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, $xinmathbb{R}$, is infrared (IR) truncated and considered as Sturm-Liouville problem over a finite interval $xinleft[-L,Lright]subsetmathbb{R}$. 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 $xinleft[-L,Lright]subsetmathbb{R}$ 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 $Lto 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 $xinmathbb{R}$ 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 $xinmathbb{R}$ 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.