# Regular reflection to Mach reflection (RR-MR) transition in Short wedges

Vinod Yeldho Baby1<EMAIL_ADDRESS>Vinoth Paramanantham1  G. Rajesh1
1Indian institute of Technology Madras, Chennai, India.

###### Abstract

Regular reflection (RR) to Mach reflection(MR) transitions
(RR$\leftrightarrow$MR) on long wedges in steady supersonic flows have been
well studied and documented. However, in a short wedge where the wedge length
is small, the transition prediction becomes really challenging owing to the
interaction of the expansion fan emanating from the trailing edge of the wedge
with the incident shock and the triple/reflection point. The extent of this
interaction depends on the distance between the wedge trailing edge and the
symmetry line (Ht). This distance is a geometric combination of the distance
of the wedge leading edge from the symmetry line (H), the wedge angle
($\theta$), and the wedge length (w). In the present work, the
RR$\leftrightarrow$MR transitions have been studied on short wedges using
analytical and computational methods, and the transition lines are plotted for
various Mach numbers and wedge lengths. The transition criterion strongly
depends on the wedge length, which can be so adjusted even to eliminate the
RR$\leftrightarrow$MR transitions till the wedge angle reaches the no-
reflection domain.

Figure 1: Ht,min and Ht,max for (a) MR and (b) RR.

##  1 Introduction

When a supersonic flow passes over symmetric wedges, a shock
interaction/reflection occurs due to the flow turning. The shock reflection
can be either a Mach reflection (MR) or a regular reflection (RR), depending
on the flow parameters and wedge geometry. Figure 1 depicts the possible shock
reflections between the maximum and minimum Ht values, which have been
thoroughly investigated (Li & Ben-Dor, 1997). The Ht value depends on the
wedge length w, wedge angle $(\theta)$ and the centre line distance H of the
wedge leading edge, as shown in figures 1 (a) and (b). The shock reflection
process is studied thoroughly by varying the wedge angle and flow Mach number.
The wedge length is also an essential parameter, and based on the non-
dimensionalized wedge length (w/H), we can classify wedges as long or short.
As depicted in figure 3, most research on wedge flow focuses on long wedges
where the expansion fan from the trailing edge does not interact with the
incident shock wave. As shown in figure 3, the expansion fan will interact
with the incident shock wave in the short wedges. Due to expansion fan
interaction, the incident shock angle in short wedges decreases and curves
continuously from the point where the leading expansion characteristic meets
the shock wave to the triple/reflection point.

Figure 2: Long wedge.

Figure 3: Short wedge.

The curvature of the incident shock essentially depends on the extent of the
expansion fan interaction. Consequently, the incident shock angle at the point
of reflection will depend not only on the wedge angle and the flow Mach number
but also on the w/H ratio. Any attempt to predict the shock transitions on
short wedges must incorporate an accurate model of the incident shock
curvature resulting from its interaction with the expansion fan.

###  1.1 Previous works in Shockwave- Expansion fan Interaction

The interactions of shock waves and expansion waves in one-dimensional flows
are the subject of several investigations, in high-speed flows (Emanuel,
2001). A comprehensive description of the interactions of shock waves,
expansion waves and contact surfaces, mainly in shock tube flows, can be seen
by the work of Glass (1991). One of the early works in expansion fan shock
wave interaction in symmetric wedge flows was carried out by Li & Ben-Dor
(1996) in which an analytical solution was obtained for the curvature of the
shock wave due to its interaction with a centered expansion wave of the
opposite family. The curvature of the shock wave was approximated as a second-
order polynomial. In the experimental and numerical validations of this
analytical model (Nel et al., 2015), it was found that though the model
prediction was accurate for small interactions, it could not predict the shock
curvature for the stronger interactions. In another study (Hillier, 2007) on
the shock-wave/expansion-wave interaction and the transition between regular
and Mach reflections, analytical and numerical simulations were carried out to
investigate the inviscid interactions of an expansion wave with an incident
shock wave of the opposite family. The study was to stabilize a Mach
reflection in a parallel duct for different flow conditions. Most of the
analytical methods developed in shock reflections focused on predicting the MR
configuration and the Mach stem height, a finite length scale in an MR in long
wedges(Azevedo & Liu, 1993; Li & Ben-Dor, 1997; Mouton & Hornung, 2007). In
these models, the slip line, which originates from the triple point in a Mach
reflection, was assumed to be a straight line. Later, the refinement of the
slip line curvature was carried out by (Bai & Wu, 2017), resulting in a better
Mach stem prediction, as they considered the expansion waves generated over
the slip line and its interaction with the transmitted expansion waves through
the reflected shock. Further, numerical studies of a shock reflection in the
presence of an upstream expansion wave and a downstream shock wave for two-
dimensional flows by (Yao et al., 2013) reported that the expansion fan
delayed the RR$\leftrightarrow$MR transition at lower Mach numbers (M=2 – 3.5)
and expedited it at higher Mach numbers.

The analytical models discussed above were all for the interaction of the
shock wave with the opposite (either right running or left running) family of
expansion waves. On the contrary, in short wedges the incident shock is curved
due to the prolonged interaction of the expansion wave of the same family. The
study of this type of interaction is significant since the
RR$\leftrightarrow$MR transition may solely depend on the incident shock angle
at the reflection point, which is determined by the shock curvature due to the
interaction. It is hence, essential to determine the parameters governing the
shock angle at the reflection point and the change in transition criteria when
these parameters vary. No works have been reported till date to investigate
the short wedge effects in the RR$\leftrightarrow$MR transitions and the MR
configurations, to the best of the authors’ knowledge. In the present
investigation, an analytical model is hence developed to study the prolonged
interaction of an expansion wave with an incident shock wave of the same
family to predict the shock wave curvature. This is used to investigate the
shock transitions in symmetric short wedges. The analytical model is validated
with an in-house inviscid, 2D, structured finite volume solver with a fifth-
order WENO scheme (Paramanantham et al., 2022). The RR$\leftrightarrow$MR
transition lines are expected to be altered by the interactions of the
expansion wave with the shock wave due to the small wedge lengths.

##  2 Shock wave expansion fan interaction - Analytical Modeling

The shock–expansion interaction is modeled with the method of characteristics
(MoC). The model assumptions are that the flow is steady, compressible,
inviscid, two-dimensional (planar), isentropic, and rotational as it is a
curved shock. The entropy is constant only along the streamlines. The basic
configuration of the incident shock-expansion fan interaction model is shown
in figure 2. The characteristic and compatibility equations for two-
dimensional, isentropic, rotational, and supersonic flow are given below in
Table 1.

The initial value line for the MOC is the first characteristic expansion line
from the trailing edge (point D), which terminates at the intersection of the
incident shock wave and the first expansion line (point B). The solution
algorithm is initiated by giving a small increment of flow turn angle
downstream from point D. The flow properties are determined using the Prandtl-
Meyer relations. The C– curve from point D and C+ curve from point B intersect
at point B1. The flow properties and location at point B1 are obtained by
solving the above-mentioned characteristic and compatibility equations from B
and D. The curved shock is obtained by solving discrete points on the shock.
Since MoC is not valid across the shock, the solution point at the shock
curvature is obtained by iteratively computing the local pressure ratio across
the shock. Post-shock properties are calculated with the local incoming Mach
number and the assumed pressure ratio across the shock. These flow properties
should satisfy the compatibility equation along the C- curve originating from
the interior point B1. The iterations stop when this condition is satisfied.
The local shock angle is obtained, and the x and y positions of the curved
shock are obtained by solving the characteristic equations. Subsequent points
on the curved shock are obtained by incrementing the next flow turn angle at
point D and repeating the above steps. One of the algorithms calculates the
interior points in the flow field, while the other determines the discrete
points along the shock curvature. $\phi_{2}$ and $\theta_{eff}$ represents the
shock angle and effective turn angle at the reflection point R respectively as
shown in figure 4. The analytical model developed does not consider the
possibility of Mach reflection and is valid only for regular reflection.

Characteristic Equations |
---|---
$(dy/dx)=\lambda_{0}=v/u$ |  (Streamline)
$(dy/dx)=\lambda_{+}=tan(\alpha+\mu)$ |  (Right running characteristic line, ‘+’ subscript)
$(dy/dx)=\lambda_{-}=tan(\alpha-\mu)$ |  (Left running characteristic line, ‘-’ subscript)
Compatibility Equations |
$\rho VdV+d\rho=0$ |  (Streamline)
$dp-a^{2}d\rho=0$ |  (Streamline)
$\frac{\sqrt{(M^{2}-1)}}{(\rho V^{2})}dp_{+}+d\alpha_{+}=0$ |  (Right running characteristic Mach, ‘+’ subscript)
$\frac{\sqrt{(M^{2}-1)}}{(\rho V^{2})}dp_{-}-d\alpha_{-}=0$ |  (Left running characteristic line, ‘-’ subscript)
Table 1: Steady,two dimensional, isentropic, rotational supersonic flow Figure
4: Shock-expansion fan interaction model using MoC. Figure 5: (a) Superimposed
image of Analytical and computational result (b) Validation curve for
$w/H$=0.33

##  3 Shockwave expansion fan interaction - Validation

The validation of the analytical model with CFD is carried out by
superimposing the curved shock locus of the analytical results and the
computational results as shown in figure 5(a). It reveals that the analytical
prediction agrees very well with the CFD results qualitatively. The same is
done by varying the Mach number, w/H ratio and the wedge angle. The Mach
number vs the non-dimensional shock reflection distance [shown in figure 2]
(R/H) for different w/H ratios, obtained from analytical and CFD results are
plotted. Figure 5(b) shows one such plot of Mach number vs R/H for the w/H
ratio 0.33, for different wedge angles. It is seen that the maximum deviation
between the analytical and CFD results is less than 2$\%$.

##  4 Results and Discussion

The transition of the shock reflection in a short wedge depends on the extent
of the interaction of the expansion fan with the incident shock wave. Due to
the interaction of the expansion fan, the incident shock is weakened,
resulting in a decrease in the shock angle at the reflection point compared to
the case where there is no interaction. Therefore, the transition angle
between RR and MR is modified. The value of the shock angle at the point of
reflection depends on the degree of interaction of the expansion fan, or the
w/H ratio, while M and $\theta$ remain constant. As stated by Li & Ben-Dor
(1997), the transition angle is typically determined using the detachment
criterion for RR$\rightarrow$MR or the von Neumann criterion for
MR$\rightarrow$RR. In the following sections, we will quantify the transition
angles for the short wedges for various w/H ratios using the MoC method
described in section 2 and validate these results with higher-order numerical
simulations.

Figure 6: (a) Transition line - Shift in Detachment criteria for w/H = 0.5.
(b) Flow configurations corresponding to points A, (c) B, (d) C (all short
wedges), and (e) case without interaction(long wedge), where M =3, $w/H=0.5$
and $\theta=20^{\circ},25^{\circ},30^{\circ}$ respectively, for short wedges,
and M=3, $w/H=0.81$ and $25^{\circ}$ for the long wedge.

###  4.1 Transition lines- Detachment criterion validation

The delay in the transition criterion, specifically the detachment condition
within the short wedge, can be attributed to the attenuation of the incident
shockwave. The extent of the delay of this transition is determined by the w/H
ratio. The determination of the detachment condition involves the solution of
MoC for a specified Mach number (M), wedge angle ($\theta$), and w/H. The
detachment condition is generally shifted to higher wedge angles for the
shorter wedges and ends at the no-reflection domain. Figure 6 (a) shows the
shift in the detachment criterion for a w/H ratio of 0.5. The blue line
represents the modified detachment criterion resulting from the interaction of
the expansion fan with the incident shockwave, whereas the black solid line
represents the detachment condition for long wedges in the absence of
expansion fan interaction. Point S in figure 6 (a) depicts the beginning of
the interaction between the expansion fan and the incident shock wave for a
given w/H ratio. The deviation of the detachment condition occurs after this
value, and for values below the interaction point, it remains the same as the
long wedge. The transition line obtained from the MoC is validated with
numerical simulations for M = 3 flow over a supersonic wedge; three cases are
considered by keeping the Mach number constant and varying the wedge angle to
comprehend the type of reflection obtained from the numerical simulations. The
wedge angles are set to $\theta_{w}=20^{\circ},25^{\circ},30^{\circ}$
respectively (points A, B, and C in the figure 6(a)), and w/H is set to 0.5
(short wedge case) and 0.81 (long wedge case). Figures 6 (b-e) depict the
numerical schlierens of the short wedge flow fields at points A, B, and C and
the long wedge flow field simulation at point B. It can be seen from figures 6
(b-e) that the shock reflections correspond to points A and B are regular
reflections, and C is the Mach reflection for the short wedge, while for point
B, it is a Mach reflection for the long wedge. This confirms the shift of the
transition line for a short wedge, making a regular reflection at point B,
which was an erstwhile Mach reflection for the long wedge.

Figure 8 demonstrates that for all w/H ratios, a combination of M-$\theta_{w}$
exists for which the interaction begins, and the transition line shifts from
that point (refer to figure 6 (a)). Figure 8 shows that w/H = 0.33 begins to
deviate from the detachment condition line at point S, where M = 1.74 and
$\theta_{w}=9.6^{\circ}$. The shift ends at the no-reflection condition at
point E corresponding to M = 3.38 and $\theta_{w}=36.3^{\circ}$. For w/H =
0.33, the region under the curve SEPD will be RR, which eliminates MR flow
configurations for any combination of Mach number and wedge angle beyond point
E. Thus, the development of transition conditions for a shorter wedge has far-
reaching implications as one can select a domain in which only a regular
reflection occurs or control the size of the Mach reflection configuration by
adjusting the w/H ratio.

Figure 7: Shift in detachment criterion for various w/H ratios.

Figure 8: Shock polar representation for w/H = 0.33.

###  4.2 Shock polar representation of the interaction effects

Figure 8 depicts the polar shock representation of a long and short wedge with
a w/H ratio 0.33. The incoming flow has a Mach number of 3 and a wedge angle
of 30∘. At the reflection point, the post-shock Mach number for a long wedge
is 1.41, while it is 2.04 for a short wedge. The shock polar yields the MR
solution corresponding to the long wedge and the RR solution corresponding to
the short wedge. The shock polar reveals that the expansion fan interaction
decreases the required flow turn angle at the reflection point in short
wedges, even though the wedge angle remains unchanged.

###  4.3 Dual solution domain in short wedges

Another significant transition criterion in the wedge flows is the von Neumann
criterion. Intuitively, it makes sense to assume that as the w/H ratio
decreases, the von Neumann criterion line shifts towards higher wedge angles,
similar to the detachment condition. The dual solution domain, the region
between the detachment criterion line and the von Neumann criterion line, also
undergoes this shift. In order to comprehend the change in the dual solution
domain, transition lines for the von Neumann criterion were also drawn for
various w/H ratios. The von Neumann criterion line for the short wedge is more
complex than the detachment criterion, and the shift in the dual solution
domain can be subdivided into three types based on the location of the von
Neumann point B, which is the weak Mach reflection point as shown in figure 9
(a). We divided the shift in the dual solution domain into three categories as
Type I, Type II, and Type III, which are discussed below.

Figure 9: Dual solution domains for short wedges for different w/H ratios (a)
w/H=0.7 and long wedge and (b) w/H=0.5025 and long wedge

####  4.3.1 Type I Dual solution domain

In the type I domain, the von Neumann point ’B’ is identical to the long wedge
flows, and the von Neumann criterion follows the long wedge von Neumann
transition line prior to expansion fan interaction, shifting to higher wedge
angles once expansion fan interaction commences. The von Neumann criterion
($\theta_{N}$) line and detachment criterion ($\theta_{D}$) line for w/H ratio
0.7 is shown in figure 9(a). The path OBSDD and OBSNN represents the
$\theta_{D}$ and $\theta_{N}$ lines for long wedges. Both the transition lines
follow the same path as that for a long wedge, up to the points of shift (SD
and SN for the detachment line and von Neumann line, respectively), then
follow a different path as (S${}_{D}-$D’ and S${}_{N}-$N’). For a short wedge,
the path OBSDD’ and OBSNN’ represents the $\theta_{D}$ and $\theta_{N}$ line.
The point of the shift of the $\theta_{D}$ line is at point SD, where M=3.03
and wedge angle $\theta_{W}=$21.66∘, whereas the point of the shift of the
$\theta_{N}$ line is at point SN, where M=3 and wedge angle
$\theta_{W}=$19.66∘. As the w/H value decreases from 0.7, the point of shifts
SD and SN moves to the direction of the von Neumann point B, and merges with
the point B, where M = 2.202, $\theta_{W}=$15.15∘ and the w/H ratio is 0.5025
as shown in figure 9(b). Thus, short wedges with w/H ratios greater than
0.5025 will give a Type I dual solution domain. The characteristic of this
type is that the weak Mach reflection or von Neumann point B remains the same
for all w/H ratios greater than 0.5025 since both the points of shift from the
detachment line and von Neumann line SD and SN occur after point B.

####  4.3.2 Type II Dual solution domain

As illustrated in figure 10, the type II dual solution domain occurs when the
detachment condition occurs prior to the von Neumann point B for w/H ratios
less than 0.5025. Consequently, the type II domain begins at w/H = 0.5025 and
ends at w/H = 0.21, where the von Neumann point (B) reaches the no-reflection
line (point B”’). The dual solution domains for w/H ratios 0.25, 0.33, 0.5025,
and long wedge are shown in figure 10. In the figure, for the long wedges,
paths OBD and OBN represent the $\theta_{D}$ and $\theta_{N}$ lines,
respectively. Path OBD’(0.5), OBN’(0.5), represents the same for short wedge
of w/H ratio 0.5025. For w/H ratio 0.33, the $\theta_{D}$ line and
$\theta_{N}$ line follow the path OB’D’(0.33) and OB’N’(0.33) respectively as
shown in figure 10. The detachment criterion line intersects the no-reflection
line at point 3 and then follows the no-reflection line. A major part of the
dual solution domain is between the no reflection line and the $\theta_{N}$
line. The $\theta_{D}$ line and $\theta_{N}$ line for w/H ratio 0.25 are shown
as path OB”1 and OB”2, respectively. The dual solution domain for w/H=0.25 is
extremely small, as seen in the zoomed-in picture in figure 10. The dual
solution domain is hardly visible for wedges with w/H ratios less than 0.25.
Thus for short wedges with w/H ratios less than 0.5025, the von Neumann point
B shifts in the direction of the no-reflection domain and eventually meets the
no-reflection line at point B”’, where the $\theta_{D}$ line of w/H ratio 0.21
intersects with the no-reflection line. Moreover, as the w/H value is reduced
from 0.5025 to 0.21, the dual solution domain diminishes and shifts towards
the no-reflection domain.

Figure 10: Dual solution domains for long and short wedges

####  4.3.3 Type III Dual solution domain

The von Neumann solution does not exist for long wedges for Mach numbers less
than 2.202, where the wedge angle $\theta_{w}=$15.15∘. However, in short
wedges, as the w/H ratio decreases, the wedge angle corresponding to the von
Neumann point moves to higher wedge angles. The limiting factor to this shift
in wedge angle is when von Neumann point intercepts the no-reflection line
shown as point B”’ in figure 10. The value of the w/H ratio whose $\theta_{D}$
line passes exactly through this point B”’ will give the limiting point of von
Neumann solutions in short wedges. The value calculated is very close to 0.21,
neglecting the numerical errors. The $\theta_{D}$ line for w/H ratio 0.21 is
shown in figure 10 as path OB”’, in which point SD(0.21) is the point of the
shift from $\theta_{D}$ line, where M=1.44 and $\theta=$5.3∘. The von Neumann
point is at point B”’, where M=2.202 and wedge angle $\theta=$25.93∘. Thus the
$M-\theta_{w}$ parameter space where the value of w/H is less than 0.21
characterizes the type III domain. Type III domain differentiates itself from
the other two domains as no dual solution domain exists in this type.

####  4.3.4 Locus of the von Neumann points

Shifting the von Neumann point is one of the defining characteristics of the
short wedge flows. It is evident from the preceding discussion that for w/H
ratios less than 0.5025, the von Neumann point moves, and figure 11
illustrates the locus of shift of the von Neumann points in the red dashed
line for various w/H ratios. The red dashed line divides the region between
the detachment and no-reflection lines into two distinct areas. Dual-domain
solutions do not exist on the left side of the locus of the line, but they do
exist on the right. The shift in von Neumann points in the $M-\theta_{w}$
parameter space follows a straight line passing through the X-intercept where
M=2.202. Although the dual solution domain exists on the right side of the
above-mentioned straight line, the domain close to this line is hardly visible
due to the interaction effects, as shown in figure 11. To confirm the shift in
von Neumann point B, shock polars of short wedges with actual wedge angle and
effective wedge angle(turn angle at reflection point) are drawn. Figure 12
shows the shock polar for w/H ratio 0.33 at the von Neumann point B0.33 (refer
figure 11). The solid blue line indicates r-polar for the actual wedge angle
used, and the dashed blue line shows the same for the effective wedge angle
$\theta_{eff}$ at the reflection point R(refer figure 2). At point B0.33, the
wedge angle is 20.18∘ and due to interaction this is reduced to 15.15∘. Thus
the von Neumann shift in short wedge flows is validated with the shock polars.
Further investigations are required to comprehend the shock reflection
configurations and the flow behaviour in Type I, Type II and Type III
interactions in the flow over short wedges.

Figure 11: Locus of bifurcation points for various w/H ratio  Figure 12: Shock
polar for M=2.202, w/H ratio 0.33

##  5 Conclusion

The shock reflection phenomena on short wedges have been investigated
analytically and computationally. The interaction of the expansion fan
emanating from the trailing edge of the short wedge with the incident shock
wave is found to curve the incident shock. The curving of the incident shock
wave leads to the change of shock angle at the reflection/triple point,
consequently leading to a shift in the transition line. The transition lines
plotted for different w/H ratios from 0.05 to 0.8 show that the shift in
transition lines depends on the extent of interaction. The plot is useful in
finding the detachment/von Neumann criterion for any Mach number between 1 and
5 for a particular w/H ratio. Moreover, we can control the appearance of MR or
its configuration, for a particular Mach number by selecting the suitable
(w/H) ratio from the transition lines generated. As the extent of expansion
fan interaction with the incident shock increases, the dual domain shifts its
location and diminishes as a function of w/H ratio facilitating the control of
the MR$\leftrightarrow$RR hysteresis process.

## References

  * Azevedo & Liu (1993) Azevedo, D. J. & Liu, Ching Shi 1993 Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA Journal 31 (1), 83–90, arXiv: https://doi.org/10.2514/3.11322.
  * Bai & Wu (2017) Bai, Chen-Yuan & Wu, Zi-Niu 2017 Size and shape of shock waves and slipline for mach reflection in steady flow. Journal of Fluid Mechanics 818, 116–140.
  * Emanuel (2001) Emanuel, George 2001 Chapter 3.1 - shock waves in gases. In Handbook of Shock Waves, Vol-1 (ed. G BEN-DOR, O IGRA & T ELPERIN), pp. 185–262. Burlington: Academic Press.
  * Glass (1991) Glass, I. I. 1991 Over forty years of continuous research at utias on nonstationary flows and shock waves. Shock Waves 1 (1), 75–86.
  * Hillier (2007) Hillier, R. 2007 Shock-wave/expansion-wave interactions and the transition between regular and mach reflection. Journal of Fluid Mechanics 575, 399–424.
  * Li & Ben-Dor (1996) Li, H. & Ben-Dor, G. 1996 Oblique-shock/expansion-fan interaction - analytical solution. AIAA Journal 34 (2), 418–421, arXiv: https://doi.org/10.2514/3.13081.
  * Li & Ben-Dor (1997) Li, H. & Ben-Dor, G. 1997 A parametric study of mach reflection in steady flows. Journal of Fluid Mechanics 341, 101–125.
  * Mouton & Hornung (2007) Mouton, Christopher A. & Hornung, Hans G. 2007 Mach stem height and growth rate predictions. AIAA Journal 45 (8), 1977–1987, arXiv: https://doi.org/10.2514/1.27460.
  * Nel et al. (2015) Nel, Lara, Skews, Beric & Naidoo, Kavendra 2015 Schlieren techniques for the visualization of an expansion fan/shock wave interaction. Journal of Visualization 18 (3), 469–479.
  * Paramanantham et al. (2022) Paramanantham, Vinoth, Janakiram, Sushmitha & Gopalapillai, Rajesh 2022 Prediction of mach stem height in compressible open jets. part 1. overexpanded jets. Journal of Fluid Mechanics 942, A48.
  * Yao et al. (2013) Yao, Y., Li, S. G. & Wu, Z. N. 2013 Shock reflection in the presence of an upstream expansion wave and a downstream shock wave. Journal of Fluid Mechanics 735, 61–90.


* [Sim90] Jacques Simon.  Nonhomogeneous Viscous Incompressible Fluids: Existence of Velocity, Density, and Pressure.  SIAM Journal on Mathematical Analysis, 21(5):1093–1117, 1990.
  * [Soh11] Vedran Sohinger.  Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations.  PhD thesis, Massachusetts Institute of Technology, 2011.
  * [SRL11] J Salort, P.-E Roche, and E Leveque.  Mesoscale equipartition of kinetic energy in quantum turbulence.  EPL, 94(2):24001, 2011.
  * [SYZ22] Yunfei Su, Lei Yao, and Mengmeng Zhu.  Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force.  Discrete and Continuous Dynamical Systems - B, 27(12):7207–7226, 2022.
  * [Tao06] Terence Tao.  Nonlinear dispersive equations: local and global analysis.  American Mathematical Society, 2006.
  * [Tem77] Roger Temam.  Navier-Stokes Equations: Theory and Numerical Analysis.  North-Holland Publishing Company, first edition, 1977.
  * [Vin04] W F Vinen.  The physics of superfluid helium.  Technical report, CERN, 2004.
  * [Vin06] W. F. Vinen.  An introduction to quantum turbulence.  Journal of Low Temperature Physics, 145(1-4):7–24, 11 2006.
  * [VSBP19] Akhilesh Kumar Verma, Vishwanath Shukla, Abhik Basu, and Rahul Pandit.  The Statistical Properties of Superfluid Turbulence in 4-He from the Hall-Vinen-Bekharevich-Khalatnikov Model.  arXiv preprint, 2019.
  * [VY16a] Alexis F Vasseur and Cheng Yu.  Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations.  Inventiones mathematicae, 206:935–974, 2016.
  * [VY16b] Alexis F. Vasseur and Cheng Yu.  Global weak solutions to the compressible quantum Navier-Stokes equations with damping.  SIAM J. Math. Anal., 48(2):1489–1511, 2016.
  * [WG20] Guangwu Wang and Boling Guo.  A blow-up criterion of strong solutions to the quantum hydrodynamic model.  Acta Mathematica Scientia, 40(3):795–804, 2020.
  * [WG21] Guangwu Wang and Boling Guo.  A new blow-up criterion of the strong solution to the quantum hydrodynamic model.  Applied Mathematics Letters, 119:107045, 2021.


# On the Pauli group on 2-qubits
in Dynamical Systems with pseudofermions

Fabio Bagarello  Fabio BagarelloDipartimento di IngegneriaUniversitá di
PalermoViale delle Scienze, I-90128, Palermo, Italyand Istituto Nazionale di
Fisica NucleareSezione di Catania, Italy<EMAIL_ADDRESS>,  Yanga
Bavuma  Yanga BavumaDepartment of Mathematical SciencesUniversity of Cape
TownPrivate Bag X1, 7701, RondeboschCape Town, South Africa
<EMAIL_ADDRESS>and  Francesco G. Russo  Francesco G. RussoDepartment of
Mathematics and Applied MathematicsUniversity of Cape Town Private Bag X1,
7701, RondeboschCape Town, South Africaand Department of Mathematics and
Applied MathematicsUniversity of the Western CapeNew CAMS Building, Private
Bag X17, 7535Bellville, South Africa<EMAIL_ADDRESS>
###### Abstract.

The group of matrices $P_{1}$ of Pauli is a finite 2-group of order 16 and
plays a fundamental role in quantum information theory, since it is related to
the quantum information on the 1-qubit. Here we show that both $P_{1}$ and the
Pauli 2-group $P_{2}$ of order 64 on 2-qubits, other than in quantum
computing, can also appear in dynamical systems which are described by non
self-adjoint Hamiltonians. This will allow us to represent $P_{1}$ and $P_{2}$
in terms of pseudofermionic operators.

###### Key words and phrases:

Pauli group; PT symmetries; LC circuits; fermionic operators; Hilbert
spaces.Mathematics Subject Classification 2020: Primary 81R05, 22E10;
Secondary 22E70, 81R30, 81Q12.

##  1\. Introduction and statement of the results

Noether’s Theorem [21, Equations (13.148) and (13.158)] is a powerful tool in
order to investigate the governing equations in dynamical systems possessing
symmetries. The role of groups of symmetries has been largely investigated in
connection with Noether’s Theorem and its applications in mathematical physics
are numerous.

In quantum information theory, one is often interested to study quantum codes
and, in quantum computing one is interested in preventing or correcting errors
that may affect dynamical systems involving qubits. This motivates several
authors to work with the Pauli group $P_{n}$ on $n$ qubits. In particular
$P_{1}$ denotes the Pauli group on 1 qubit

$I=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\ \
X=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),\ \
Y=\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right),\ \
Z=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$ (1.1)

which satisfy the well known identities

$X=iZY,\ \ Y=iXZ,\ \ Z=iYX,\ \ X^{2}=Y^{2}=Z^{2}=I.$ (1.2)

These are the well known Pauli matrices [29], which form the nonabelian group
of order $16$

$P_{1}=\\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\\}$ (1.3)

When we deal with quantum information systems, based on more than 1-qubit, one
generalizes the notion of Pauli group, considering larger $2$-groups $P_{n}$
of order $4^{n+1}$ for $n$-qubits with $n\geq 1$ big enough. These groups,
known as large Pauli groups, are prominent in the literature of dynamical
systems of the last years [23, 28]. Here we focus on the case $n=2$, that is,
on the Pauli group on 2-qubits, defined by

$P_{2}=\\{M\otimes N\mid M,N\in P_{1}\\},$ (1.4)

where the symbol $\otimes$ denotes the Kronecker product among matrices. We
will realize (1.4) via a new family of examples which come from the area of
the electromagnetism, and specifically from the LC quantum circuits.

In this spirit we are going to illustrate the notion of pseudofermionic ladder
operators, according to [6]; in fact we may reconstruct Hamiltonians from
ladder operators as originally indicated by Dirac and Dieudonné [18]. Among
other things, we mention some recent dynamical systems, whose Hamiltonian is
obtained with a construction of the same type, see [5, 10].

We shall mention that [19] presents the construction of a pair of coupled $LC$
electronic circuits, one with amplification and the other with an equivalent
attenuation. The interest behind these circuits, besides of the fact that they
produce an interesting set of matrices in the analysis of (1.4), is that they
can be used to construct an experimental setting which is connected with a PT
symmetric system, i.e. with a quantum system whose Hamiltonian satisfies
certain transformation rules with respect to the parity ($P$) and the time
reversal ($T$) operators, [11]. These systems are becoming more and more
studied, both by physicists, and by mathematicians, for their very peculiar
properties, [11, 12]. In fact there exists a dynamical system $\mathcal{S}$ in
[19], realized by coupled LC electronic circuits, possessing PT symmetries and
the dynamics of $\mathcal{S}$ is governed by the differential equation

$i\frac{\mathrm{d}\Psi(t)}{\mathrm{d}t}=H_{\mathcal{S}}\Psi(t),$ (1.5)

where $H_{\mathcal{S}}=iL_{\mathcal{S}}$ is the (formal) Hamiltonian of
$\mathcal{S}$, the symbols $\alpha,\mu,\gamma$ are parameters defining the
circuits,

$L_{\mathcal{S}}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\
-\alpha&\mu\alpha&\gamma&0\\\ \mu\alpha&-\alpha&0&-\gamma\\\
\end{array}\right)\ \qquad\mbox{and}\
\qquad\Psi(t)=\left(\begin{array}[]{c}Q_{1}(t)\\\ Q_{2}(t)\\\
\dot{Q}_{1}(t)\\\ \dot{Q}_{2}(t)\\\ \end{array}\right)$ (1.6)

is the vector of charges $Q_{1}(t)$ and $Q_{2}(t)$ on the capacitors of the
two circuits with corresponding derivatives $\dot{Q}_{1}(t)$ and
$\dot{Q}_{2}(t)$. It might be useful to stress that (1.5) looks like a
Schrödinger equation, but this is just a formal identification. In fact, the
four components of $\Psi(t)$ are not all independent, as they should.

Our main result is listed below.

######  Theorem 1.1.

The dynamical system $\mathcal{S}$ above, realized by coupled LC electronic
circuits, satisfies the following properties:

  * (i).

$H_{\mathcal{S}}$ can be decomposed in the linear combination of 12 bounded
operators $X_{k}$ for $k=1,2,\ldots,12$. Moreover, the operators which commute
with all the $X_{k}$’s are only those of diagonal type;

  * (ii).

The operators $X_{k}$ can be identified with matrices in
$\mathrm{SL}_{4}(\mathbb{C})$ and it is possible to select six operators among
the $X_{k}$ which are generators of the Pauli group $P_{2}$ on 2-qubits and
are simultenously realized by pseudofermionic operators;

  * (iii).

There exist two subgroups $U$ and $V$ of $P_{2}$, generated by sets of
pseudofermionic operators $\Gamma_{\mu}$ and $\Gamma_{\nu}$ respectively, such
that $P_{2}=UV$ and the derived subgroup $[U,V]$ is trivial.

While Section 2 discusses some general parts of the theory of pseudohermitian
operators in dynamical systems and their relevance in Quantum Mechanics,
Section 3 focuses specifically on the theory of pseudofermionic operators and
on their recent developments. The main proofs can be found in Section 4, while
Section 5 describes further investigations which might be possible for larger
Pauli groups.

##  2\. Some folklore on pseudohermitian physical systems

Self-adjoint and non-self-adjoint operators are adequate to describe many
properties that are useful for physical systems in quantum mechanics, see
[21]. Note also that the eigenvalues of operators a, which are significant in
many quantum mathematical models, turn out to be real;
$\langle\psi,\textbf{a}\psi\rangle$ is the expectation value for the
measurements of the observable a in the quantum state $\psi$, and its
eigenvalue $\lambda$ represents one of the possible values for this
measurement.

Wave functions can evolve in time. In fact one of the main assumptions in
quantum mechanics is that there exists an operator H on the Hilbert space
$\mathcal{H}$, called the Hamiltonian operator for the system, producing the
well known Schrödinger Equation:

######  Proposition 2.1 (See [22], Axiom 5, Claim 3.17).

The time-evolution of the wave function $\psi$ in a quantum system is governed
by the Schrödinger Equation,

$\frac{d\psi}{dt}=\frac{1}{i\hbar}\textbf{H}\psi,$ (2.1)

where H is the Hamiltonian and $\hbar$ is the constant of Planck. In
particular, if H is time independent, then (2.1) can be formally solved by
setting

$\psi(t)=e^{-it\textbf{H}/\hbar}\psi_{0}.$ (2.2)

where $\psi_{0}$ is the initial condition.

Heisenberg had a different view of the dynamics of a quantum system. He
thought of the operators (quantum observable) as evolving in time instead of
the quantum states (vectors in the Hilbert space). In his interpretation each
operator a evolves in time according to the operator-valued differential
equation

$\frac{d\textbf{a}(t)}{dt}=\frac{1}{i\hbar}\left[\textbf{a}(t),\textbf{H}\right],$
(2.3)

where H is the Hamiltonian operator of the system (time-independent, here, as
in the previous proposition), and where

$\left[\textbf{a},\textbf{b}\right]=\textbf{a}\textbf{b}-\textbf{b}\textbf{a}$
(2.4)

is the commutator between the operators  a and b according to [22, Definition
3.20]. Note that since H commutes with itself, the operator H remains constant
in time, even in this interpretation. However this turns out to have the same
physical meaning in Schrödinger’s interpretation.

An increasing number of physicists and mathematicians started to be recently
interested to situations in which the Hamiltonian of a mathematical model
under consideration is not necessarily self-adjoint. Examples of a non-self-
adjoint Hamiltonian operators have been studied by Bender [11, 13, 14, 15]),
Mostafazadeh [26, 27] and other authors [2, 3, 4], introducing, for instance,
generalized versions of the harmonic oscillator.

##  3\. Elementary theory of pseudofermionic and pseudobosonic operators

In particle physics, elementary particles and composite particles are
essentially broken up into two (non-overlapping) classes: bosons and
fermions. Roughly speaking fermions are particles that are associated with the
matter and pseudofermions are generalizations of fermions. The bosons are
usually used to represent the interactions (in terms of fields), while
pseudobosons generalize bosons. In this paper, we will only focus on
pseudofermions. The reader can find more information the theory of the
pseudobosons and of the pseudofermions in [6, 9].

Typically on a two dimensional Hilbert space $\mathcal{H}=\mathbb{C}^{2}$ we
can define lowering and raising operators, which will lower or raise the
eigenvalues (respectively) associated with an eigenstate by acting on the
state itself. These come as a pair, say c and $\textbf{c}^{*}$ respectively.
The lowering operator, c lowers the eigenvalue of a given state by acting on
it, and the raising operator $\textbf{c}^{*}$ raises the eigenvalue of a given
state by acting on it, and it is the adjoint of the lowering operator.

######  Remark 3.1.

The fermionic operators satisfy the  canonical anticommutation relations (CAR)

$\\{\textbf{c},\textbf{c}^{*}\\}:=\textbf{c}\textbf{c}^{*}+\textbf{c}^{*}\textbf{c}=\mathbb{I},\
\ \\{\textbf{c},\textbf{c}\\}=\\{\textbf{c}^{*},\textbf{c}^{*}\\}=0.$ (3.1)

However pseudofermionic operators satisfy more general anticommutation rules:

$\\{\textbf{a},\textbf{b}\\}=\mathbb{I},\ \
\\{\textbf{a},\textbf{a}\\}=\\{\textbf{b},\textbf{b}\\}=0,$ (3.2)

where $\textbf{b}\neq\textbf{a}^{*}$ a priori. Note that (3.1) is motivated by
Pauli’s Principle.

Note that fermionic operators are bounded. A nonzero vector
$\varphi_{0}\in\mathcal{H}$ such that $\textbf{a}\varphi_{0}=0$ surely exists,
as well as a nonzero vector $\Psi_{0}\in\mathcal{H}$ such that
$\textbf{b}^{*}\Psi_{0}=0$. This is because a and $\textbf{b}^{*}$ have
nontrivial kernels. We are going to illustrate better this aspect, providing a
minimal framework of functional analysis in the remaining part of the present
section.

######  Definition 3.2.

A set $\mathcal{E}=\\{e_{n}\in\mathcal{H},n\geq 0\\}$ is a Schauder basis for
$\mathcal{H}$ if any vector $f\in\mathcal{H}$ can be written uniquely as

$f=\sum\limits^{\infty}_{n=0}c_{n}(f)e_{n},$

that is, as linear combination of $e_{n}$ (eventually infinite) with
$c_{n}(f)\in\mathbb{C}$ depending only on $f$.

A particular type of bases is given by the orthonormal bases, that is,
Schauder bases such that $\langle e_{n},e_{m}\rangle=\delta_{n,m}$, where
$\delta_{n,m}$ denotes the Kronecker delta and $n,m$ are positive integers. In
this particular case, we recover $c_{n}(f)=\langle e_{n},f\rangle$ via the
scalar product. Note that the scalar product is linear in its second variable.
A more specific version of orthonormal bases is given by the Riesz bases:

######  Definition 3.3.

In $\mathcal{H}$ the set $\mathcal{F}=\\{f_{n}\in\mathcal{H},n\geq 0\\}$ is a
Riesz basis if there exists a bounded operator $T$ on $\mathcal{H}$ with
bounded inverse $T^{-1}$ and an orthonormal basis
$\mathcal{E}=\\{e_{n}\in\mathcal{H},n\geq 0\\}$ such that $f_{n}=Te_{n}$, for
all $n\geq 0$.

A detailed analysis on operators defined by Riesz bases can be found in [8,
17], but it is clear that $\langle f_{n},f_{m}\rangle\neq\delta_{n,m}$ in the
context of Definition 3.3. In this case, however, one can introduce a second
set $\mathcal{G}$, which is another Riesz basis biorthonormal to
$\mathcal{F}$, whose vectors are simply $g_{n}=(T^{-1})^{*}e_{n}$. We go ahead
to illustrate some results that are crucial in the present framework.

First we define the following vectors

$\varphi_{1}:=\textbf{b}\varphi_{0},\ \ \Psi_{1}:=\textbf{a}^{*}\Psi_{0},$
(3.3)

as well as the following non-self-adjoint operators

$\textbf{N}:=\textbf{b}\textbf{a},\ \
\textbf{N}^{*}=\textbf{a}^{*}\textbf{b}^{*}.$ (3.4)

Note that for $n\geq 2$ the vectors $\textbf{b}^{n}\varphi_{0}$ and
${\textbf{a}^{*}}^{n}\Psi_{0}$ are automatically equal to zero. Then the
following equations are satisfied:

$\textbf{a}\varphi_{1}=\varphi_{0},\ \ \textbf{b}^{*}\Psi_{1}=\Psi_{0}$ (3.5)
$\textbf{N}\varphi_{n}=n\varphi_{n},\ \
\textbf{N}^{*}\Psi_{n}=n\Psi_{n},\text{ for $n=0,1$}.$ (3.6)

If the normalizations of $\varphi_{0}$ and $\Psi_{0}$ are chosen such that
$\langle\varphi_{0},\Psi_{0}\rangle=1$, then we have also

$\langle\varphi_{k},\Psi_{n}\rangle=\delta_{k,n}\text{ for $k,n=0,1$}.$ (3.7)

Now we introduce the self-adjoint operators $\textbf{S}_{\varphi}$ and
$\textbf{S}_{\Psi}$ via their action on a generic $f\in\mathcal{H}$:

$\textbf{S}_{\varphi}f=\sum\limits^{1}_{n=0}\langle\varphi_{n},f\rangle\varphi_{n},\
\ \textbf{S}_{\Psi}f=\sum\limits^{1}_{n=0}\langle\Psi_{n},f\rangle\Psi_{n}.$
(3.8)

Note also that the operators $\textbf{S}_{\varphi}$ and $\textbf{S}_{\Psi}$
are bounded, strictly positive, self-adjoint, and invertible. They satisfy

$\|\textbf{S}_{\varphi}\|\leq\|\varphi_{0}\|^{2}+\|\varphi_{1}\|^{2},\ \
\|\textbf{S}_{\Psi}\|\leq\|\Psi_{0}\|^{2}+\|\Psi_{1}\|^{2},$ (3.9)
$\textbf{S}_{\varphi}\Psi_{n}=\varphi_{n},\ \
\textbf{S}_{\Psi}\varphi_{n}=\Psi_{n},$ (3.10)

for $n=0,1$, as well as $\textbf{S}_{\varphi}=\textbf{S}_{\Psi}^{-1}$ and the
following intertwining relations

$\textbf{S}_{\Psi}\textbf{N}=\textbf{N}^{*}\textbf{S}_{\Psi},\ \
\textbf{S}_{\varphi}\textbf{N}^{*}=\textbf{N}\textbf{S}_{\varphi}.$ (3.11)

The vectors of $\mathcal{F}_{\varphi}=\\{\varphi_{0},\varphi_{1}\\}$ and
$\mathcal{F}_{\Psi}=\\{\Psi_{0},\Psi_{1}\\}$ are biorthogonal, linearly
independent in a two dimensional complex Hilbert space so that
$\mathcal{F}_{\varphi}$ is a Schauder basis for $\mathcal{H}$. The same
argument applies to $\mathcal{F}_{\Psi}$ and in fact it turns out that both
these sets are Riesz bases for $\mathcal{H}$.

A connection between pseudofermions and fermions is reported below:

######  Proposition 3.4 (See [6], Theorem 3.5.1).

Let c and $\textbf{T}=\textbf{T}^{*}$ be two operators on $\mathcal{H}$ such
that (3.1) are satisfied and in addition T is positive. Then the operators
$\textbf{a}=\textbf{TcT}^{-1}$ and $\textbf{b}=\textbf{Tc}^{*}\textbf{T}^{-1}$
satisfy (3.2). Viceversa given two operators a and b acting on $\mathcal{H}$,
satisfying (3.2), it is possible to construct two operators c and T with the
above properties.

Examples of pseudofermions (and generalizations) can be found in [1, 2, 3, 4,
6, 7].

In particular [6, Chapter 3.5.1] shows an effective non-selfadjoint
hamiltonian of a mathematical model of a two-level atom which interacts with
an electromagnetic field. Let’s recall a few facts from [6, Chapter 3.5.1]
with more details; it will be useful in the proof of Theorem 1.1. Maamache and
others [16] described an effective non self-adjoint Hamiltonian involved in a
two levels atom interacting with an electromagnetic field. This followed an
intuition of Mostafazadeh in [26, 27]. Writing (2.1), one gets

$i\dot{\Phi}(t)=H_{eff}\Phi(t),\qquad
H_{eff}=\frac{1}{2}\left(\begin{array}[]{cc}-i\delta&\overline{\omega}\\\
\omega&i\delta\\\ \end{array}\right),$ (3.12)

where $\delta$ is a real quantity, related to the decay rates for the two
levels, while the complex parameter $\omega$ describes the interaction due to
radiations of the atom. In particular we may write

$\omega=|\omega|e^{i\theta}$ (3.13)

Of course, the effective Hamiltonian cannot be self-adjoint in the present
situation, but let’s see better some details in connection with the framework
of functional analysis which we have seen in (3.3)-(3.11). Now introduce the
operators

$\textbf{a}=\frac{1}{2\Omega}\left(\begin{array}[]{cc}-|\omega|&-e^{-i\theta}(\Omega+i\delta)\\\
e^{i\theta}(\Omega-i\delta)&|\omega|\\\
\end{array}\right),\quad\textbf{b}=\frac{1}{2\Omega}\left(\begin{array}[]{cc}-|\omega|&e^{-i\theta}(\Omega-i\delta)\\\
-e^{i\theta}(\Omega+i\delta)&|\omega|\\\ \end{array}\right),$ (3.14)

where $\Omega=\sqrt{|\omega|^{2}-\delta^{2}}$ can be assumed to be real and
strictly positive. One can check easily that (3.14) are pseudofermionic
operators, that is, they satisfy (3.2) and that

$H_{eff}=\Omega\left(\textbf{b}\textbf{a}-\frac{1}{2}\mathbb{I}\right).$
(3.15)

In order to visualize the notions in (3.3)–(3.11), we may consider

$\varphi_{0}=k\left(\begin{array}[]{c}1\\\
-\,\frac{e^{i\theta}(\Omega-i\delta)}{|\omega|}\\\
\end{array}\right),\qquad\Psi_{0}=k^{\prime}\left(\begin{array}[]{c}1\\\
-\,\frac{e^{i\theta}(\Omega+i\delta)}{|\omega|}\\\ \end{array}\right),$ (3.16)

where $k$ and $k^{\prime}$ are normalization constants such that the
orthogonality condition below is satisfied

$\left<\varphi_{0},\Psi_{0}\right>=\overline{k}\,k^{\prime}\left(1+\frac{1}{|\omega|^{2}}(\Omega+i\delta)^{2}\right)=1.$
(3.17)

Then we introduce

$\varphi_{1}=\textbf{b}\varphi_{0}=k\left(\begin{array}[]{c}\frac{i\delta-\Omega}{|\omega|}\\\
-e^{i\theta}\\\
\end{array}\right),\qquad\Psi_{1}=\textbf{a}^{\dagger}\Psi_{0}=k^{\prime}\left(\begin{array}[]{c}\frac{-i\delta-\Omega}{|\omega|}\\\
-e^{i\theta}\\\ \end{array}\right).$ (3.18)

In particular, we find that $\mathcal{F}_{\varphi}$ and $\mathcal{F}_{\Psi}$
are biorthonormal bases of $\mathcal{H}$, and that

$H_{eff}\varphi_{0}=-\,\frac{\Omega}{2}\,\varphi_{0},\quad
H_{eff}\varphi_{1}=\frac{\Omega}{2}\,\varphi_{1},\quad
H_{eff}^{\dagger}\Psi_{0}=-\,\frac{\Omega}{2}\,\Psi_{0},\quad
H_{eff}^{\dagger}\Psi_{1}=\frac{\Omega}{2}\,\Psi_{1}.$ (3.19)

It is evident that $H_{eff}$ and $H_{eff}^{\dagger}$ are not self-adjoint. In
order to find (3.10) and (3.11), now

$\mathbf{S}_{\varphi}=2|k|^{2}\left(\begin{array}[]{cc}1&\frac{-i\delta}{|\omega|}\,e^{-i\theta}\\\
\frac{i\delta}{|\omega|}\,e^{i\theta}&1\\\
\end{array}\right),\quad\mathbf{S}_{\Psi}=\frac{|\omega|^{2}}{2|k|^{2}\Omega^{2}}\left(\begin{array}[]{cc}1&\frac{i\delta}{|\omega|}\,e^{-i\theta}\\\
\frac{-i\delta}{|\omega|}\,e^{i\theta}&1\\\ \end{array}\right)$ (3.20)

and one is the inverse of the other. In the present situation Proposition 3.4
may be applied to the operator $\mathbf{S}_{\varphi}^{\pm
1/2}=\pm\frac{1}{2}\mathbf{S}_{\varphi}$ to define two standard fermionics
operators c and $\textbf{c}^{\dagger}$ such that

$\mathbf{N}_{0}=\textbf{c}^{\dagger}\textbf{c}\ \ \mbox{and}\ \
H_{eff}=\mathbf{S}_{\varphi}^{1/2}\
\left(\Omega\left(\mathbf{N}_{0}-\frac{1}{2}\mathbb{I}\right)\right)\ \
\mathbf{S}_{\varphi}^{-1/2},$ (3.21)

but now we have clearly produced another Hamiltonian which is self-adjoint and
similar to $H_{eff}$.

More recently, [5, Theorem 1.2] describes the Pauli group (1.1) in terms of
(3.14). In fact, it is possible to introduce the matrices

$\mu_{1}=\frac{1}{\Omega}\left(\begin{array}[]{cc}-|\omega|&-i\delta
e^{-i\theta}\\\ -i\delta e^{i\theta}&|\omega|\\\
\end{array}\right),\quad\mu_{2}=i\left(\begin{array}[]{cc}0&e^{-i\theta}\\\
e^{i\theta}&0\\\
\end{array}\right),\quad\mu_{3}=\frac{1}{\Omega}\left(\begin{array}[]{cc}i\delta&-|\omega|e^{-i\theta}\\\
-|\omega|e^{i\theta}&-i\delta\\\ \end{array}\right),$ (3.22)

where

$\mu_{1}=\mu_{1}(\theta,\delta)=\textbf{b}+\textbf{a},\qquad\mu_{2}=\mu_{2}(\theta,\delta)=i(\textbf{b}-\textbf{a}),\qquad\mu_{3}=\mu_{3}(\theta,\delta)=[\textbf{a},\textbf{b}]=\textbf{a}\textbf{b}-\textbf{b}\textbf{a}.$
(3.23)

Of course, $\mu_{1}=\mu_{1}(\theta,\delta)$, that is, $\mu_{1}$ depends on
$\theta$ and $\delta$, similarly this happens for $\mu_{2}$ and $\mu_{3}$, but
also for $\textbf{a}=\textbf{a}(\theta,\delta)$ and
$\textbf{b}=\textbf{b}(\theta,\delta)$ and their linear combinations. As shown
in [5, Theorem 1.2] independently on the choice of $\theta$ and $\delta$ we
may consider the set

$P_{\mu}=\\{\mu_{1},\mu_{2},\mu_{3}\\}$ (3.24)

which is a concrete realization of the generators of the first Pauli group
$P_{1}$ via pseudofermionic operators. On the other hand, specific choices of
$\theta$ and $\delta$ give more precise proportionality relations between
(3.22) and (1.2). For instance, the choice $\theta=\pi/2$ and $\delta=0$
allows us to consider

$\mu_{1}\left(\frac{\pi}{2},0\right)=-Z,\quad\mu_{2}\left(\frac{\pi}{2},0\right)=-iY,\quad\mu_{3}\left(\frac{\pi}{2},0\right)=-Y,$
(3.25)

and using $X=iZY$ in (1.2) we find that

$X=\mu_{1}\left(\frac{\pi}{2},0\right)\
\mu_{2}\left(\frac{\pi}{2},0\right),\quad
Y=-\mu_{3}\left(\frac{\pi}{2},0\right),\quad
Z=-\mu_{1}\left(\frac{\pi}{2},0\right).$ (3.26)

Of course, we may use (3.14) and (3.23), getting in correspondence with
$\theta=\pi/2$ and $\delta=0$

$X=i(\textbf{a}\textbf{b}-\textbf{b}\textbf{a})\left(\frac{\pi}{2},0\right),\quad
Y=(\textbf{b}\textbf{a}-\textbf{a}\textbf{b})\left(\frac{\pi}{2},0\right),\quad
Z=-(\textbf{b}+\textbf{a})\left(\frac{\pi}{2},0\right).$ (3.27)

The above computations play a fundamental role in the proof of Theorem 1.1.

##  4\. Proof of the main theorem

Before we prove our main result, we recall separately some terminology from
[30]:

######  Definition 4.1 (See [30], p.145).

A group $G$ is (the internal) central product of its subgroups $H$ and $K$, if
$G=HK$ and the derived subgroup of $H$ and $K$ is trivial, that is,

$[H,K]=\\{h^{-1}k^{-1}hk\mid h\in H,k\in K\\}=1.$ (4.1)

For groups as in Definition 4.1, one has that both the subgroups realizing the
central product are normal in the group, that is, $H$ and $K$ are normal in
$G$. Incidentally, (4.1) is notationally similar to (2.4). While in the first
case one uses the algebraic structure of group (in multiplicative notation),
in order to produce a new object which still has the structure of group, in
the second case one uses two functional operators, in order to produce a new
functional operator.

###### Proof of Theorem 1.1.

(i). Let us consider the set of matrices $\\{X_{j},j=1,2,\ldots,12\\}$, where

$X_{1}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\
\end{array}\right),X_{2}=\left(\begin{array}[]{cccc}0&0&-i&0\\\ 0&0&0&-i\\\
i&0&0&0\\\ 0&i&0&0\\\
\end{array}\right),X_{3}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&-1&0\\\ 0&0&0&-1\\\ \end{array}\right),$ (4.2)
$X_{4}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\\\ \end{array}\right),X_{5}=\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{array}\right),X_{6}=\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\
0&0&0&1\\\ 0&0&-1&0\\\ \end{array}\right),$ (4.3)
$X_{7}=\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
\end{array}\right),X_{8}=\left(\begin{array}[]{cccc}0&0&-i&0\\\ 0&0&0&i\\\
i&0&0&0\\\ 0&-i&0&0\\\
\end{array}\right),X_{9}=\left(\begin{array}[]{cccc}0&0&0&-i\\\ 0&0&-i&0\\\
0&i&0&0\\\ i&0&0&0\\\ \end{array}\right),$ (4.4)
$X_{10}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\\\ \end{array}\right),X_{11}=\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\ 0&0&-1&0\\\
\end{array}\right),X_{12}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\ 0&-1&0&0\\\ \end{array}\right).$ (4.5)

Now observe that

$[X_{j},X_{k}]=X_{j}X_{k}-X_{k}X_{j}\ \ \ \ \mbox{and}\ \ \ \
\\{X_{j},X_{k}\\}=X_{j}X_{k}+X_{k}X_{j}.$ (4.6)

The matrices $X_{j}$ satisfy a mixed set of rules. For instance we have

$\\{X_{1},X_{2}\\}=\\{X_{1},X_{3}\\}=0\ \ \mbox{and}\ \
[X_{2},X_{4}]=[X_{2},X_{5}]=0.$ (4.7)

This aspect of the matrices is relevant when deducing explicitly the equation
of motion, [24, 25]. To calculate the matrices that commute with all the
$X_{i}$’s we consider a general expression for the following matrix below with
coefficients $f,g,h,\ldots,u\in\mathbb{C}$

$M=\begin{pmatrix}f&g&h&d\\\ j&k&l&m\\\ n&o&p&q\\\ r&s&t&u\end{pmatrix}$ (4.8)

and calculate the commutator of the matrix $M$ with each of the $X_{i}$’s.
From a detailed (and long) analysis of the various $[M,X_{j}]$ it follows that
the only matrices which commute with all the $X_{j}$’s are those proportional
to the identity matrix. In fact we have to solve the linear system of 192 (12
$\times$ 16) equations in 16 complex variables. Of course, all the operators
$X_{i}$ turns out to be bounded, since they are expressed in terms of finite
dimensional complex matrices.

It is now an easy computation to check that $L_{\mathcal{S}}$ in (1.6) can be
written as a linear combination of the $X_{j}$’s above. In particular, we can
check that

$L_{\mathcal{S}}=\sum_{k=1}^{12}\alpha_{k}X_{k},$ (4.9)

where the only nonzero $\alpha_{k}$ are the following:

$\alpha_{1}=\frac{1-\alpha}{2},\quad\alpha_{2}=\frac{i(1+\alpha)}{2},\quad\alpha_{4}=\frac{\gamma}{2},\quad\alpha_{7}=\frac{\alpha\mu}{2},\quad\alpha_{9}=-\frac{\alpha\mu}{2i},\quad\alpha_{10}=-\frac{\gamma}{2}.$
(4.10)

Note that the solution of the differential equation for $\Psi(t)$ in (1.5)
turns out to be

$\Psi(t)=e^{L_{\mathcal{S}}\ t}\Psi(0),$ (4.11)

which of course can be rewritten in terms of the $X_{j}$’s. Therefore,
recalling that $H_{\mathcal{S}}=iL_{\mathcal{S}}$, (i) is proved completely.

(ii). First of all, one can check from (4.2) that all the matrices $X_{k}$
belongs to $\mathrm{SL}_{4}(\mathbb{C})$.

Now the idea is that we begin to construct two pseudofermionic operators a and
b on $\mathbb{C}^{2}$ satisfying (3.2) and realizing (1.3). The various
matrices $X_{j}$ can be written in terms of pseudofermionic operators A, B,
$\widetilde{\textbf{A}}$ and $\widetilde{\textbf{B}}$, which are deduced as
tensor product of matrices in $\mathrm{SL}_{2}(\mathbb{C})$. Interestingly
enough, A, B, $\widetilde{\textbf{A}}$ and $\widetilde{\textbf{B}}$ can be
deduced from some matrices, which appear in the analysis of the two levels
atom interacting with an electromagnetic field as in Section 3. Here we may
use ${\mathbf{a}}$ and ${\mathbf{b}}$ as in (3.14) and introduce

$\textbf{A}:=\textbf{a}\otimes\mathbb{I},\qquad\textbf{B}:=\textbf{b}\otimes\mathbb{I}.$
(4.12)

It is easy to check that also A and B give a concrete realization for
$P_{\mu}$, but now A and B are expressed by matrices in
$\mathrm{SL}_{4}(\mathbb{C})$ and no longer by matrices in
$\mathrm{SL}_{2}(\mathbb{C})$ as it was for a and b. We have created in this
way two pseudofermionic operators A and B on the Hilbert space
$\mathbb{C}^{4}$, via the notion of Kronecker product from (3.14). We can find
another two pseudofermionic operators

$\widetilde{\textbf{A}}:=\mathbb{I}\otimes\textbf{a},\qquad\widetilde{\textbf{B}}:=\mathbb{I}\otimes\textbf{b}.$
(4.13)

always in the same Hilbert space $\mathbb{C}^{4}$, where we described A and B.
In fact, one can check that also (4.13) satisfy (3.2). We may conclude that
the set

$\Gamma_{\mu}=\\{\mu_{1}\otimes\mathbb{I},\mu_{2}\otimes\mathbb{I},\mu_{3}\otimes\mathbb{I}\\}$
(4.14)

represents a concrete realization for $P_{1}$ via the operators A and B in
$\mathbb{C}^{4}$, moreover in the same Hilbert space we find also the set

$\Gamma_{\nu}=\\{\mathbb{I}\otimes\mu_{1},\mathbb{I}\otimes\mu_{2},\mathbb{I}\otimes\mu_{3}\\}$
(4.15)

which represents another concrete realization for $P_{1}$, but via the
operators $\widetilde{\textbf{A}}$ and $\widetilde{\textbf{B}}$. Note that the
Pauli matrices $X$, $Y$ and $Z$ in (1.1) may be expressed by (3.26) and
(3.27). Consequently, $\textbf{A}=\textbf{A}(\theta,\delta)$,
$\textbf{B}=\textbf{B}(\theta,\delta)$,
$\widetilde{\textbf{A}}=\widetilde{\textbf{A}}(\theta,\delta)$ and
$\widetilde{\textbf{B}}=\widetilde{\textbf{B}}(\theta,\delta)$, where $\theta$
and $\delta$ are introduced in (3.13). Therefore we get

$\mu_{1}\otimes\mathbb{I}=\textbf{B}+\textbf{A},\
\mu_{2}\otimes\mathbb{I}=i(\textbf{B}-\textbf{A}),\
\mu_{3}\otimes\mathbb{I}=\textbf{A}\textbf{B}-\textbf{B}\textbf{A},$ (4.16)
$\mathbb{I}\otimes\mu_{1}=\widetilde{\textbf{B}}+\widetilde{\textbf{A}},\
\mathbb{I}\otimes\mu_{2}=i(\widetilde{\textbf{B}}-\widetilde{\textbf{A}}),\
\mathbb{I}\otimes\mu_{3}=\widetilde{\textbf{A}}\widetilde{\textbf{B}}-\widetilde{\textbf{B}}\widetilde{\textbf{A}}.$

We are going to report some computations for the matrices
$X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}$. In fact we may write in correspondence
with $\theta=\pi/2$ and $\delta=0$

$X_{1}=X\otimes\mathbb{I}=\left(\mu_{1}\left(\frac{\pi}{2},0\right)\mu_{2}\left(\frac{\pi}{2},0\right)\right)\otimes\mathbb{I}=(i(\textbf{a}\textbf{b}-\textbf{b}\textbf{a})\otimes\mathbb{I})\left(\frac{\pi}{2},0\right)$
(4.17)
$=i((\textbf{a}\otimes\mathbb{I})(\textbf{b}\otimes\mathbb{I})-(\textbf{b}\otimes\mathbb{I})(\textbf{a}\otimes\mathbb{I}))\left(\frac{\pi}{2},0\right)=i(\textbf{A}\textbf{B}-\textbf{B}\textbf{A})\left(\frac{\pi}{2},0\right),$
$X_{2}=Y\otimes\mathbb{I}=-\mu_{3}\left(\frac{\pi}{2},0\right)\otimes\mathbb{I}=(\textbf{B}\textbf{A}-\textbf{A}\textbf{B})\left(\frac{\pi}{2},0\right),$
(4.18)
$X_{3}=Z\otimes\mathbb{I}=-\mu_{1}\left(\frac{\pi}{2},0\right)\otimes\mathbb{I}=-(\textbf{B}+\textbf{A})\left(\frac{\pi}{2},0\right),$
(4.19) $X_{4}=\mathbb{I}\otimes
Z=\mathbb{I}\otimes\left(-\mu_{1}\left(\frac{\pi}{2},0\right)\right)=-(\widetilde{\textbf{B}}+\widetilde{\textbf{A}})\left(\frac{\pi}{2},0\right),$
(4.20) $X_{5}=\mathbb{I}\otimes
X=\mathbb{I}\otimes\left(\mu_{1}\left(\frac{\pi}{2},0\right)\mu_{2}\left(\frac{\pi}{2},0\right)\right)=\mathbb{I}\otimes(i(\textbf{a}\textbf{b}-\textbf{b}\textbf{a}))\left(\frac{\pi}{2},0\right)$
(4.21)
$=i((\mathbb{I}\otimes\textbf{a})(\mathbb{I}\otimes\textbf{b})-(\mathbb{I}\otimes\textbf{b})(\mathbb{I}\otimes\textbf{a}))\left(\frac{\pi}{2},0\right)=i(\widetilde{\textbf{A}}\widetilde{\textbf{B}}-\widetilde{\textbf{B}}\widetilde{\textbf{A}})\left(\frac{\pi}{2},0\right),$
$X_{6}=i(\mathbb{I}\otimes
Y)=i\left(\mathbb{I}\otimes\left(-\mu_{3}\left(\frac{\pi}{2},0\right)\right)\right)=i(\widetilde{\textbf{B}}\widetilde{\textbf{A}}-\widetilde{\textbf{A}}\widetilde{\textbf{B}})\left(\frac{\pi}{2},0\right).$
(4.22)

This means that $\\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\\}$ is a set, which is
realized by pseudofermionic operators. Now we claim that the same set is also
a set of generators for $P_{2}$.

Consider that

$U\ \mbox{is the Pauli group generated by}\ \Gamma_{\mu}\ \ \ \mbox{and}\ \ V\
\mbox{is the Pauli group generated by}\ \Gamma_{\nu},$ (4.23)

referring to (4.14) and (4.15). Of course, at the level of isomorphism of
groups we have

$U\simeq V\simeq P_{1}$ (4.24)

and both $U$ and $V$ are subgroups of $\mathrm{SL}_{4}(\mathbb{C})$. Having in
mind the usual matrix product in $\mathrm{SL}_{4}(\mathbb{C})$ and the rules
in (1.2) in terms of (4.16), we have that for (4.16) with $\theta=\pi/2$ and
$\delta=0$

$P_{2}=UV=\langle\mu_{1}\otimes\mathbb{I},\mu_{2}\otimes\mathbb{I},\mu_{3}\otimes\mathbb{I}\rangle\
\langle\mathbb{I}\otimes\mu_{1},\mathbb{I}\otimes\mu_{2},\mathbb{I}\otimes\mu_{3}\rangle$
(4.25)
$=\langle\mu_{1}\otimes\mathbb{I},\mu_{2}\otimes\mathbb{I},\mu_{3}\otimes\mathbb{I},\mathbb{I}\otimes\mu_{1},\mathbb{I}\otimes\mu_{2},\mathbb{I}\otimes\mu_{3}\rangle=\langle
X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\rangle.$ (4.26)

Note that (4.25) reflects at the level of operators the fact that each element
of $P_{2}$ may be written as $M\otimes
N=(M\otimes\mathbb{I})(\mathbb{I}\otimes N)$ with $M$ and $N$ which are Pauli
matrices, see (1.4). Consequently, each element of $P_{2}$ can be written as
product of elements of $\Gamma_{\mu}$ and $\Gamma_{\nu}$, that is,
$\Gamma_{\mu}\cup\Gamma_{\nu}$ generates $P_{2}$. This means that the claim is
true in particular for (3.25), (3.26), (3.27), hence
$\\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\\}$ is a set of generators for $P_{2}$
in terms of pseudofermionic operators.

(iii). From what we have seen in (ii) above, $P_{2}=UV$, where $U$ and $V$ are
in (4.23). We have exactly the conditions in Definition 4.1 if we look at
(4.23). In fact, the derived subgroup of $U$ and $V$ (according to (4.1)) is
given by

$[U,V]=[\langle\mu_{1}\otimes\mathbb{I},\mu_{2}\otimes\mathbb{I},\mu_{3}\otimes\mathbb{I}\rangle,\langle\mathbb{I}\otimes\mu_{1},\mathbb{I}\otimes\mu_{2},\mathbb{I}\otimes\mu_{3}\rangle]$
(4.27)

and it is enough to check that the generators of $U$ and $V$ commute at the
group level, in order to conclude that $[U,V]$ is trivial. Now if
$i,j\in\\{1,2,3\\}$, then a generic element of (4.27) may be written as

${(\mu_{i}\otimes\mathbb{I})}^{-1}\ {(\mathbb{I}\otimes\mu_{j})}^{-1}\
(\mu_{i}\otimes\mathbb{I})\
(\mathbb{I}\otimes\mu_{j})=(\mu_{i}^{-1}\otimes\mathbb{I}^{-1})(\mathbb{I}^{-1}\otimes\mu_{j}^{-1})(\mu_{i}\otimes\mathbb{I})(\mathbb{I}\otimes\mu_{j})$
(4.28)
$=(\mu_{i}\otimes\mathbb{I})(\mathbb{I}\otimes\mu_{j})(\mu_{i}\otimes\mathbb{I})(\mathbb{I}\otimes\mu_{j})=(\mu_{i}\otimes\mu_{j})(\mu_{i}\otimes\mu_{j})=(\mu_{i}\mu_{i})\otimes(\mu_{j}\mu_{j})=\mathbb{I}\otimes\mathbb{I}=\mathbb{I}$

and so $[U,V]$ is trivial as claimed. We may conclude that $P_{2}$ is central
product of $U$ and $V$. ∎

We should make some comments on the ideas in the previous arguments.

######  Remark 4.2.

It is worth noting that the specific values of $\theta$ and $\delta$ in (3.25)
are not the only values that realise $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ as
proportional to the Pauli matrices. We note the following examples which can
be produced with a different choice of $\theta$ and $\delta$ :

$\mu_{1}(0,0)=-Z,\quad\mu_{2}(0,0)=iX,\quad\mu_{3}(0,0)=-X,$ (4.29)

In fact $\mu_{1},\mu_{2},\mu_{3}$ are proportional to $X,Y,Z$, if
$2\theta=n\pi$ for $n\in\mathbb{Z}$ and either $\omega=0$ or $\delta=0$.

Furthermore we note explicitly that the choice of the pseudofermionic
operators a and b in the proof of Theorem 1.1 has been done, in order to show
analogies with the models in [6, 16, 26, 27].

Of course, different choices of pseudofermionic operators a and b can be used
in the proof of Theorem 1.1, or, we can keep a and b and get different
expressions of (4.17)–(4.21) in terms of A, B, $\widetilde{\textbf{A}}$ and
$\widetilde{\textbf{B}}$. For instance, this last case happens when we use
(4.29) instead of (3.25); we would get a different formulation of
(4.17)–(4.21) always in terms of A, B, $\widetilde{\textbf{A}}$ and
$\widetilde{\textbf{B}}$.

Let’s note another relevant fact:

######  Corollary 4.3.

The statement of Theorem 1.1 (ii) is true for six operators $X_{k}$ which are
fermionic, up to similarity in $\mathrm{GL}_{4}(\mathbb{C})$.

###### Proof.

Repeat the argument of the proof of Theorem 1.1, using Proposition 3.4 along
with (3.20) and (3.21), in order to replace a and b with fermionic operators c
and $\textbf{c}^{*}$. ∎

Another important aspect that we isolate in form of corollary is due to the
presence of the structure of central product and to its description via
pseudofermionic operators:

######  Corollary 4.4.

There exist two subgroups $U$ and $V$ of $P_{2}$ such that $P_{2}$ is the
central product of $U$ and $V$. Moreover $U$ and $V$ are generated by
pseudofermionic operators.

###### Proof.

Application of Theorem 1.1 (iii). ∎

##  5\. Conclusions and analogies with other dynamical systems

The dynamical system $\mathcal{S}$ of Theorem 1.1 is only one of the systems
whose dynamics can be described in terms of the matrices
$X_{1},X_{2},\ldots,X_{12}$ in (4.2).

Another dynamical system $\mathcal{T}$ is discussed in [20], involving the
same matrices (4.2) for the description of its Hamiltonian. Again, the
physical system is an electronic circuit, and again the main interest is in
the possibility of having a simple experimental device connected with PT
symmetry in quantum mechanics. In this case the dynamical system is described
in analogy with $\mathcal{S}$ of Theorem 1.1, but with some minor differences,
see [20] for more details.

What is interesting is the fact that the operator $L_{\mathcal{T}}$ is now
replaced by the following symmetric, but not self-adjoint, matrix:

$H_{\mathcal{T}}=\left(\begin{array}[]{cccc}0&b+ir&d+ir&0\\\ b+ir&0&0&d-ir\\\
d+ir&0&0&d-ir\\\ 0&d-ir&d-ir&0\\\ \end{array}\right).$ (5.1)

Interestingly enough, $\\{X_{k}\ \mid\ k=1,2,\ldots,12\\}$ allows us to
decompose also $H_{\mathcal{T}}$ as follows:

$H_{\mathcal{T}}=\sum_{k=1}^{12}\beta_{k}X_{k},$ (5.2)

where the only non zero $\beta_{k}$ are the following:

$\alpha_{1}=d,\qquad\beta_{5}=\frac{b+d}{2},\qquad\beta_{6}=ir,\qquad\beta_{11}=\frac{b-d}{2}+ir.$
(5.3)

This means that Theorem 1.1 can be proved also beginning from the dynamical
system $\mathcal{T}$ instead of $\mathcal{S}$. The argument of the proof of
Theorem 1.1 applies in fact to $X_{k}$ which appear also in (5.2).

More examples can be constructed from the circuits considered in [1, 7]. Again
we will find a description for the matrices (4.2)–(4.6) in terms of
pseudofermionic operators, for instance, arising from the model described in
(3.14)-(3.21). This suggests the presence of PT symmetries both in
$\mathcal{S}$ and $\mathcal{T}$, but the formalization of the PT symmetry can
be difficult to write explicitly; this is made for instance in [16], but not
in [1].

Another aspect, which deserves interest, is due to the presence of the
structure of central product in $P_{2}$; this peculiar structure has been
already investigated in [5, 10]. Generally, larger Pauli groups, i.e.: $P_{n}$
for $n\geq 2$, may be decomposed in central products (see [31]) so it may be
interesting to generalize Theorem 1.1 to dynamical systems involving larger
Pauli groups.

## References

  * [1] G. Alicata, F. Bagarello, F. Gargano and S. Spagnolo, Quantum mechanical settings inspired by RLC circuits,  J. Math. Phys. 59 (2018), 042112.
  * [2] F. Bagarello and F.G. Russo, A description of pseudo-bosons in terms of nilpotent Lie algebras, J. Geom. Phy. 125 (2018), 1–11.
  * [3] F. Bagarello and F.G. Russo, On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank, J. Geom. Phy. 125 (2019), 124–131.
  * [4] F. Bagarello and F.G. Russo, Realization of Lie Algebras of High Dimension via Pseudo-Bosonic Operators, J. Lie Theory 30 (2020), 925–938.
  * [5] F. Bagarello, Y. Bavuma and F. G. Russo, Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems, Math. Phys. Anal. Geom. 24 (2021), 16.
  * [6] F. Bagarello, Deformed canonical (anti-)commutation relations and non hermitian Hamiltonians, in Non-selfadjoint operators in quantum physics: Mathematical aspects, M. Znojil et al. (Eds.), John Wiley and Sons, Hoboken, 2015, pp.121–188.
  * [7] F. Bagarello and G. Pantano, Pseudo-fermions in an electronic loss-gain circuit, Int. J. Theor. Phys. 52 (2013), 4507–4518.
  * [8] F. Bagarello, A. Inoue and C. Trapani, Non-self-adjoint Hamiltonians defined by Riesz bases, J. Math. Phys. 55 (2014), 033501.
  * [9] F. Bagarello, Pseudo-bosons and their coherent states, Mathematical Physics Studies, Springer, Cham, 2022.
  * [10] Y. Bavuma, A short note on the decomposition of the central product of groups, Trans. Comb. 11 (2022), 123–129.
  * [11] C. M. Bender, PT Symmetry in Quantum and Classical Physics, World Scientific, Singapore, 2019.
  * [12] C. M. Bender, F. Correa and A. Fring, Proceedings for “Pseudo-Hermitian Hamiltonians in Quantum Physics”,  J. Physics: Conference Series 2038 (2021), 012001.
  * [13] C.M. Bender, M. De Kieviet and S.P. Klevansky, PT quantum mechanics, Philos Trans R Soc London Ser A 371 (2013), 20120523.
  * [14] C.M. Bender, A. Fring, U. Günther and H. Jones, Quantum physics with non-Hermitian operators, J. Phys. A Math. Theor. 45 (2012), 440301.
  * [15] C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947–1018.
  * [16] O. Cherbal, M. Drir, M. Maamache and D. A. Trifonov, Fermionic coherent states for pseudo-Hermitian two-level systems, J. Phys. A 40 (2007), 1835–1844.
  * [17] O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, Boston, 2003.
  * [18] J. Dieudonné, Quasi-Hermitian operators, Proceedings of the International Symposium on Linear Spaces, Jerusalem 1960, Oxford: Pergamon Press (1961), pp. 115–122.
  * [19] F. M. Ellis, U. Günther, T. Kottos, H. Ramezani and J. Schindler, Bypassing the bandwidth theorem with $PT$ symmetry, Phys. Rev. A 85 (2012), 062122.
  * [20] F. M. Ellis, T. Kottos, J. M. Lee, H. Ramezani and J. Schindler, $PT$-Symmetric Electronics, J. Phys. A Math. and Theor. 45 (2012), 444029.
  * [21] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, Addison Wesley, New York, 2000.
  * [22] B. C. Hall. Quantum Theory For Mathematicians, Springer, 2013, New York.
  * [23] M.R. Kibler, Variations on a theme of Heisenberg, Pauli and Weyl, J. Phys. A 41 No. 37 (2008).
  * [24] E. Merzbacher, Quantum Mechanics, Wiley, New York (1970),
  * [25] P. Roman, _Advanced quantum mechanics_ , Addison–Wesley, New York (1965)
  * [26] A. Mostafazadeh, Pseudo-Hermitian quantum mechanics with unbounded metric operators, Philos Trans R Soc London Ser A, 2013; 45:115311.
  * [27] A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanic, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1191–1306.
  * [28] M. A. Nielsen and I. L. Chuang, Quantum computing and quantum information, Cambridge University Press, Cambridge, 2004.
  * [29] W. Pauli, Zur Quantenmechanik des magnetischen Elektrons, Z. Phys. 43 (1927), 601–623.
  * [30] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, 1996, New York.
  * [31] A. Rocchetto and F.G. Russo, Decomposition of Pauli groups via weak central products, preprint, ArXiv, 2019, https://arxiv.org/abs/1911.10158.