IR-truncated $\mathcal{PT}-$symmetric $ix^3$ model and its asymptotic spectral scaling graph

The $\mathcal{PT}-$symmetric quantum mechanical $V=ix^3$ model over the real line, $x\in\mathbb{R}$, is infrared (IR) truncated and considered as Sturm-Liouville problem over a finite interval $x\in\left[-L,L\right]\subset\mathbb{R}$. Via WKB and Stokes graph analysis, the location of the complex spectral branches of the $V=ix^3$ model and those of more general $V=-(ix)^{2n+1}$ models over $x\in\left[-L,L\right]\subset\mathbb{R}$ are obtained. The corresponding eigenvalues are mapped onto $L-$invariant asymptotic spectral scaling graphs $\mathcal{R}\subset \mathbb{C}$. These scaling graphs are geometrically invariant and cutoff-independent so that the IR limit $L\to \infty $ can be formally taken. Moreover, an increasing $L$ can be associated with an $\mathcal{R}-$constrained spectral UV$\to$IR renormalization group flow on $\mathcal{R}$. The existence of a scale-invariant $\mathcal{PT}$ symmetry breaking region on each of these graphs allows to conclude that the unbounded eigenvalue sequence of the $ix^3$ Hamiltonian over $x\in\mathbb{R}$ can be considered as tending toward a mapped version of such a $\mathcal{PT}$ symmetry breaking region at spectral infinity. This provides a simple heuristic explanation for the specific eigenfunction properties described in the literature so far and clear complementary evidence that the $\mathcal{PT}-$symmetric $V=-(ix)^{2n+1}$ models over the real line $x\in\mathbb{R}$ are not equivalent to Hermitian models, but that they rather form a separate model class with purely real spectra. Our findings allow us to hypothesize a possible physical interpretation of the non-Rieszian mode behavior as a related mode condensation process.