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 grap
h 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.
The helical magnetorotational instability is known to work for resistive rotational flows with comparably steep negative or extremely steep positive shear. The corresponding lower and upper Liu limits of the shear are continuously connected when some
axial electrical current is allowed to flow through the rotating fluid. Using a local approximation we demonstrate that the magnetohydrodynamic behavior of this dissipation-induced instability is intimately connected with the nonmodal growth and the pseudospectrum of the underlying purely hydrodynamic problem.
Using a homotopic family of boundary eigenvalue problems for the mean-field $alpha^2$-dynamo with helical turbulence parameter $alpha(r)=alpha_0+gammaDeltaalpha(r)$ and homotopy parameter $beta in [0,1]$, we show that the underlying network of diabol
ical points for Dirichlet (idealized, $beta=0$) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, $beta=1$) boundary conditions. In the $(alpha_0,beta,gamma)-$space the Arnold tongues of oscillatory solutions at $beta=1$ end up at the diabolical points for $beta=0$. In the vicinity of the diabolical points the space orientation of the 3D tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space induced geometry of the resonance zones explains the subtleties in finding $alpha$-profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.
A new class of semi-analytically solvable MHD alpha^2-dynamos is found based on a global diagonalization of the matrix part of the dynamo differential operator. Close parallels to SUSY QM are used to relate these models to the Dirac equation and to e
xtract non-numerical information about the dynamo spectrum.
