
\[+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}\,M(a-b+1,2-b,z)\,. \tag{28}\]  
The choice of this function with the power-law asymptotical behavior \(U(a,b,z)\sim z^{-a}\) valid in the complex plane of \(z=x+\mu^{2}\) with the left cut from \(z=0\) to \(-\infty\) yields the correct, asymptotically vanishing wave functions at positive \(x\geq 0\) in closed form,  
\[\psi(x)=\psi(x,E)=\]  
\[=\exp(-(x+\mu^{2})^{2}/2)\ U\!\left((1-E)/4,1/2,(x+\mu^{2})^{2}\right). \tag{29}\]  
The formula remains manifestly dependent on the value of energy \(E\).