7680\Delta^{2}Eg^{8}-4096E^{5}g^{6}-16384E^{3}g^{8}+3840Eg^{10}+64\Delta^{10}-64\Delta^{8}E^{2}$
$\displaystyle+$ $\displaystyle
256\Delta^{8}g^{2}-768\Delta^{6}E^{2}g^{2}+512\Delta^{6}g^{4}+1024\Delta^{4}E^{2}g^{4}+2176\Delta^{4}g^{6}$
$\displaystyle-$ $\displaystyle
1536\Delta^{2}E^{4}g^{4}+2304\Delta^{2}E^{2}g^{6}+2208\Delta^{2}g^{8}-10240E^{4}g^{6}-2688E^{2}g^{8}$
$\displaystyle+$ $\displaystyle
2944g^{10}-64\Delta^{8}E-224\Delta^{6}Eg^{2}+256\Delta^{4}E^{3}g^{2}+2048\Delta^{4}Eg^{4}$
$\displaystyle-$ $\displaystyle
3072\Delta^{2}E^{3}g^{4}+672\Delta^{2}Eg^{6}-3840E^{3}g^{6}+5504Eg^{8}-64\Delta^{8}+96\Delta^{6}E^{2}$
$\displaystyle+$ $\displaystyle
16\Delta^{6}g^{2}+384\Delta^{4}E^{2}g^{2}-52\Delta^{4}g^{4}-1440\Delta^{2}E^{2}g^{4}-48\Delta^{2}g^{6}+960E^{4}g^{4}$
$\displaystyle+$ $\displaystyle
4480E^{2}g^{6}+1024g^{8}+96\Delta^{6}E-40\Delta^{4}Eg^{2}+96\Delta^{2}E^{3}g^{2}+96\Delta^{2}Eg^{4}$
$\displaystyle+$ $\displaystyle
1920E^{3}g^{4}+1744Eg^{6}+20\Delta^{6}-52\Delta^{4}E^{2}-84\Delta^{4}g^{2}+144\Delta^{2}E^{2}g^{2}$
$\displaystyle+$ $\displaystyle
60\Delta^{2}g^{4}+672E^{2}g^{4}-216g^{6}-52\Delta^{4}E+72\Delta^{2}Eg^{2}-48E^{3}g^{2}-288Eg^{4}$
$\displaystyle-$ $\displaystyle
2\Delta^{4}+12\Delta^{2}E^{2}+12\Delta^{2}g^{2}-72E^{2}g^{2}-25g^{4}+12\Delta^{2}E-26Eg^{2}-E^{2}$
$\displaystyle-$ $\displaystyle g^{2}-E),$

$\mathcal{P}_{0}=g^{-4}(16Eg^{2}+8\Delta^{2}+8g^{2}-5),$

and

$\mathcal{Q}_{0}=-\frac{8}{g^{8}}(8\Delta^{2}g^{4}+8\Delta^{2}Eg^{2}+2\Delta^{4}+4\Delta^{2}g^{2}-8Eg^{2}-3\Delta^{2}-4g^{2}+1).$

Now, having the functions $\mathcal{D}_{0}=\mathcal{D}_{0}(\Delta,E,g)$,
$\mathcal{P}_{0}=\mathcal{P}_{0}(\Delta,E,g)$, and
$\mathcal{Q}_{0}=\mathcal{Q}_{0}(\Delta,E,g)$ depending on the physical
parameters of Rabi problem in hand, we can use J. L. Lagrange theorem as it
was stated in [3, Chapter IV, Section 7], to identify which of the types I,
II, or III of Stokes graphs and domain configurations of $Q_{0}(z)\,dz^{2}$
described in Section 4 corresponds to a given choice of the parameters
$\Delta$, $E$, and $g$.

######  Proposition 5.1 (Generic cases).

Suppose that the parameters $\Delta$ and $E$ of the Rabi problem are real and
that $g^{2}\not=0$ is also real. Then the following holds:

  1. I.

Quadratic differential $Q_{0}(z)\,dz^{2}$ has four distinct complex zeros,
which are in conjugate pairs, if and only if the discriminant
$\mathcal{D}_{0}(\Delta,E,g)$ is positive and at least one of the functions
$\mathcal{P}_{0}(\Delta,E,g)$ and $\mathcal{Q}_{0}(\Delta,E,g)$ is also
positive. In this case, possible Stokes graphs and domain configurations of
$Q_{0}(z)\,dz^{2}$ are described in cases 1, 2, and 3 in part I of Section 4.

  2. II.

Quadratic differential $Q_{0}(z)\,dz^{2}$ has a pair of complex conjugate
zeros and two distinct real zeros not equal $\pm 1$ if and only if the
discriminant $\mathcal{D}_{0}(\Delta,E,g)$ is negative and $E\not=-g^{2}$,
$E\not=-g^{2}\pm 1$, $E\not=-g^{2}+2$. In this case, possible Stokes graphs
and domain configurations of $Q_{0}(z)\,dz^{2}$ are described in part II of
Section 4.

  3. III.

Quadratic differential $Q_{0}(z)\,dz^{2}$ has four distinct real zeros not
equal $\pm 1$ if and only if the discriminant $\mathcal{D}_{0}(\Delta,E,g)$ is
positive, both functions $\mathcal{P}_{0}(\Delta,E,g)$ and
$\mathcal{Q}_{0}(\Delta,E,g)$ are negative, and $E\not=-g^{2}$,
$E\not=-g^{2}\pm 1$, $E\not=-g^{2}+2$. In this case, possible Stokes graphs
and domain configurations of $Q_{0}(z)\,dz^{2}$ are described in part III of
Section 4.

To specify possible positions of multiple zeros, we follow J. L. Lagrange work
cited above. For this, we need two more characteristics, $\mathcal{R}_{0}$ and
$\mathcal{S}_{0}$, of the polynomial $P_{0}(z)$, which can be expressed in
terms of the coefficients $c_{k}$ of $P_{0}(z)$ and in terms of the parameters
of the Rabi problem as follows:

$\mathcal{R}_{0}=c_{3}^{3}+8c_{1}-4c_{3}c_{2}=g^{-6}(1-16g^{4}-16Eg^{2}-4\Delta^{2}-7g^{2})$

and

$\displaystyle\mathcal{S}_{0}$ $\displaystyle=$ $\displaystyle
c_{2}^{2}-3c_{3}c_{1}+12c_{0}=\frac{1}{16g^{8}}(1-192\Delta^{2}g^{4}+256E^{2}g^{4}+64\Delta^{2}Eg^{2}$
$\displaystyle+$ $\displaystyle
256Eg^{4}+16\Delta^{4}+32\Delta^{2}g^{2}+64g^{4}+32Eg^{2}-8\Delta^{2}+16g^{2}+1).$

######  Proposition 5.2 (Degenerate cases with full set of critical points and
multiple zeros of $P_{0}(z)$).

Suppose that the parameters $\Delta$, $E$, and $g^{2}\not=0$ of the Rabi
problem are real such that $E\not=-g^{2}$, $E\not=-g^{2}\pm 1$,
$E\not=-g^{2}+2$. Then $P_{0}(z)$ has a multiple zero $\not=\pm 1$ if and only
if $\mathcal{D}_{0}(\Delta,E,g)=0$.

Furthermore, if $\mathcal{D}_{0}(\Delta,E,g)=0$, then the following subcases
refining positions of zeros of $P_{0}(z)$ happen:

  1. 1.

If $\mathcal{Q}_{0}(\Delta,E,g)=0$, $\mathcal{P}_{0}(\Delta,E,g)>0$, and
$\mathcal{R}_{0}(\Delta,E,g)=0$, then there are two double complex conjugate
zeros. In this case, possible Stokes graphs and domain configurations of
$Q_{0}(z)\,dz^{2}$ are described in the case 4 in part I of Section 4.

  2. 2.

If $\mathcal{Q}_{0}(\Delta,E,g)=0$ and $\mathcal{P}_{0}(\Delta,E,g)<0$, then
there are two double real zeros $\not=\pm 1$.

  3. 3.

If $\mathcal{Q}_{0}(\Delta,E,g)=0$ and $\mathcal{S}_{0}(\Delta,E,g)=0$, then
$P_{0}(z)$ has a real zero of order four at $z=-\frac{1}{4g^{2}}$.

  4. 4.

If $\mathcal{P}_{0}(\Delta,E,g)<0$, $\mathcal{Q}_{0}(\Delta,E,g)<0$, and
$\mathcal{S}_{0}(\Delta,E,g)\not=0$, then there is a double real zero
$\not=\pm 1$ and two simple real zeros $\not=\pm 1$.

  5. 5.

If $\mathcal{Q}_{0}(\Delta,E,g)>0$ or if $\mathcal{P}_{0}(\Delta,E,g)>0$ and
at least one of the quantities $\mathcal{Q}_{0}(\Delta,E,g)$ and
$\mathcal{R}_{0}(\Delta,E,g)$ is not zero, then there are a double real zero
$\not=\pm 1$ and two complex conjugate zeros.

  6. 6.

If $\mathcal{S}_{0}(\Delta,E,g)=0$ and $\mathcal{Q}_{0}(\Delta,E,g)\not=0$,
then there are a triple real zero $\not=\pm 1$ and a simple real zero
$\not=\pm 1$.

We stress here that Proposition 5.2 describes all possible cases when
$P_{0}(z)$ has multiple zeros and the quadratic differential
$Q_{0}(z)\,dz^{2}$ has double poles at the points $\pm 1$.

Equations (2.9) and (2.10) provide a parametric description of the set of all
quadruples $(c_{3},c_{2},c_{1},c_{0})\in\mathbb{R}^{4}$, which coordinates are
the coefficients of the numerator $P_{0}(z)$ of the quadratic differential
$Q_{0}(z)\,dz^{2}$ that appears in the framework of the Rabi problem. Next, we
prove two results, which provide more explicit description of this set.

######  Theorem 5.3.

Let
$Q_{0}(z)\,dz^{2}=-\frac{z^{4}+c_{3}z^{3}+c_{2}z^{2}+c_{1}z+c_{0}}{(z-1)^{2}(z+1)^{2}}\,dz^{2}$be
a quadratic differential associated with the Rabi problem for some choice of
the parameters $\Delta$, $E$, and $g$, such that
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$. Then

(5.1) $(c_{3},c_{2},c_{1},c_{0})\in\\{a\\}\times S(a),\quad{\mbox{where
$a=g^{-2}$,}}$

and $S(a)$ denotes a parabolic cylinder in $\mathbb{R}^{3}$ defined by

(5.2)
$S(a)=\\{(X,Y,Z)\in\mathbb{R}^{3}:\,(Y+a)^{2}-a^{2}X-a^{2}Z-(1/4)a^{2}(4+3a^{2})=0\\}.$

Moreover, there is a constant $c\geq 0$ such that the coordinates $X$, $Y$,
$Z$ in (5.2), representing the coefficients $c_{2}$, $c_{1}$, $c_{0}$, are
related via the following equations:

(5.3) $X=-\frac{1}{4a}(8Y+a^{3}+16a)+c,\quad\quad
Z=\frac{1}{2a^{2}}(2Y^{2}+8aY-a^{4}+8a^{2})-c.$

_Proof._ To prove the first part of this theorem, we put $a=g^{-2}$ and
suppose that the quadratic differential $Q_{0}(z)\,dz^{2}$ is associated with
the Rabi problem having parameters $\Delta$, $E$, and $g$, such that
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$. Since $c_{3}=a=g^{-2}$ by the first
equation in (2.9), we have to show that $(c_{2},c_{1},c_{0})\in S(a)$.
Substituting expressions for $c_{2}$, $c_{1}$, and $c_{0}$ given by equations
(2.9) and (2.10) for the variables $X$, $Y$, and $Z$ in (5.2), we obtain

$\displaystyle(c_{1}+a)^{2}-a^{2}c_{2}-a^{2}c_{0}-\frac{a^{2}}{4}(4+3a^{2})$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{2g^{4}}(4g^{2}+2E+1)+g^{-2}\right)^{2}$
$\displaystyle-$ $\displaystyle\frac{1}{4g^{8}}(8Eg^{2}+4\Delta^{2}+4g^{2}-1)$
$\displaystyle+$ $\displaystyle\frac{1}{4g^{8}}(4\Delta^{2}-4E^{2}-4E+1)$
$\displaystyle-$ $\displaystyle\frac{1}{4g^{8}}(4g^{4}+3).$

Simplifying the latter equation, we find that the right-hand side of this
equality equals zero and therefore the point $(c_{2},c_{1},c_{0})$ lies on the
surface of the parabolic cylinder (5.2).

To prove relations (5.3), we substitute $Y$ for $c_{1}$ in the first equation
in (2.9) and then solve it for $E$ to get

(5.4) $E=-\frac{1}{2a^{2}}(2Y+a^{2}+4a).$

Substituting this expression for $E$, $X$ for $c_{2}$, and $Z$ for $c_{0}$ in
equations (2.9) and (2.10), we obtain relations (5.3) with
$c=a^{2}\Delta^{2}\geq 0$. $\Box$

Theorem 5.3 shows that equality $a=g^{-2}$ and relations (5.2) and (5.3) are
necessary for the point $(X,Y,Z)\in\mathbb{R}^{3}$ to represent coefficients
$c_{2}$, $c_{1}$, and $c_{0}$ of the quadratic differential $Q_{0}(z)\,dz^{2}$
associated with the Rabi problem. Our next result shows that these conditions
are also sufficient.

######  Theorem 5.4.

If $X$, $Y$, and $Z$ satisfy equations (5.2) and (5.3) with some $a\not=0$ and
$c\geq 0$, then there are parameters $\Delta$, $E$, and $g$ of the Rabi
problem such that $g^{-2}=a$ and

(5.5) $X=c_{2}(\Delta,E,g)\quad Y=c_{1}(E,g),\quad Z=c_{0}(\Delta,E,g)$

with $c_{2}(\Delta,E,g)$, $c_{1}(E,g)$, and $c_{0}(\Delta,E,g)$ defined by
(2.9) and (2.10).

_Proof._ We can choose $g$ so that $a=g^{-2}$. Then, for a given $Y$, we
choose $E$ as in (5.4). Solving (5.4) for $Y$, we obtain

(5.6) $Y=-(a/2)(2aE+a+4).$

Substituting this expression for $Y$ and $g^{-2}$ for $a$ in equations (5.3),
we find

(5.7) $X=\frac{1}{4g^{4}}(8g^{2}E+4g^{2}-1)+c,\quad
Z=\frac{1}{4g^{4}}(4E^{2}+4E-1)-c.$

Choosing $\Delta$ so that $c=\Delta^{2}/g^{4}>0$, substituting this in the
latter equations, and taking into account equations (2.9), (2.10), we obtain
the desired equations (5.5). $\Box$

Figure 1. Portion of the cylinder $S(a)$ associated to the Rabi problem with
parameters $g^{2}=a^{-1}=1$, $\Delta\geq 0$, and $E\in\mathbb{R}$.

Since the four real coefficients $c_{k}$, $k=0,1,2,3$, of the polynomial
$P_{0}(z)$ depend on only three real parameters, $\Delta$, $E$ and $g^{2}$, of
the Rabi model, it is reasonable to expect that some of the Stokes graphs and
domain configurations of $Q_{0}(z)\,dz^{2}$ described in Section 4 will not
appear in the framework of the Rabi problem. We start our discussion of
possible and impossible types of Stokes graphs with a simple result, which
excludes the possibility of graphs symmetric with respect to the imaginary
axis.

######  Lemma 5.5.

There are no Stokes graphs and domain configurations of $Q_{0}(z)\,dz^{2}$
symmetric with respect to the imaginary axis, which are associated with the
Rabi problem having the parameters $\Delta$, $E$, and $g$ such that
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$.

_Proof._ Since the Stokes graphs appeared in this study already posses
symmetry with respect to the real axis, the symmetry with respect to the
imaginary axis occurs when the zeros of $P_{0}(z)$ are in pairs symmetric with
respect to the origin. In this case, the polynomial $P_{0}(z)$ is biquadratic;
i.e. $P_{0}(z)=z^{4}+c_{2}z^{2}+c_{0}$. In particular, $c_{3}=0$ in this case.
Under our assumptions on the Rabi parameters, $c_{3}(g)=g^{-2}$ represents the
boson-fermion coupling $g$. Thus, the case $c_{3}=0$ does not occur for finite
values of $g$ and therefore Stokes graphs symmetric with respect to the
imaginary axis do not appear in the Rabi problem under our assumptions. $\Box$

Although the symmetry of Stokes graphs with respect to the imaginary axis does
not occur for finite values of $g$, the symmetric cases are possible as
“asymptotic cases”, which appear in the Rabi model when the boson-fermion
coupling tends to infinity. Possible structures of Stokes graphs in these
asymptotic cases will be discussed in Section 6.

Next, we examine possibility of so-called “breaks of symmetry” in the Rabi
model. Let $\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$ denote the domain
configuration of the quadratic differential $Q_{0}(z)\,dz^{2}$ defined by
(2.8) with the coefficients $c_{k}=c_{k}(\Delta,E,g)$, $k=0,1,2,3$, given by
equations (2.9) and (2.10). By “break of symmetry” in the Rabi model we
understand a situation, when a certain domain configuration
$\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$ corresponds to some values of
the Rabi parameters $\Delta$, $E$, and $g$, such that
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$, but its mirror configuration, call
it $\widetilde{\mathbf{\Omega}}$, does not correspond to any choice of such
$\Delta$, $E$, and $g$. As the following lemma shows, such breaks of symmetry
never occur in the settings of the Rabi problem with
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$.

######  Lemma 5.6.

For any domain configuration $\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$
corresponding to the Rabi problem with $\Delta,E,g^{2}\in\mathbb{R}$,
$g\not=0$, there exists a mirror domain configuration
$\widetilde{\mathbf{\Omega}}$, which corresponds to the Rabi problem with the
parameters $\Delta$, $-(E+1)$, and $ig$; i.e.
$\widetilde{\mathbf{\Omega}}=\mathbf{\Omega}(\Delta,-(E+1),ig)$.

_Proof._ Let $\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$ be the domain
configuration of $Q_{0}(z)\,dz^{2}$ with coefficients
$c_{k}=c_{k}(\Delta,E,g)$ defined by formulas (2.9), (2.10) with
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$. Let $e_{k}=e_{k}(\Delta,E,g)$,
$k=1,2,3,4$, denote zeros of $Q_{0}(z)\,dz^{2}$. Since the coefficients
$c_{k}$ are real, it follows that the mirror domain configuration
$\widetilde{\mathbf{\Omega}}$ corresponds to the quadratic differential
$\widetilde{Q}_{0}(z)\,dz^{2}$, which has zeros at the points
$\widetilde{e}_{k}=-e_{k}(\Delta,E,g)$, $k=1,2,3,4$. From this, using Vieta’s
formulas (see, for instance, [18, Section 26, formula (1)] or [19, Remark
3.14]), we conclude that the coefficients $\widetilde{c}_{k}$, $k=0,1,2,3$, of
the numerator $\widetilde{P}_{0}(z)$ of $\widetilde{Q}_{0}(z)$ satisfy the
following equations:

$\widetilde{c}_{3}=-c_{3}(g),\ \widetilde{c}_{2}=c_{2}(\Delta,E,g),\
\widetilde{c}_{1}=-c_{1}(E,g),\ \widetilde{c}_{0}=c_{0}(\Delta,E,g).$

Matching these equations with appropriate equations for the coefficients
$c_{k}$ given by formulas (2.9), (2.10), we obtain the following:

$\displaystyle\widetilde{c}_{3}$ $\displaystyle=$
$\displaystyle-g^{-2}=(ig)^{-2},$ $\displaystyle\widetilde{c}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4g^{4}}\left(8Eg^{2}+4\Delta^{2}+4g^{2}-1\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4(ig)^{4}}\left(8(-(E+1))(ig)^{2}+4\Delta^{2}+4(ig)^{2}-1\right),$
$\displaystyle\widetilde{c}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2g^{4}}\left(4g^{2}+2E+1\right)=-\frac{1}{2(ig)^{4}}\left(4(ig)^{2}+2(-(E+1))+1\right),$
$\displaystyle\widetilde{c}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4g^{4}}\left(4\Delta^{2}-4E^{2}-4E+1\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{4(ig)^{4}}\left(4\Delta^{2}-4(-(E+1))^{2}-4(-(E+1))+1\right).$

As the latter equations show, the domain configuration
$\widetilde{\mathbf{\Omega}}=\mathbf{\Omega}(\Delta,-(E+1),ig)$ corresponding
to the Rabi parameters $\Delta$, $-(E+1)$, and $ig$ is the mirror domain
configuration for the domain configuration
$\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$ for a given set of the Rabi
parameters $\Delta$, $E$, and $g$. $\Box$

As Lemma 5.5 shows, the Stokes graphs of $Q_{0}(z)\,dz^{2}$ symmetric with
respect to the imaginary axis do not appear in the Rabi problem. Next, we
discuss possibility of cases when critical points of $Q_{0}(z)\,dz^{2}$
possess some “partial symmetries”. In our next lemma, we study the case when
all zeros of $Q_{0}(z)\,dz^{2}$ lie on the same vertical line
$\\{z:\,\operatorname{Re}z=\alpha\\}$. Since the mirror configuration always
exists, in this lemma we assume without loss of generality that $\alpha>0$.

######  Lemma 5.7.

For every $\alpha>0$ and $\beta_{1}\geq 0$, there is a unique
$\beta_{2}>\beta_{1}$ such that the quadratic differential $Q_{0}(z)\,dz^{2}$
with zeros $e_{1}=\alpha+i\beta_{1}$, $e_{2}=\alpha+i\beta_{2}$,
$e_{3}=\alpha-i\beta_{1}$, and $e_{4}=\alpha-i\beta_{2}$ is associated with
the Rabi problem for some $\Delta$, $E$, and $g$ such that
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$. Precisely,
$\beta_{2}=\beta_{2}(\alpha,\beta_{1})$ is given by the following equation:

(5.8) $\beta_{2}=\sqrt{\beta_{1}^{2}-8\alpha}.$

Furthermore, the Rabi parameters $g$, $E$, and $\Delta$ corresponding to the
given values $\alpha$ and $\beta_{1}$ are defined by the following equations:

(5.9) $g^{-2}=-4\alpha,\quad
E=\frac{1}{4\alpha}(\alpha^{2}-6\alpha+2+\beta_{1}^{2}),\quad\Delta^{2}=\frac{1}{4\alpha^{2}}(3\alpha^{2}-4\alpha+1+\beta_{1}^{2}).$

_Proof._ Let $e_{1}=\alpha+i\beta_{1}$, $e_{2}=\alpha+i\beta_{2}$,
$e_{3}=\alpha-i\beta_{1}$, and $e_{4}=\alpha-i\beta_{2}$ with $\alpha>0$,
$0<\beta_{1}<\beta_{2}$, be zeros of $P_{0}(z)$. Then

(5.10) $P_{0}(z)=z^{4}-4\alpha
z^{3}+(6\alpha^{2}+\beta_{1}^{2}+\beta_{2}^{2})z^{2}-2\alpha(2\alpha^{2}+\beta_{1}^{2}+\beta_{2}^{2})z+(\alpha^{2}+\beta_{1}^{2})(\alpha^{2}+\beta_{2}^{2}).$

We define $a=g^{-2}-4\alpha$ and

(5.11)
$\delta_{1}=\beta_{1}^{2}+\beta_{2}^{2},\quad\delta_{2}=\beta_{1}^{2}\beta_{2}^{2}.$

As in Theorems 5.3 and 5.4, we will use notations $X=c_{2}$, $Y=c_{1}$, and
$Z=c_{0}$ for the appropriate coefficients of $P_{0}(z)$. With these
notations, the coefficients of the polynomial (5.10) are the following:

(5.12) $X=\frac{3}{8}a^{2}+\delta_{1},\quad
Y=\frac{a^{3}}{16}+\frac{1}{2}a\delta_{1},\quad
Z=\frac{a^{4}}{16^{2}}+\frac{1}{16}a^{2}\delta_{1}+\delta_{2}.$

Equating the right-hand side of the second equation in (5.12) to the right-
hand side of equation (5.6) and solving the resulting equation for
$\delta_{1}$, we find that

(5.13) $\delta_{1}=-2aE-\frac{1}{8}a^{2}-a-4.$

Substituting this expression for $\delta_{1}$ in the first equation in (5.12)
and then equating its right-hand side to the right-hand side of the first
equation in (5.7) and then solving the resulting equation for $c$, we find
that

(5.14) $c=a\Delta^{2}=-4aE+\frac{1}{2}a^{2}-2a-4.$

To find an expression for $\delta_{2}$, we replace $c$ in the second equation
in (5.7) with the right-hand side of equation (5.14) and we replace
$\delta_{1}$ in the third equation in (5.12) with the right-hand side of
equation (5.13). Then we equate the resulting expressions and solve this
equation for $\delta_{2}$ to get, after some algebra, the following:

(5.15)
$\delta_{2}=\left(aE+\frac{1}{16}a^{2}-\frac{1}{2}a+2\right)\left(aE+\frac{1}{16}a^{2}+\frac{3}{2}a+2\right).$

It follows from equations (5.11) that $\beta_{1}^{2}$ and $\beta_{2}^{2}$ are
solutions of the quadratic equation

(5.16) $\tau^{2}-\delta_{1}\tau+\delta_{2}=0$

with $\delta_{1}$ and $\delta_{2}$ given in (5.13)) and (5.15). Calculating
the discriminant $\nabla$ of this equation, we find that

$\nabla=\delta_{1}^{2}-4\delta_{2}=4a^{2}>0.$

Solving equation (5.16) for $\tau=\beta_{k}^{2}$, $k=1,2$, we find that

(5.17)
$\beta_{1}^{2}=-aE-\frac{1}{16}a^{2}-\frac{3}{2}a-2,\quad\beta_{2}^{2}=\beta_{1}^{2}+2a=-aE-\frac{1}{16}a^{2}+\frac{1}{2}a-2.$

The second of these equations implies (5.8). Solving the first of these
equations for $E$, we obtain the second equation in (5.9).

Next, we recall that $c=a^{2}\Delta^{2}$. Substituting this expression for $c$
and the expression for $E$ given by the second equation in (5.9) in (5.14) and
then solving the resulting equation for $\Delta^{2}$, we obtain the third
equation in (5.9)). $\Box$

######  Corollary 5.8.

As equation (5.8) shows, degenerate configurations, when $Q_{0}(z)\,dz^{2}$
has two conjugate double zeros or one real zero $\not=\pm 1$ of order four, do
not appear in the framework of the Rabi problem with
$\Delta,E,g^{2}\in\mathbb{R}$, $g\not=0$.

Another case, when a ”partial symmetry” may be important, is when zeros of
$Q_{0}(z)\,dz^{2}$ lie on two horizontal lines
$\\{z:\,\operatorname{Im}z=\pm\beta\\}$, $\beta>0$. But, as the following
lemma shows, this case do not appear in solutions of the Rabi problem with
$\Delta,E,g^{2}\in\mathbb{R}$.

######  Lemma 5.9.

Suppose that the quadratic differential $Q_{0}(z)\,dz^{2}$ with complex zeros
$e_{1}=\alpha_{1}+i\beta_{1}$, $e_{2}=\alpha_{2}+i\beta_{2}$,
$e_{3}=\alpha_{1}-i\beta_{1}$, $e_{4}=\alpha_{2}-i\beta_{2}$, such that
$\beta_{1}>0$, $\beta_{2}>0$, is associated with the Rabi problem for some
values of the parameters $\Delta,E,g^{2}\in\mathbb{R}$. Then
$\beta_{1}\not=\beta_{2}$.

_Proof._ Suppose, by contradictions, that $\beta_{1}=\beta_{2}$ for some
choice of $\Delta,E,g^{2}\in\mathbb{R}$. Using the ”mirror configuration”
argument once more, we may assume without loss of generality that in this case
the zeros are $e_{1}=(\alpha-\delta)+i\beta$, $e_{2}=(\alpha+\delta)+i\beta$,
$e_{3}=(\alpha-\delta)-i\beta$, and $e_{4}=(\alpha+\delta)-i\beta$ with
$\alpha<0$, $\beta>0$, and $\delta>0$. With these zeros, the polynomial
$P_{0}(z)$, which is the numerator of $Q_{0}(z)$, is the following:

$\displaystyle P_{0}(z)$ $\displaystyle=$ $\displaystyle z^{4}-4\alpha
z^{3}+2(3\alpha^{2}-\delta^{2}+\beta^{2})z^{2}-4\alpha(\alpha^{2}-\delta^{2}+\beta^{2})z$
$\displaystyle+$
$\displaystyle(\alpha^{2}-\delta^{2})^{2}+\beta^{2}(2\alpha^{2}+2\delta^{2}+\beta^{2}).$

We define $a=-4\alpha$ and, as before, we use $X$, $Y$, and $Z$ to denote the
coefficients $c_{2}$, $c_{1}$, and $c_{0}$ of $P_{0}(z)$. Then

$X=2((3/16)a^{2}+\beta^{2}-\delta^{2}),\quad
Y=a((1/16)a^{2}+\beta^{2}-\delta^{2}),\quad$
$Z=((1/16)a^{2}-\delta^{2})^{2}+\beta^{2}((1/8)a^{2}+2\delta^{2}+\beta^{2}).$

Equating these expressions for $X$, $Y$, and $Z$ to the corresponding
expressions in the formulas (5.6) and (5.7), and then solving the resulting
equations for $E$, $c$, and $\delta$, we find that

$E=-\frac{a^{2}\beta^{2}+16\beta^{4}+8a\beta^{2}+4a^{2}+32\beta^{2}}{16a\beta^{2}},\quad
c=\frac{16\beta^{4}+(3a^{2}+16)\beta^{2}+4a^{2}}{4\beta^{2}},$

and

$\delta^{2}=-\frac{1}{4}\frac{a^{2}}{\beta^{2}}.$

Since $a^{2}>0$, $\beta^{2}>0$, the latter equation contradicts the assumption
that $\delta>0$, which proves the lemma. $\Box$

Next, we examine a possibility of two real zeros symmetric with respect to the
origin and two complex conjugate zeros.

######  Lemma 5.10.

The quadratic differential $Q_{0}(z)\,dz^{2}$ with zeros $e_{1}=-\alpha$,
$e_{2}=\alpha$, $e_{3}=\delta+i\beta$, $e_{4}=\delta-i\beta$, where
$\alpha>0$, $\beta\geq 0$, is associated with the Rabi problem for some
$\Delta,E,g^{2}\in\mathbb{R}$ if and only if one of the following conditions
holds true:

  1. (a)

$\alpha\in(1,\sqrt{8/5})\cup(\sqrt{2},2)$ and
$\delta^{2}>\frac{\alpha^{2}(\alpha^{2}-1)}{4-\alpha^{2}}$,

  2. (b)

$\sqrt{8/5})\leq\alpha\leq\sqrt{2}$ and
$\delta^{2}>\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}$,

and if and only if

(5.18)
$\beta^{2}=\frac{(4-\alpha^{2})\delta^{2}-\alpha^{2}(\alpha^{2}-1)}{\alpha^{2}-1}.$

Furthermore, the Rabi parameters $g$, $E$, and $\Delta$ corresponding to the
given values $\alpha$ and $\delta$, satisfying conditions (a) and (b), are
defined by the following equations:

(5.19) $g^{-2}=-2\delta,\quad
E=-\frac{\delta+\alpha^{2}-2}{2\delta},\quad\Delta^{2}=\frac{(\alpha^{2}+2)\delta^{2}-4(\alpha^{2}-1)^{2}}{4\delta^{2}(\alpha^{2}-1)}.$

_Proof._ With zeros defined in this lemma, the numerator of the quadratic
differential $Q_{0}(z)\,dz^{2}$ has the form

(5.20) $P_{0}(z)=z^{4}-2\delta
z^{3}+(\delta^{2}-\alpha^{2}+\beta^{2})z^{2}+2\alpha^{2}\delta
z-\alpha^{2}(\delta^{2}+\beta^{2}).$

Thus, in this case $c_{3}=g^{-2}=-2\delta$. As before, using notations
$X=c_{2}$, $Y=c_{1}$, and $Z=c_{0}$ for the appropriate coefficients of
$P_{0}(z)$, we find that

(5.21) $X=\delta^{2}-\alpha^{2}+\beta^{2},\quad Y=2\alpha^{2}\delta,\quad
Z=-\alpha^{2}(\delta^{2}+\beta^{2}).$

Equating these expressions for $X$, $Y$, and $Z$ to the corresponding
expressions in formulas (5.6) and (5.7), and then solving the resulting
equations for $E$, $\Delta^{2}=c/(4\delta^{2})$, and $\beta^{2}$, we obtain
the second and the third equations in (5.19) and equation (5.18).

Now, simple algebra shows that the right-hand side of equation (5.18) is
positive if and only if $1<\alpha<2$ and
$\delta^{2}>(\alpha^{2}(\alpha^{2}-1))/(4-\alpha^{2})$. Furthermore, if
$1<\alpha<2$, then the right-hand side of the third equation in (5.19) is
positive if and only if $\delta^{2}>(4(\alpha^{2}-1)^{2})/(\alpha^{2}+2)$.
Combining these cases, we conclude that relation (5.18) and the inequalities

(5.22)
$1<\alpha<2,\quad\delta^{2}\geq\max\left\\{\frac{\alpha^{2}(\alpha^{2}-1)}{4-\alpha^{2}},\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}\right\\}$

are necessary and sufficient for the quadratic differential $Q_{0}(z)\,dz^{2}$
to be associated with the Rabi problem and the corresponding physical
parameters are given by formulas (5.19).

Finally, comparing functions $\frac{\alpha^{2}(\alpha^{2}-1)}{4-\alpha^{2}}$
and $\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}$, one can easily find that
conditions (a) and (b) of Lemma 5.10 are satisfied if and only if inequalities
(5.22) are satisfied. $\Box$

A similar result, for the quadratic differential $Q_{0}(z)\,dz^{2}$ with four
real zeros such that two of them are symmetric with respect to the origin, is
presented in the following lemma.

######  Lemma 5.11.

The quadratic differential $Q_{0}(z)\,dz^{2}$ with zeros $e_{1}=-\alpha$,
$e_{2}=\alpha$, $e_{3}=\delta-\beta$, $e_{4}=\delta+\beta$, where $\alpha>0$,
$\beta>0$, $\delta>0$, is associated with the Rabi problem for some
$\Delta,E,g^{2}\in\mathbb{R}$ if and only if one of the following conditions
holds true:

  1. (a)

$0<\alpha<1$ and $\delta^{2}\leq\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}$,

  2. (b)

$\alpha\in(1,\sqrt{8/5}]\cup[\sqrt{2},2]$ and
$\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}\leq\delta^{2}<\frac{\alpha^{2}(\alpha^{2}-1)}{4-\alpha^{2}}$,

  3. (c)

$\alpha>2$ and $\delta^{2}\geq\frac{4(\alpha^{2}-1)^{2}}{\alpha^{2}+2}$,

and if and only if

(5.23)
$\beta^{2}=-\frac{(4-\alpha^{2})\delta^{2}-\alpha^{2}(\alpha^{2}-1)}{\alpha^{2}-1}.$

Furthermore, the Rabi parameters $g$, $E$, and $\Delta$ corresponding to the
given values $\alpha$ and $\delta$, satisfying conditions (a), (b), (c), are
defined by equations (5.19).

_Proof._ The proof is similar to the proof of the previous lemma. Under the
assumptions, the numerator of the quadratic differential $Q_{0}(z)\,dz^{2}$
has the form

(5.24) $P_{0}(z)=z^{4}-2\delta
z^{3}+(\delta^{2}-\alpha^{2}-\beta^{2})z^{2}+2\alpha^{2}\delta
z-\alpha^{2}(\delta^{2}-\beta^{2}).$

Thus, in this case $c_{3}=g^{-2}=-2\delta$. With the notations $X=c_{2}$,
$Y=c_{1}$, and $Z=c_{0}$, we have the following:

(5.25) $X=\delta^{2}-\alpha^{2}-\beta^{2},\quad Y=2\alpha^{2}\delta,\quad
Z=-\alpha^{2}(\delta^{2}-\beta^{2}).$

Equating these expressions for $X$, $Y$, and $Z$ to the corresponding
expressions in the formulas (5.6) and (5.7), and then solving the resulting
equations for $E$, $\Delta^{2}=c/(4\delta^{2})$, and $\beta^{2}$, we conclude
that $g$, $E$, and $\Delta$ are given by equations (5.19) as in Lemma 5.10 and
that $\beta$ is given by equation (5.23).

After simple algebra, left to the interested reader, we conclude that, under
the assumptions of Lemma 5.11, the right-hand side of equation (5.23) and the
right-hand side of the third equation in (5.19) are non-negative if and only
if $\alpha$ and $\delta$ satisfy conditions (a), (b), (c) of the lemma. $\Box$

As we have shown earlier, the quadratic differentials $Q_{0}(z)\,dz^{2}$
associated with the Rabi problem may have all zeros on the same vertical line,
but may not have all zeros on two horizontal lines. Next, we show that a
similar effect happens for quadratic differentials with all zeros on the same
circle centered at the origin and all zeros on two rays issuing from the
origin.

######  Lemma 5.12.

For every $r>0$ and $0\leq\theta_{1}<\frac{\pi}{2}$, there is a unique
$\theta_{2}$, $0\leq\theta_{2}\leq\pi$, such that the quadratic differential
$Q_{0}(z)\,dz^{2}$ with zeros $e_{1}=re^{i\theta_{1}}$,
$e_{2}=re^{i\theta_{2}}$, $e_{3}=re^{-i\theta_{1}}$, and
$e_{4}=re^{-i\theta_{2}}$ is associated with the Rabi problem for some
$\Delta,E,g^{2}\in\mathbb{R}$. This unique $\theta_{2}$ does not depend on $r$
and it is given by the following equation:

(5.26) $\theta_{2}=\pi-\arccos(\frac{1}{3}\cos\theta_{1}).$

Furthermore, the Rabi parameters $g$, $E$, and $\Delta$ corresponding to the
given values $r$ and $\theta_{1}$ are defined by equations

(5.27) $g^{-2}=-4r\alpha,\quad
E=\frac{1}{4r\alpha}(r^{2}-2r\alpha+2),\quad\Delta^{2}=\frac{r^{2}+1-2r^{2}\alpha^{2}}{4r^{2}\alpha^{2}},$

where $\alpha=\frac{1}{3}\cos\theta_{1}$.

_Proof._ Suppose that $Q_{0}(z)\,dz^{2}$ has zeros $e_{1}=re^{i\theta_{1}}$,
$e_{2}=re^{i\theta_{2}}$, $e_{3}=re^{-i\theta_{1}}$, and
$e_{4}=re^{-i\theta_{2}}$. Then, the numerator of this quadratic differential
has the form

(5.28) $P_{0}(z)=z^{4}-4r\delta
z^{3}+2r^{2}(1+2(\delta^{2}-\beta^{2}))z^{2}-4r^{3}\delta z+r^{4},$

where

(5.29)
$\delta=(1/2)(\cos\theta_{1}+\cos\theta_{2}),\quad\quad\beta=(1/2)(\cos\theta_{1}-\cos\theta_{2}).$

Thus, in this case $c_{3}=g^{-2}=-4r\delta$. Identifying the coefficients
$c_{2}$, $c_{1}$, and $c_{0}$ with the coordinates $X$, $Y$, and $Z$ of
$\mathbb{R}^{3}$, we obtain the following relations:

(5.30) $X=2r^{2}(1+2(\delta^{2}-\beta^{2})),\quad Y=-4r^{3}\delta,\quad
Z=r^{4}.$

Equating these expressions for $X$, $Y$, and $Z$ to the corresponding
expressions in formulas (5.6) and (5.7), and then solving the resulting
equations for $E$, $\Delta^{2}=c/(4\delta^{2})$, and $\beta^{2}$, we obtain

(5.31)
$E=\frac{1}{4r\delta}(r^{2}-2r\delta+2),\quad\Delta^{2}=\frac{r^{2}+1-2r^{2}\delta^{2}}{4r^{2}\delta^{2}},$

and

$\beta^{2}=4\delta^{2}.$

The latter equation together with (5.29) leads to the following quadratic
equation for the quotient $y=\frac{\cos\theta_{2}}{\cos\theta_{1}}$:

$3y^{2}+10y+3=0,$

which solutions are $y=-\frac{1}{3}$ and $y=-3$. The solutions correspond to
two configurations, which are mirror configurations to each other. Thus,
without loss of generality, we assume that $y=-\frac{1}{3}$. Then,
$\cos\theta_{2}=-\frac{1}{3}\cos\theta_{1}$, which gives (5.26).

Furthermore, substituting $-\frac{1}{3}\cos\theta_{1}$ for $\cos\theta_{2}$
into the first equation in (5.29), we find that
$\delta=(1/2)(\cos\theta_{1}+\cos\theta_{2})=\frac{1}{3}\cos\theta_{1}=\alpha$.
This, together with the relation $g^{-2}=-4r\delta$ and equations (5.31),
gives equations (5.27).

It remains to verify that the right-hand side in the third equation in (5.27)
is non-negative. The later is immediate from the following obvious inequality:
$\alpha^{2}=\frac{1}{9}\cos^{2}\theta_{1}<\frac{1}{2}+\frac{1}{2r^{2}}$.
$\Box$

For the quadratic differential $Q_{0}(z)\,dz^{2}$ with zeros on two rays
issuing from the origin, which are symmetric to each other with respect to the
real axis, we have the following result.

######  Lemma 5.13.

Suppose that the quadratic differential $Q_{0}(z)\,dz^{2}$ with complex zeros
$e_{1}=r_{1}e^{i\theta_{1}}$, $e_{2}=r_{2}e^{i\theta_{2}}$,
$e_{3}=r_{1}e^{-i\theta_{1}}$, $e_{4}=r_{2}e^{-i\theta_{2}}$, such that
$r_{k}>0$, $0<\theta_{k}<\pi$, $k=1,2$, is associated with the Rabi problem
for some values of the parameters $\Delta,E,g^{2}\in\mathbb{R}$. Then
$\theta_{1}\not=\theta_{2}$.

_Proof._ Suppose, by contradictions, that $0<\theta_{1}=\theta_{2}=\theta<\pi$
for some choice of $\Delta,E,g^{2}\in\mathbb{R}$. We may assume without loss
of generality that the zeros are $e_{1}=r_{1}e^{i\theta}$,
$e_{2}=r_{2}e^{i\theta}$, $e_{3}=r_{1}e^{-i\theta}$, $e_{4}=r_{2}e^{-i\theta}$
with $0<\theta<\pi/2$. Then, the polynomial $P_{0}(z)$ has the form:

$P_{0}(z)=z^{4}-4\delta t\,z^{3}+2(2\delta^{2}-\beta+2\beta
t^{2})z^{2}-4\delta\beta t\,z+\beta^{2},$

where

(5.32) $\delta=(1/2)(r_{1}+r_{2}),\quad\beta=r_{1}r_{2},\quad t=\cos\theta.$

As before, we use the coordinates $X$, $Y$, and $Z$ to denote the coefficients
$c_{2}$, $c_{1}$, and $c_{0}$ of $P_{0}(z)$. Thus,

$X=2(2\delta^{2}-\beta+2\beta t^{2}),\quad Y=-4\delta\beta,\quad Z=\beta^{2}.$

Equating these expressions for $X$, $Y$, and $Z$ to the corresponding
expressions for the coordinates $X$, $Y$, $Z$ in formulas (5.6) and (5.7), one
can solve the resulting equations for $E$, $c$, and $\beta$. We only need the
following resulting expression for $\beta$:

$\beta=\frac{1+3t^{2}}{1-t^{2}}\,\delta^{2}.$

Using this equations and relations (5.32), we obtain the following quadratic
equation for the ratio $y=r_{1}/r_{2}$:

$(1+3t^{2})y^{2}+2(5t^{2}-1)y+(1+3t^{2})=0.$

We recall that $t=\cos\theta$ and therefore the discriminant
$\bigtriangledown=16t^{2}(t^{2}-1)$ of the latter equation is negative.
Therefore, the ratio $r_{1}/r_{2}$ is not real contradicting our assumption
that $r_{k}>0$, $k=1,2$. This proves the lemma. $\Box$

##  6\. Asymptotic behavior for rescaled Rabi problem

In this section, we describe possible limit cases of the quadratic
differential $Q_{0}(z)dz^{2}$, when the boson-fermion coupling $g$ grows
without bounds; i.e. when $|g|\to\infty$. To guarantee the existence of the
limit quadratic differential, we impose the following conditions on the level
of separation of the fermion mode $\Delta$ and on the eigenvalue $E$ of the
Hamiltonian:

(6.1) $E/g^{2}\to
E_{a},\quad\Delta^{2}/g^{4}\to\Delta_{a}^{2}\quad\quad{\mbox{as
$|g|\to\infty$.}}$

Here, $E_{a}\in\mathbb{R}$, $\Delta_{a}\geq 0$ and the subscript “$a$” stands
for “asymptotic”.

Under these conditions, the polynomial $P_{0}(z)$ defined by equations (2.8) -
(2.10) reduces to the biquadratic polynomial $P_{a}(z)=z^{4}+c_{2}z^{2}+c_{0}$
with the coefficients

(6.2) $c_{2}=2E_{a}+\Delta^{2}_{a},\quad c_{0}=E_{a}^{2}-\Delta^{2}_{a},$

which zeros can be calculated as follows:

$\displaystyle e_{1}$ $\displaystyle=$
$\displaystyle\sqrt{-\frac{1}{2}\,\Delta_{a}^{2}-E_{a}-\frac{\Delta_{a}}{2}\,\sqrt{\Delta_{a}^{2}+4\,E_{a}+4}},$
$\displaystyle e_{2}$ $\displaystyle=$
$\displaystyle\sqrt{-\frac{1}{2}\,\Delta_{a}^{2}-E_{a}+\frac{\Delta_{a}}{2}\,\sqrt{\Delta_{a}^{2}+4\,E_{a}+4}},$
$\displaystyle e_{3}$ $\displaystyle=-$
$\displaystyle\sqrt{-\frac{1}{2}\,\Delta_{a}^{2}-E_{a}+\frac{\Delta_{a}}{2}\,\sqrt{\Delta_{a}^{2}+4\,E_{a}+4}},$
$\displaystyle e_{4}$ $\displaystyle=-$
$\displaystyle\sqrt{-\frac{1}{2}\,\Delta_{a}^{2}-E_{a}-\frac{\Delta_{a}}{2}\,\sqrt{\Delta_{a}^{2}+4\,E_{a}+4}}.$

Accordingly, the quadratic differential $Q_{0}(z)\,dz^{2}$ reduces to the
quadratic differential

(6.3)
$Q_{a}(z)dz^{2}=-\frac{z^{4}+c_{2}z^{2}+c_{0}}{(z-1)^{2}(z+1)^{2}}\,dz^{2}=-\frac{(z-e_{1})(z-e_{2})(z-e_{3})(z-e_{4})}{(z-1)^{2}(z+1)^{2}}\,dz^{2}.$

We stress here that the expressions for the coefficients $c_{2}$, $c_{0}$ in
(6.2) and the numeration of zeros $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$ in this
section may differ from those used in Section 4. Notice that the coefficients
of $Q_{a}(z)\,dz^{2}$ are real and its set of zeros is symmetric with respect
to both real and imaginary axis and therefore its Stokes graph and domain
configuration are also symmetric with respect to both axis. Since the zeros of
$Q_{a}(z)\,dz^{2}$ are given explicitly, the type of the associated Stokes
graph can be easily identified in terms of the asymptotic parameters $E_{a}$
and $\Delta_{a}$. Precisely, the type of the Stokes graph of
$Q_{a}(z)\,dz^{2}$ is determined by the numbers of real and complex zeros of
$Q_{a}(z)\,dz^{2}$ and therefore it is determined by the combination of signs
of the expressions inside of the radicals in the formulas for $e_{1}$,
$e_{2}$, $e_{3}$, $e_{4}$ given above; i.e. the type depends on the functions

$\Delta_{a}^{2}+4E_{a}+4\quad{\mbox{and}}\quad-\frac{1}{2}\Delta_{a}^{2}-E_{a}\pm\frac{\Delta_{a}}{2}\sqrt{\Delta_{a}^{2}+4E_{a}+4}.$

Next, we describe the sets of the parameters $E_{a}$ and $\Delta_{a}$, which
correspond to the types of Stokes graphs and domain configurations introduced
in Section 4. Our designation of possible cases here is the same as in Section
4.

I-1. The quadratic differential $Q_{a}(z)\,dz^{2}$ has four distinct pure
imaginary zeros if and only if the following conditions hold:

$\Delta_{a}>0,\quad\Delta_{a}^{2}+4E_{a}+4>0,\quad-\frac{1}{2}\Delta_{a}^{2}-E_{a}+\frac{\Delta_{a}}{2}\sqrt{\Delta_{a}^{2}+4E_{a}+4}<0.$

Using these relations, we conclude, after simple algebra, that this case
occurs if and only if $(\Delta_{a},E_{a})\in\mathcal{I}_{4}$, (here
$\mathcal{I}_{4}$ stands for “four distinct pure imaginary zeros), where the
set $\mathcal{I}_{4}\subset\mathbb{R}^{2}$ is defined as the following union:

$\mathcal{I}_{4}=\\{(X,Y):\,Y>X>0\\}\cup\\{(X,Y):\,X>2,\,-1-\frac{1}{4}X^{2}<Y<-X\\},$

see Figure 2, where the set $\mathcal{I}_{4}$ is shown in the yellow color.

Under these conditions,
$\operatorname{Im}e_{1}>\operatorname{Im}e_{2}>0>\operatorname{Im}e_{3}>\operatorname{Im}e_{4}$.
Then the intervals $(-i\infty,e_{4})$, $(e_{3},e_{2})$, $(e_{1},i\infty)$ of
the imaginary axis are critical trajectories of $Q_{a}(z)\,dz^{2}$ and the
intervals $(e_{4},e_{3})$ and $(e_{2},e_{1})$ are critical orthogonal
trajectories of $Q_{a}(z)\,dz^{2}$. This implies that the domain configuration
of $Q_{a}(z)\,dz^{2}$ has a ring domain $\Omega_{r}$. Therefore, in this case
the Stokes graph and domain configuration belong to the type described in part
I-1 of Section 4, see Figure 2.

I-2. The quadratic differential $Q_{a}(z)\,dz^{2}$ has four distinct complex
zeros, each having non-zero real and imaginary parts, if and only if

$\Delta_{a}>0,\quad\Delta_{a}^{2}+4E_{a}+4<0$

or, equivalently, if and only if $(\Delta_{a},E_{a})\in\mathcal{C}_{4}$, where
$\mathcal{C}_{4}$ stands for “four complex zeros” and the set
$\mathcal{C}_{4}\subset\mathbb{R}^{2}$ is defined as
$\mathcal{C}_{4}=\\{(X,Y):\,X\not=0,\,Y<-\frac{1}{4}X^{2}-1\\}$; see Figure 2,
where the set $\mathcal{C}_{4}$ is shown in the blue color.

In this case, the imaginary axis is a trajectory of $Q_{a}(z)\,dz^{2}$. This
implies that the domain configuration has a strip domain
$\Omega_{s}(-i\infty,i\infty)$. Therefore, in this case the Stokes graph and
domain configuration belong to the type described in part I-2 of Section 4,
see Figure 2.

In the case I-3 discussed in Section 4, the Stokes graph is not symmetric with
respect to the imaginary axis and therefore this type of graphs do not appear
in the asymptotic cases considered in this section. Also, the case I-4 of
Section 4 is a degenerate case when $Q_{0}(z)\,dz^{2}$ has a multiple zero.
These cases will be considered later in this section.

Figure 2. Sets $\mathcal{I}_{4}$, $\mathcal{C}_{4}$, $\mathcal{IR}$ and
$\mathcal{R}_{4}$ with different types of Stokes graphs of $Q_{a}(z)\,dz^{2}$.

II. The polynomial $P_{a}(z)$ has four distinct zeros, two real and two pure
imaginary, if and only if

(6.4)
$\Delta_{a}>0,\quad\Delta_{a}^{2}+4E_{a}+4>0,\quad\frac{\Delta_{a}}{2}\sqrt{\Delta_{a}^{2}+4E_{a}+4}>\left|\frac{1}{2}\Delta_{a}^{2}+E_{a}\right|.$

Notice that these inequalities hold if and only if $0<|E_{a}|<\Delta_{a}$ and
therefore this case occurs if and only if
$(\Delta_{a},E_{a})\in\mathcal{I}\mathcal{R}$, where the set
$\mathcal{I}\mathcal{R}\subset\mathbb{R}^{2}$ is defined as
$\mathcal{I}\mathcal{R}=\\{(X,Y):\,0<|Y|<X\\}$; see Figure 2, where the set
$\mathcal{IR}$ is shown in the pink color. Furthermore, using inequalities
(6.4) and elementary algebra one can show that $e_{2}$ and $e_{3}$ defined
earlier are real zeros such that $-1\leq e_{3}<0<e_{2}\leq 1$. In the case
when $e_{2}=-e_{3}=1$, which occurs if and only if $E_{a}=-1$,
$Q_{a}(z),dz^{2}$ reduces to the depressed quadratics differential
$Q_{s}(z)\,dz^{2}=-\frac{z^{2}-c_{0}}{(z-1)(z+1)}\,dz^{2}$ with
$c_{0}==1-\Delta_{a}^{2}$, which has the points simple poles at $\pm 1$. In
the case under consideration, the Stokes graphs as in the subcases II-1, II-3
and II-4 of Section 4 do not appear as an asymptotic case.

Also, the subcases II-1 and II-3 can be excluded because the Stokes graphs in
these cases are not symmetric with respect to the imaginary axis and therefore
these type of graphs do not appear as asymptotic cases.

Thus, the only possible types of Stokes graphs and domain configurations in
the case under consideration are those discussed in part II-2 of Section 4.

In this case, the intervals $(-i\infty,e_{4})$ and $(e_{1},i\infty)$ of the
imaginary axis and the interval $(e_{3},e_{2})$ of the real axis are critical
trajectories of $Q_{a}(z)\,dz^{2}$ and the interval $(e_{4},e_{1})$ is a
critical orthogonal trajectory. This implies that the domain configuration has
a ring domain $\Omega_{r}$. Therefore, in this case the Stokes graph and
domain configuration belong to the type described in part II-2-a of Section 4,
as shown in Figure 2, and the cases II-2-b and II-2-c do not occur as
asymptotic cases.

III. The polynomial $P_{a}(z)$ has four distinct real zeros if and only if

(6.5)
$\Delta_{a}>0,\quad\Delta_{a}^{2}+4E_{a}+4>0,\quad-\frac{1}{2}\Delta_{a}^{2}-E_{a}-\frac{\Delta_{a}}{2}\sqrt{\Delta_{a}^{2}+4E_{a}+4}>0.$

We perform algebraic operations to find that inequalities (6.5) hold if and
only if $(\Delta_{a},E_{a})\in\mathcal{R}_{4}$, where the set
$\mathcal{R}_{4}\subset\mathbb{R}^{2}$ is defined as
$\mathcal{R}_{4}=\\{(X,Y):\,0<X<2,\,-\frac{1}{4}X^{2}-1<Y<-X\\}$; see Figure
2, where the set $\mathcal{R}_{4}$ is shown in the green color. Furthermore,
using inequalities (6.5) and elementary algebra one can show that $e_{1}$,
$e_{2}$, $e_{3}$, and $e_{4}$ defined above are real zeros such that $-1\leq
e_{3}<e_{4}<0<e_{1}<e_{2}\leq 1$. As in the previous case, if
$e_{2}=-e_{3}=1$, then $Q_{a}(z),dz^{2}$ reduces to the depressed quadratics
differential $Q_{s}(z)\,dz^{2}$ with simple poles at the points $\pm 1$. This
implies that, if $e_{3}=-e_{2}<1$, then only Stokes graphs described in the
subcase III-2 of Section 4 and shown in Figure 2 appear as an asymptotic case
while all other subcases described in part III of Section 4 do not appear as
asymptotic cases.

Turning to the depressed and degenerate cases, we first mention that if
$(\Delta_{a},E_{a})\in\mathcal{L}(-1)$, where $L(-1)=\\{(X,Y):\,X\geq
0,\,Y=-1\\}$ is the dash line shown in Figure 2, then $P_{a}(z)$ has zeros at
the points $\pm 1$ and therefore $Q_{a}(z)\,dz^{2}$ reduces to the depressed
quadratic differential $Q_{s}(z)\,dz^{2}$ defined above. Thus, if
$\Delta_{a}>1$, then $Q_{s}(z)\,dz^{2}$ has two pure imaginary zeros and the
Stokes graph as in Figure 2. If $\Delta_{a}=1$, then $Q_{s}(z)\,dz^{2}$ has a
double zero at $z=0$ and the Stokes graph as in Figure 2. If $0<\Delta_{a}<1$,
then $Q_{s}(z)\,dz^{2}$ has two real zeros and the Stokes graph as in Figure
2. Finally, if $\Delta_{a}=0$, then $Q_{s}(z)\,dz^{2}=-dz^{2}$ and therefore
its Stokes graph is empty and the vertical lines are the trajectories of
$Q_{s}(z)\,dz^{2}$ in this case.

Next, we mention that if the point $(\Delta_{2},E_{a})$ lies on the half-
parabola $L_{1}=\\{(X,Y):\,Y=-\frac{1}{4}X^{2}-1,\,X>0\\}$ or on one of the
half-lines $L_{2}=\\{(X,Y):\,Y=X,\,X>0\\}$, $L_{3}=\\{(X,Y):\,Y=-X,\,X>0\\}$,
$L_{4}\\{(0,Y):\,Y>-1\\}$, $L_{5}=\\{(0,Y):\,Y<-1\\}$ (all of them are shown
in red color, except four black points, in Figure 2), then $Q_{a}(z)\,dz^{2}$
has two double zeros if $(\Delta_{1},E_{a})\not=(0,0)$ and
$(\Delta_{1},E_{a})\not=(2,-2)$ and it has zero of order $4$ at $z=0$ when
$(\Delta_{1},E_{a})=(0,0)$ and when $(\Delta_{1},E_{a})=(2,-2)$. If
$(\Delta_{a},E_{a})\in L_{k}$, $k=1,\ldots,5$, then the corresponding
quadratic differential $Q_{a}(z)\,dz^{2}$ has the Stokes graph of the type
shown in Figures 2–6, respectively.

##  Appendix A List of notation

  * •

$\Delta$ \- level of separation of the fermion mode in the Rabi problem.

  * •

$g$ \- boson-fermion coupling in the Rabi problem.

  * •

$E$ \- eigenvalue of the Hamiltonian in the Rabi problem.

  * •

$\mathbb{C}$ \- complex plane.

  * •

$\overline{\mathbb{C}}$ \- Riemann sphere.

  * •

$\mathbb{H}_{+}$ \- upper half-plane.

  * •

$\mathbb{H}_{-}$ \- lower half-plane.

  * •

$(a,b)$ \- open interval from $a$ to $b$.

  * •

$[a,b]$ \- closed interval from $a$ to $b$.

  * •

$Q(z)\,dz^{2}$ \- general notation for a quadratic differential.

  * •

$G_{Q}$ \- Stokes graph of $Q(z)\,dz^{2}$.

  * •

$[a,b]_{Q}$ \- integral $\int_{a}^{b}\sqrt{Q(z)}\,dz$ taken over the interval
$[a,b]$.

  * •

$Q_{0}(z)\,dz^{2}$ \- quadratic differential
$-\frac{z^{4}+c_{3}z^{3}+c_{2}z^{2}+c_{1}z+c_{0}}{(z-1)^{2}(z+1)^{2}}\,dz^{2}$.

  * •

$P_{0}(z)=z^{4}+c_{3}z^{3}+c_{2}z^{2}+c_{1}z+c_{0}$ \- numerator of
$Q_{0}(z)\,dz^{2}$.

  * •

$c_{k}$, $k=0,1,2,3$ \- coefficients of $P_{0}(z)$.

  * •

$e_{k}$, $k=1,2,3,4$ \- zeros of $P_{0}(z)$.

  * •

$e_{j,k}$ \- double zero of $P_{0}(z)$ obtained by merging zeros $e_{j}$ and
$e_{k}$.

  * •

$\mathbf{\Omega}=\mathbf{\Omega}(\Delta,E,g)$ \- domain configuration of
$Q_{0}(z)\,dz^{2}$ associated with the Rabi parameters $\Delta$, $E$, and $g$.

  * •

$\delta_{k}$ \- $Q_{0}$-length of a trajectory/orthogonal trajectory around
$k=-1,1$.

  * •

$\gamma_{a,b}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
oriented from $a$ to $b$.

  * •

$\gamma_{a,b}^{l}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $b$ intersecting $(-\infty,-1)$.

  * •

$\gamma_{a,b}^{c}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $b$ intersecting $(-1,1)$.

  * •

$\gamma_{a,b}^{r}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $b$ intersecting $(1,\infty)$.

  * •

$\gamma_{a}^{l}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $a$ intersecting $(-\infty,-1)$, anticlockwise oriented.

  * •

$\gamma_{a}^{c}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $a$ intersecting $(-1,1)$, anticlockwise oriented.

  * •

$\gamma_{a}^{r}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $a$ intersecting $(1,\infty)$, anticlockwise oriented.

  * •

$\gamma_{a}^{l-}$ \- closed curve $\gamma_{a}^{l}$ with reversed orientation.

  * •

$\gamma_{a}^{c-}$ \- closed curve $\gamma_{a}^{c}$ with reversed orientation.

  * •

$\gamma_{a}^{r-}$ \- closed curve $\gamma_{a}^{r}$ with reversed orientation.

  * •

$\gamma_{a,i\infty}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
starting at $a$ and approaching $\infty$ along positive direction of some
vertical line.

  * •

$\gamma_{a,-i\infty}$ \- closure of a critical trajectory of
$Q_{0}(z)\,dz^{2}$ starting at $a$ and approaching $\infty$ along negative
direction of some vertical line.

  * •

$\gamma_{-i\infty,i\infty}$ \- closure of a critical trajectory of
$Q_{0}(z)\,dz^{2}$ from $\infty$ to $\infty$, which approaches its initial
point along negative direction of some vertical line and approached its
terminal point along positive direction of the same vertical line.

  * •

$\gamma_{i\infty,a}$ \- arc $\gamma_{a,i\infty}$ with reversed orientation.

  * •

$\gamma_{a,-i\infty}$ \- arc $\gamma_{-i\infty,a}$ with reversed orientation.

  * •

$\gamma_{i\infty,-i\infty}$ \- closed curve $\gamma_{-i\infty,i\infty}$ with
reversed orientation.

  * •

$\gamma_{a,b}^{+}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $b$, $a,b\in\mathbb{R}\cup\\{\infty\\}$ lying in
$\overline{\mathbb{H}}_{+}$.

  * •

$\gamma_{a,b}^{-}$ \- closure of a critical trajectory of $Q_{0}(z)\,dz^{2}$
from $a$ to $b$, $a,b\in\mathbb{R}\cup\\{\infty\\}$ lying in
$\overline{\mathbb{H}}_{-}$.

  * •

$\Omega_{e}^{l}$ \- left end domain of $Q_{0}(z)\,dz^{2}$.

  * •

$\Gamma_{e}^{l}$ \- boundary of $\Omega_{e}^{l}$ positively oriented.

  * •

$\Omega_{e}^{r}$ \- right end domain of $Q_{0}(z)\,dz^{2}$.

  * •

$\Gamma_{e}^{r}$ \- boundary of $\Omega_{e}^{r}$ positively oriented.

  * •

$\Omega_{c}(k)$, $k=-1,1$ \- circle domain of $Q_{0}(z)\,dz^{2}$ centered at
$z=k$.

  * •

$\Gamma_{c}(k)$, $k=-1,1$ \- boundary of $\Omega_{c}(k)$ positively oriented.

  * •

$\Omega_{r}$ \- ring domain of $Q_{0}(z)\,dz^{2}$.

  * •

$\Gamma_{r}^{(out)}$ \- outer boundary component of $\Omega_{r}$ oriented
counterclockwise.

  * •

$\Gamma_{r}^{(inn)}$ \- inner boundary component of $\Omega_{r}$ oriented
counterclockwise.

  * •

$\Omega_{s}(a,b)$ \- strip domain of $Q_{0}(z)\,dz^{2}$ with vertices $a$ and
$b$.

  * •

$\Gamma_{s}^{l}(a,b)$ with $a=-i\infty$ and/or $b=i\infty$ \- left side of
$\Omega_{s}(a,b)$.

  * •

$\Gamma_{s}^{r}(a,b)$ with $a=-i\infty$ and/or $b=i\infty$ \- right side of
$\Omega_{s}(a,b)$.

  * •

$\Gamma_{s}^{(out)}(a,a)$ with $a=-1$ or $a=1$ \- outer side of
$\Omega_{s}(a,a)$.

  * •

$\Gamma_{s}^{(inn)}(a,a)$ with $a=-1$ or $a=1$ \- inner side of
$\Omega_{s}(a,a)$.

  * •

$\Gamma_{s}^{+}(-1,1)$ \- side of $\Omega_{s}(-1,1)$ lying in the upper half-
plane.

  * •

$\Gamma_{s}^{-}(-1,1)$ \- side of $\Omega_{s}(-1,1)$ lying in the lower half-
plane.

  * •

$S(a)$ \- parabolic cylinder
$\\{(X,Y,Z)\in\mathbb{R}^{3}:\,(Y+a)^{2}-a^{2}X-a^{2}Z-(1/4)a^{2}(4+3a^{2})=0\\}$.

  * •

$Q_{a}(z)\,dz^{2}$ \- asymptotic quadratic differential
$-\frac{z^{4}+c_{2}z^{2}+c_{0}}{(z-1)^{2}(z+1)^{2}}\,dz^{2}$.

##  Appendix B Zoo of Stokes graphs.

$e_{1}$$e_{3}$$e_{2}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$
$\Omega_{r}$$\gamma_{e_{2},e_{4}}^{l}$$\gamma_{e_{2},e_{4}}^{r}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{1},e_{3}}^{l}$$\gamma_{e_{1},e_{3}}^{r}$$\gamma_{e_{1},e_{3}}^{c}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\Omega_{s}(-i\infty,i\infty)$$\gamma_{e_{1},e_{3}}^{l}$$\gamma_{e_{1},e_{3}}^{c}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},e_{4}}^{c}$$\gamma_{e_{2},e_{4}}^{r}$$\gamma_{e_{4},-i\infty}$

Figure 3. Case I-1 and case I-2.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$ $\gamma_{e_{1},e_{3}}^{l}$
$\gamma_{e_{2},e_{4}}^{c}$$\gamma_{e_{2},e_{4}}^{r}$ $\gamma_{e_{1},e_{2}}$
$\gamma_{e_{3},e_{4}}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{1},i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$
$\Omega_{c}(-1)$ $\gamma_{e_{1},e_{3}}^{r}$
$\gamma_{e_{2},e_{4}}^{c}$$\gamma_{e_{2},e_{4}}^{l}$ $\gamma_{e_{1},e_{2}}$
$\gamma_{e_{3},e_{4}}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{1},i\infty}$

Figure 4. Case I-3 and case I-3-m
$e_{1,2}$$e_{3,4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$
$\gamma_{e_{1,2},e_{3,4}}^{l}$$\gamma_{e_{1,2},e_{3,4}}^{c}$$\gamma_{e_{1,2},e_{3,4}}^{r}$$\gamma_{e_{3,4},-i\infty}$$\gamma_{e_{1,2},i\infty}$
Figure 5. Case I-3-deg.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$ $\Omega_{s}(-i\infty,i\infty)$
$\gamma_{e_{2}}^{c}$$\gamma_{e_{3},e_{4}}^{r}$
$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$
$\Omega_{c}(-1)$ $\Omega_{s}(-i\infty,i\infty)$
$\gamma_{e_{2}}^{c}$$\gamma_{e_{3},e_{4}}^{l}$
$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 6. Case II-1-a and case II-1-a-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$ $\Omega_{r}$$\gamma_{e_{3},e_{4}}^{l}$
$\gamma_{e_{3},e_{4}}^{r}$ $\gamma_{e_{3},e_{4}}^{c}$
$\gamma_{e_{2}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$
$\Omega_{c}(-1)$ $\Omega_{r}$$\gamma_{e_{3},e_{4}}^{r}$
$\gamma_{e_{3},e_{4}}^{l}$ $\gamma_{e_{3},e_{4}}^{c}$
$\gamma_{e_{2}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 7. Case II-1-b and case II-1-b-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$
$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2},e_{3}}^{+}$
$\gamma_{e_{2},e_{4}}^{-}$ $\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$
$\Omega_{c}(-1)$
$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2},e_{3}}^{+}$
$\gamma_{e_{2},e_{4}}^{-}$ $\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 8. Case II-1-c and case II-1-c-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\Omega_{r}$$\gamma_{e_{3},e_{4}}^{l}$
$\gamma_{e_{2}}^{c}$ $\gamma_{e_{3},e_{4}}^{r}$ $\gamma_{e_{1}}^{r}$
$\gamma_{e_{4},-i\infty}$$\gamma_{e_{3},i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\Omega_{r}$$\gamma_{e_{3},e_{4}}^{r}$
$\gamma_{e_{2}}^{c}$ $\gamma_{e_{3},e_{4}}^{l}$ $\gamma_{e_{1}}^{l}$
$\gamma_{e_{4},-i\infty}$$\gamma_{e_{3},i\infty}$

Figure 9. Case II-1-d and case II-1-d-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{2}}^{c}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{2}}^{c}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 10. Case II-1-e and case II-1-e-m
$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$
$\Omega_{r}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{1}}^{l}$
$\gamma_{e_{2}}^{r}$ $\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$ Figure
11. Case II-2-a.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$
$\Omega_{c}(1)$ $\Omega_{s}(-i\infty,i\infty)$
$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$
$\Omega_{c}(-1)$ $\Omega_{s}(-i\infty,i\infty)$
$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 12. Case II-2-b and case II-2-b-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{3},i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{3},i\infty}$

Figure 13. Case II-2-c and case II-2-c-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,-1)$
$\Omega_{c}(1)$ $\Omega_{s}(-1,i\infty)$ $\Omega_{s}(-1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2}}^{r}$$\gamma_{-1,e_{3}}^{+}$$\gamma_{-1,e_{4}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$
$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,-1)$
$\Omega_{c}(-1)$ $\Omega_{s}(1,i\infty)$ $\Omega_{s}(1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2}}^{l}$$\gamma_{1,e_{3}}^{+}$$\gamma_{1,e_{4}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$
$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 14. Case II-3-a-$\alpha$ and case II-3-a-$\alpha$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$
$\Omega_{c}(1)$ $\Omega_{s}(-1,i\infty)$ $\Omega_{s}(-1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-i\infty,i\infty)$
$\Omega_{c}(-1)$ $\Omega_{s}(1,i\infty)$ $\Omega_{s}(1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},e_{4}}^{c}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$

Figure 15. Case II-3-a-$\beta$ and case II-3-a-$\beta$-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$
$\Omega_{s}(-1,i\infty)$ $\Omega_{s}(-1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2},e_{3}}^{+}$
$\gamma_{e_{2},e_{4}}^{-}$
$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(-1)$
$\Omega_{s}(1,i\infty)$ $\Omega_{s}(1,-i\infty)$
$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2},e_{3}}^{+}$
$\gamma_{e_{2},e_{4}}^{-}$
$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 16. Case II-3-a-$\gamma$ and case II-3-a-$\gamma$-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$
$\Omega_{s}(-1,-1)$$\Omega_{r}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$ $e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(-1)$
$\Omega_{s}(-1,-1)$$\Omega_{r}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 17. Case II-3-b and case II-3-b-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$
$\Omega_{s}(-1,-1)$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(-1)$
$\Omega_{s}(1,1)$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 18. Case II-3-c and case II-3-c-m
$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,-i\infty)$
$\Omega_{s}(1,-i\infty)$ $\Omega_{s}(-1,i\infty)$ $\Omega_{s}(1,i\infty)$
$\Omega_{s}(-1,1)$
$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{3},1}^{+}$$\gamma_{e_{4},-1}^{-}$$\gamma_{e_{4},1}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$
Figure 19. Case II-4-a-$\alpha$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,i\infty)$
$\Omega_{s}(-1,-i\infty)$$\Omega_{s}(-1,-1)$ $\Omega_{s}(-1,1)$
$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{4},-1}^{-}$$\gamma_{e_{2},-1}^{+}$$\gamma_{e_{2},-1}^{-}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(1,i\infty)$
$\Omega_{s}(1,-i\infty)$$\Omega_{s}(1,1)$ $\Omega_{s}(-1,1)$
$\gamma_{e_{3},1}^{+}$$\gamma_{e_{4},1}^{-}$$\gamma_{e_{2},1}^{+}$$\gamma_{e_{2},1}^{-}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 20. Case II-4-a-$\beta$ and case II-4-a-$\beta$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{s}(-1,1)$$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{4},-1}^{-}$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{s}(-1,1)$$\gamma_{e_{3},1}^{+}$$\gamma_{e_{4},1}^{-}$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 21. Case II-4-a-$\gamma$ and case II-4-a-$\gamma$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}(-1,-1)$
$\Omega_{s}(-1,1)$ $\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{2},1}^{+}$
$\gamma_{e_{2},1}^{-}$
$\gamma_{e_{1}}^{r}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{s}(1,1)$
$\Omega_{s}(-1,1)$ $\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{2},-1}^{+}$
$\gamma_{e_{2},-1}^{-}$
$\gamma_{e_{1}}^{l}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 22. Case II-4-b-$\alpha$ and case II-4-b-$\alpha$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,-1)$
$\Omega_{s}(-1,1)$
$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{-1,e_{2}}^{+}$$\gamma_{-1,e_{2}}^{-}$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(1,1)$
$\Omega_{s}(-1,1)$
$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{2},1}^{+}$$\gamma_{e_{2},1}^{-}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 23. Case II-4-c-$\alpha$ and case II-4-c-$\alpha$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}(-1,1)$$\gamma_{e_{3},e_{4}}^{r}$$\gamma_{e_{1},e_{2}}^{+}$$\gamma_{e_{1},e_{2}}^{-}$$\gamma_{e_{3},e_{4}}^{l}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,1)$$\gamma_{e_{1},e_{3}}^{+}$$\gamma_{e_{1},e_{4}}^{-}$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 24. Case II-4-b-$\beta$ and case II-4-c-$\beta$.

$e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$
$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{3}}^{r}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$
$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{3}}^{l}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 25. Case III-1 and case III-1-m
$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{1}}^{l}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$
Figure 26. Case III-2

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}^{-1}$$\Omega_{c}(1)$
$\gamma_{e_{2}}^{r}$$\gamma_{e_{3}}^{r}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{s}^{1}$$\Omega_{c}(-1)$
$\gamma_{e_{2}}^{l}$$\gamma_{e_{3}}^{l}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 27. Case III-3 and case III-3-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,1)$$\Omega_{s}(1,1)$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\gamma_{e_{2},1}^{+}$$\gamma_{e_{3},1}^{+}$$\gamma_{e_{2},1}^{-}$$\gamma_{e_{3},1}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,1)$$\Omega_{s}(-1,-1)$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\gamma_{e_{2},-1}^{+}$$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{2},-1}^{-}$$\gamma_{e_{3},-1}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 28. Case III-4-a-$\alpha$ and case III-4-a-$\alpha$-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}(1,1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{r}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{3},1}^{-}$$\gamma_{e_{3},1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{s}(-1,-1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{l}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{3},-1}^{-}$$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 29. Case III-4-a-$\beta$ and case III-4-a-$\beta$-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{3},1}^{-}$$\gamma_{e_{3},1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,-1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{3},-1}^{-}$$\gamma_{e_{3},-1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 30. Case III-4-a-$\gamma$ and case III-4-a-$\gamma$-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}(-1,-1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{r}$$\gamma_{e_{3}}^{r}$$\gamma_{e_{4},-1}^{-}$$\gamma_{e_{4},-1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{s}(1,1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{l}$$\gamma_{e_{3}}^{l}$$\gamma_{e_{4},1}^{-}$$\gamma_{e_{4},1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 31. Case III-4-b and case III-4-b-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{r}$$\gamma_{e_{3},e_{4}}^{-}$$\gamma_{e_{3},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{s}(-1,1)$$\gamma_{e_{2}}^{l}$$\gamma_{e_{3},e_{4}}^{-}$$\gamma_{e_{3},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 32. Case III-4-c and case III-4-c-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{c}(1)$$\gamma_{e_{4}}^{r}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{c}(-1)$$\gamma_{e_{4}}^{l}$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$

Figure 33. Case III-5 and case III-5-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$

Figure 34. Case III-6-a and case III-6-a-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{3}}^{l}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{3}}^{r}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 35. Case III-6-b and case III-6-b-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{3}}^{-}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{3}}^{-}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 36. Case III-6-c and case III-6-c-m
$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{2}}^{c}$$\gamma_{e_{3}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$
Figure 37. Case III-7-a.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{r}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{r}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 38. Case III-7-b and case III-7-b-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 39. Case III-7-c and case III-7-c-m

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{c}(-1)$$\Omega_{s}(-i\infty,i\infty)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{c}(1)$$\Omega_{s}(-i\infty,i\infty)$$\gamma_{e_{3}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 40. Case III-8-a and case III-8-a-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{c}(-1)$$\Omega_{s}(1,1)$$\gamma_{e_{2},1}^{+}$$\gamma_{e_{2},1}^{-}$$\gamma_{e_{3}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{c}(1)$$\Omega_{s}(-1,-1)$$\gamma_{e_{2},-1}^{+}$$\gamma_{e_{2},-1}^{-}$$\gamma_{e_{3}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 41. Case III-8-b-$\alpha$ and case III-8-b-$\alpha$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,1)$$\Omega_{c}(-1)$$\Omega_{r}$$\gamma_{e_{2}}^{r}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{3}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,-1)$$\Omega_{c}(1)$$\Omega_{r}$$\gamma_{e_{2}}^{l}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{3}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 42. Case III-8-b-$\beta$ and case III-8-b-$\beta$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,1)$$\Omega_{c}(-1)$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{3}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,-1)$$\Omega_{c}(1)$$\gamma_{e_{2},e_{4}}^{+}$$\gamma_{e_{2},e_{4}}^{-}$$\gamma_{e_{3}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 43. Case III-8-b-$\gamma$ and case III-8-b-$\gamma$-m.

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{c}(-1)$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{3}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{c}(1)$$\gamma_{e_{2},e_{3}}^{+}$$\gamma_{e_{2},e_{3}}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 44. Case III-8-c and III-8-c-m.
$e_{1}$$e_{2}$$e_{3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-i\infty,i\infty)$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\gamma_{e_{2},i\infty}$$\gamma_{e_{2},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{3},i\infty}$$\gamma_{e_{3},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$
Figure 45. Case III-9.

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(-1)$$\Omega_{c}(1)$$\gamma_{e_{2,3}}^{r}$$\gamma_{e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{2,3}}^{l}$$\gamma_{e_{4}}^{c}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 46. Case III-1-deg and case III-1-deg-m
$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{c}(1)$$\Omega_{c}(-1)$$\gamma_{e_{1}}^{l}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{2,3},i\infty}$$\gamma_{e_{2,3},-i\infty}$
Figure 47. Case III-2-deg

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,1)$$\Omega_{c}(1)$$\gamma_{e_{2,3}}^{r}$$\gamma_{e_{4}}^{r}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,1)$$\Omega_{c}(-1)$$\gamma_{e_{2,3}}^{l}$$\gamma_{e_{4}}^{l}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 48. Case III-3-deg and case III-3-deg-m

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,1)$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\gamma_{e_{2,3},1}^{+}$$\gamma_{e_{2,3},1}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,1)$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\gamma_{e_{2,3},-1}^{+}$$\gamma_{e_{2,3},-1}^{-}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$

Figure 49. Case III-4-a-$\alpha$-deg and case III-4-a-$\alpha$-deg-m

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,-1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2,3}}^{r}$$\gamma_{e_{4},-1}^{-}$$\gamma_{e_{4},-1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(1,1)$$\Omega_{s}(-1,1)$$\gamma_{e_{2,3}}^{l}$$\gamma_{e_{4},1}^{-}$$\gamma_{e_{4},1}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 50. Case III-4-b-deg and case III-4-b-deg-m

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,1)$$\gamma_{e_{2,3},e_{4}}^{-}$$\gamma_{e_{2,3},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(-1,1)$$\gamma_{e_{2,3},e_{4}}^{-}$$\gamma_{e_{2,3},e_{4}}^{+}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 51. Case III-4-c-deg and case III-4-c-deg-m

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{c}(1)$$\gamma_{e_{4}}^{r}$$\gamma_{e_{2,3},i\infty}$$\gamma_{e_{2,3},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{l}_{e}$$\Omega^{r}_{e}$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\Omega_{c}(-1)$$\gamma_{e_{4}}^{l}$$\gamma_{e_{2,3},i\infty}$$\gamma_{e_{2,3},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$

Figure 52. Case III-5-deg and case III-5-deg-m
$e_{1}$$e_{2,3}$$e_{4}$$\Omega^{r}_{e}$$\Omega^{l}_{e}$$\Omega_{s}(-1,i\infty)$$\Omega_{s}(-1,-i\infty)$$\Omega_{s}(1,i\infty)$$\Omega_{s}(1,-i\infty)$$\gamma_{e_{2,3},i\infty}$$\gamma_{e_{2,3},-i\infty}$$\gamma_{e_{1},i\infty}$$\gamma_{e_{1},-i\infty}$$\gamma_{e_{4},i\infty}$$\gamma_{e_{4},-i\infty}$
Figure 53. Case III-9-deg.

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