# Perturbation of discriminant for one-dimensional discrete Schrödinger
operator with sparse periodic potential

Masahiro Kaminaga

###### Abstract

We consider the one-dimensional discrete Schrödinger operator with complex-
valued sparse periodic potential. The spectrum for a complex-valued periodic
potential is a complicated compact set in the complex plane represented by
real intersections of algebraic curves determined by a discriminant. We
represent the discriminant by Chebyshev polynomials and use perturbations of
the discriminant to study the spectrum.

##  1 Introduction

In this paper, we consider the one-dimensional discrete Schrödinger operator:

$(Hu)(n)=u(n+1)+u(n-1)+V(n)u(n)\quad\mbox{on $l^{2}(\mathbb{Z})$},$

where $V(n)$ is a complex-valued periodic potential with period $L$ containing
only one nonzero value $v$ within a period. As the period $L$ increases, this
potential becomes sparse, which we call the sparse potential. If $v$ is real,
$H$ is a bounded self-adjoint operator. For real $v$, the spectrum $\sigma(H)$
of $H$ is purely absolutely continuous and consists of at most $L$ closed
intervals on the real axis. These results are proved in general dimension
using the method of direct integral decomposition (see, e.g., Reed-Simon [12])
following Gel’fand [5]. It describes the behavior of electrons or holes in a
one-dimensional crystal (e.g., Kittel [9]). Each closed interval of the
spectrum is called a spectral band. In condensed matter physics, it is
essential to determine the location of the bands, especially the band-to-band
gap (band gap). Estimating the band gap is applied to study the electrical
properties of semiconductors.

Even if $v$ is not real, $H$ is still a bounded operator. Thus, while its
spectrum $\sigma(H)$ is a compact set on the complex plane, its shape often
becomes more complicated. In the case of self-adjoint operators, a nice
theory, including the spectral decomposition theorem, can be applied. On the
other hand, for non-self-adjoint cases, no such nice general theory exists and
must be analyzed on a problem-by-problem basis. Because of this inconvenience,
the spectral theory for complex-valued potential literature has been
relatively few.

Non-self-adjoint Schrödinger-type operators (non-Hermitian Hamiltonian in
physics) have emerged naturally in $\mathcal{PT}$-symmetric (parity-time
symmetric) quantum theory (see, e.g., Bender [2]), providing a strong
incentive for their study. For differential operators with complex-valued
periodic potentials, Valiev [14] is an excellent guide for researchers in this
field. In the discrete (Jacobi matrix) case, a theory similar to that of
differential operators can be constructed. Moreover, the direct and inverse
spectral theory for more general Jacobi matrices with complex periodic
coefficients has also been obtained (Hochstadt [7], Papanicolaou [11]). An
example of the significant difference between complex-valued periodic
potentials and real-valued periodic potentials is given by Gasimov [4](see
also Guillemin and Uribe [6]). In continuous case, if $V(x)\in
L^{2}_{\text{\rm loc}}(\mathbb{R})$ is real-valued, a famous theorem of Borg
[3], that is, $\sigma(-\frac{d^{2}}{dx^{2}}+V)=[0,\infty)$ if and only if
$V(x)=0$ a.e.. In the case of complex-valued periodic potentials, the results
are very different: Gasimov [4] showed that if

$V(x)=\sum_{k=1}^{\infty}c_{k}e^{ikx},\quad\sum_{k=1}^{\infty}|c_{k}|<\infty,$

then $\sigma(-\frac{d^{2}}{dx^{2}}+V)=[0,\infty)$. Papanicolaou [11] showed a
discrete version of this Gasimov’s theorem.

The discriminant (defined in Section 2) determines the spectrum of the one-
dimensional Schrödinger operator with periodic potential. The discriminant is
a polynomial in $E$ with the period of the potential as its degree.
Papanicolaou [11] proved there exists at most $L!$ different $L$-periodic
potentials whose discriminants are the same. As a result, the spectra of these
operators coincide. Papanicolaou [11] used the fact that the coefficients of
the discriminant are represented by the elementary symmetric polynomials of
$V(1),V(2),\ldots,V(L)$ to show this result. This representation is suitable
for inverse problems but not for the perturbation theory of discriminant
because of the difficulty in obtaining information about the value of the
discriminant.

The most fundamental problem in the study of operators is determining the
operator’s spectrum. As described in Section 2, the problem of determining the
spectrum for a periodic potential with period $L$ is equivalent to the problem
of finding the intersection of two algebraic curves of degree $L$. It becomes
more difficult as $L$ increases; thus, a few examples have been studied in
detail. The sparse potentials treated in this paper can be analyzed even for
large $L$. This paper presents a Chebyshev polynomial (defined in Section 3)
representation of the discriminant, which is then applied to the spectrum
analysis.

This paper is organized as follows. Section 2 defines the discriminant, states
that the discriminant describes the spectrum of $H$, and introduces the
Floquet spectrum. Section 3 presents the Chebyshev polynomial representation
of the discriminant, which is the main theorem of this paper, after
introducing Chebyshev polynomials and listing their properties. Furthermore,
we apply this theorem to derive two properties concerning integrals in
$[-2,2]$ of the discriminant. Section 4 derives the first-order Taylor
polynomials of the discriminant using the main theorem. Section 5 shows that
exactly $L$ spectral bands appear for nonzero real $v$ and that the band
outside $[-2,2]$ converges to a point as $L$ is large, applying the result of
Section 4. Section 6 studies the Floquet spectrum by perturbation method for
$v\in\mathbb{C}$ with small and large $|v|$.

##  2 Discriminant and Spectrum of $H$

The spectrum of the one-dimensional discrete Schrödinger operator with
periodic potential is represented by the Hill discriminant.

For every $E\in\mathbb{C}$, the equation $Hu=Eu$ can be uniquely solved by
giving initial values $u(0)$ and $u(1)$. Precisely, the solution $u(n)$ can be
represented as

$\left(\begin{array}[]{c}u(n+1)\\\
u(n)\end{array}\right)=\Phi_{n}(E)\left(\begin{array}[]{c}u(1)\\\
u(0)\end{array}\right),$

where

$\Phi_{n}(E)=\left\\{\begin{array}[]{ll}A_{n}(E)\cdots A_{1}(E)&(n\geq 1)\\\
I&(n=0)\\\ A_{n+1}(E)^{-1}\cdots A_{0}(E)^{-1}&(n\leq-1)\end{array}\right.$

and

$A_{n}(E)=\left(\begin{array}[]{cc}E-V(n)&-1\\\ 1&0\end{array}\right).$

Note that $\det\Phi_{n}(E)=1$ since $\det A_{n}(E)=1$. The spectrum
$\sigma(H)$ of $H$ can be characterized as

$\sigma(H)=\\{E\in\mathbb{C}:\Delta_{L}(E)\in[-2,2]\\},$

where $\Delta_{L}(E)=\text{\rm tr}\Phi_{L}(E)$ is the Hill discriminant
(“discriminant” for short) of $H$. Note that $\Delta_{L}(E)$ is a monic
polynomial. This result was proved by Rofe-Beketov [13] for continuous
Schrödinger operators with complex coefficients, but it can also be
established for the discrete case. We note that in the case $v=0$, i.e., free
Laplacian $H_{0}$, $\Delta_{L}(E)=2T_{L}(E/2)$ for the first kind Chebyshev
polynomial $T_{L}$ of degree $L$ (described in Section 3). In this case,
$\Delta_{L}(E)=2\cos(L\cos^{-1}(E/2))$ by the definition of the Chebyshev
polynomial, which shows that $\sigma(H_{0})=[-2,2]$. We introduce the Floquet
spectrum

$\sigma_{\kappa}(H)=\\{E\in\mathbb{C}:\Delta_{L}(E)=2\cos\kappa\\}$

for $\kappa\in[0,\pi]$, then $\sigma(H)$ can be written as the disjoint union

$\sigma(H)=\bigcup_{\kappa\in[0,\pi]}\sigma_{\kappa}(H).$ (2.1)

$\Delta_{L}(E)=2\cos\kappa$ is equivalent to $\Phi_{L}(E)$ having eigenvalues
$e^{i\kappa},e^{-i\kappa}$. Since the discriminant $\Delta_{L}(E)$ is a
polynomial of degree $L$, $\Delta_{L}(E)-2\cos\kappa$ has $L$ roots
$E_{1}(\kappa),\ldots,E_{L}(\kappa)$ with multiplicity(these are called
$\kappa$-Floquet eigenvalues), each $E_{j}(\kappa)$ of which is continuous
with respect to $\kappa$. Therefore, (2.1) leads that $\sigma(H)$ consists of
$L$ closed bounded analytic arcs lying in the complex plane at most.

Figure 1: The spectrum for $L=3$ and $v=2i$

######  Example 1.

By setting $E=x+iy~{}(x,y)\in\mathbb{R}^{2}$, the Floquet spectrum
$\sigma_{\kappa}(H)$ can be represented as the intersection of the curves
$\text{{\rm Re}}\Delta_{L}(E)=\text{{\rm Re}}\Delta_{L}(x+iy)=2\cos\kappa$ and
$\text{{\rm Im}}\Delta_{L}(E)=\text{{\rm Im}}\Delta_{L}(x+iy)=0$. These two
curves are real zeros of polynomials of total degree $L$ in $x$ and $y$. The
two curves represented by the real and imaginary parts of a holomorphic
function on the complex plane intersect perpendicularly. Therefore,
$\text{{\rm Re}}\Delta_{L}(x+iy)=2\cos\kappa$ and $\text{{\rm
Im}}\Delta_{L}(x+iy)=0$ have perpendicular intersections. For example,
consider the case $L=3$ and $v=2i$. In this case, $\text{{\rm
Re}}\Delta_{3}(x+iy)=x^{3}-3xy^{2}-3x+4xy$ and $\text{{\rm
Im}}\Delta_{3}(x+iy)=3x^{2}y-y^{3}-3y-2(x^{2}-y^{2}-1)$. In Figure 1, the
dashed line represents $\text{{\rm Re}}\Delta_{3}(x+iy)=\pm 2$ and the dotted
line represents $\text{{\rm Im}}\Delta_{3}(x+iy)=0$. The part of the dotted
line that is solid is the arc of the spectrum, and the three curves that
intersect these curves are $\text{{\rm
Re}}\Delta_{3}(x+iy)=0.8~{}(\kappa=\cos^{-1}0.4)$. These intersections (three
filled circles in Figure 1) correspond to the Floquet spectrum
$\sigma_{\cos^{-1}0.4}(H)$.

######  Remark 1.

It is easy to verify that the spectrum is symmetric about the imaginary axis
for pure imaginary $v$.

##  3 Representation of the discriminant using Chebyshev polynomials

In this section, we list several formulas for the Chebyshev polynomials (see,
e.g., Mason-Handscomb [10]) to be used later, and represent the discriminant
$\Delta_{L}(E)$ using Chebyshev polynomials.

######  Definition 1.

The Chebyshev polynomials of the first kind $T_{n}(x)$ are defined by
$T_{n}(\cos\theta)=\cos(n\theta)~{}(n\geq 0)$. Similarly, define the Chebyshev
polynomials of the second kind $U_{n}(x)$ are defined by
$U_{n}(\cos\theta)\sin\theta=\sin(n+1)\theta~{}(n\geq 0)$.

It is easy to verify that $T_{n}(-x)=(-1)^{n}T_{n}(x)$ and
$U_{n}(-x)=(-1)^{n}U_{n}(x)$. The first few Chebyshev polynomials of the first
and second kind are $T_{0}(x)=1$, $T_{1}(x)=x$, $T_{2}(x)=2x^{2}-1$,
$T_{3}(x)=4x^{3}-3x$, $U_{0}(x)=1$, $U_{1}(x)=2x$, $U_{2}(x)=4x^{2}-1$,
$U_{3}(x)=8x^{3}-4x$. It is also easily seen that

$\displaystyle T_{n}(\cosh\xi)$ $\displaystyle=$ $\displaystyle\cosh n\xi,$
$\displaystyle U_{n}(\cosh\xi)$ $\displaystyle=$
$\displaystyle\frac{\sinh(n+1)\xi}{\sinh\xi}.$

######  Proposition 1.

The Chebyshev polynomials satisfy the following recursive relations:

$\displaystyle T_{n+1}(x)$ $\displaystyle=$ $\displaystyle
2xT_{n}(x)-T_{n-1}(x)~{}(n\geq 1),$ $\displaystyle U_{n+1}(x)$
$\displaystyle=$ $\displaystyle 2xU_{n}(x)-U_{n-1}(x)~{}(n\geq 1),$
$\displaystyle T_{n}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(U_{n}(x)-U_{n-2}(x))~{}(n\geq 2).$

######  Proposition 2.

The following formulas hold for the derivative of Chebyshev polynomials.

$\displaystyle T^{\prime}_{n}(x)$ $\displaystyle=$ $\displaystyle
nU_{n-1}(x)~{}(n\geq 1),$ $\displaystyle U^{\prime}_{n}(x)$ $\displaystyle=$
$\displaystyle\frac{(n+1)T_{n+1}(x)-xU_{n}(x)}{x^{2}-1}~{}(n\geq 0)$

######  Proposition 3.

Both $\\{T_{n}(x)\\}_{n=0}^{\infty}$ and $\\{U_{n}(x)\\}_{n=0}^{\infty}$ form
a sequence of orthogonal polynomials in the following sense:

$\displaystyle\frac{2}{\pi}\int_{-1}^{1}T_{n}(x)T_{m}(x)\frac{1}{\sqrt{1-x^{2}}}dx$
$\displaystyle=$ $\displaystyle\delta_{nm}$
$\displaystyle\frac{2}{\pi}\int_{-1}^{1}U_{n}(x)U_{m}(x)\sqrt{1-x^{2}}dx$
$\displaystyle=$ $\displaystyle\delta_{nm},$

where $\delta_{nm}$ is the Kronecker Delta.

######  Proposition 4.

$\displaystyle T_{n}(x)$ $\displaystyle=$ $\displaystyle
2^{n-1}\prod_{k=1}^{n}\left(x-\cos\frac{(2k-1)\pi}{2n}\right),$ $\displaystyle
U_{n}(x)$ $\displaystyle=$ $\displaystyle
2^{n}\prod_{k=1}^{n}\left(x-\cos\frac{k\pi}{n+1}\right)$

######  Lemma 1.

Let $\Phi_{0}(E)$ be the matrix

$\Phi_{0}(E)=\left(\begin{array}[]{cc}E&-1\\\ 1&0\end{array}\right),$

then, $\text{\rm tr}(\Phi_{0}(E)^{n})=2T_{n}(E/2)$.

###### Proof.

We first consider the case of $\Phi_{0}(E)$ has eigenvalues
$\lambda,\lambda^{-1}$ satisfying $\lambda^{2}\neq 1$. By direct computation,
we learn

$\Phi_{0}(E)^{n}=\frac{\lambda^{n}-\lambda^{-n}}{\lambda-\lambda^{-1}}\Phi_{0}(E)-\frac{\lambda^{n-1}-\lambda^{-(n-1)}}{\lambda-\lambda^{-1}}I.$
(3.1)

Let $\lambda=e^{i\theta}$, i.e., $E=2\cos\theta$ ($\theta$ need not be a real
number). Taking trace of both sides of (3.1), combining the second and the
third identities in Proposition 1 yields

$\displaystyle\text{\rm tr}(\Phi_{0}(E)^{n})$ $\displaystyle=$
$\displaystyle\frac{e^{in\theta}-e^{-in\theta}}{e^{i\theta}-e^{-i\theta}}E-2\frac{e^{i(n-1)\theta}-e^{-i(n-1)\theta}}{e^{i\theta}-e^{-i\theta}}$
$\displaystyle=$ $\displaystyle EU_{n-1}(E/2)-2U_{n-2}(E/2)$ $\displaystyle=$
$\displaystyle U_{n}(E/2)-U_{n-2}(E/2)=2T_{n}(E/2).$

We consider the case $\lambda^{2}=1$, i.e., $E=\pm 2$. In this case, $\lambda$
is a double root of the eigenpolynomial of $\Phi_{0}(E)$, then we have

$\Phi_{0}(E)^{n}=n\lambda^{n-1}\Phi_{0}(E)-(n-1)\lambda^{n}I.$ (3.2)

Taking trace of both sides of (3.2), we have

$\text{\rm tr}(\Phi_{0}(E)^{n})=n(\pm 1)^{n-1}(\pm 2)-2(n-1)(\pm 1)^{n}.$

Here is the double sign in the same order. Then, $\text{\rm
tr}(\Phi_{0}(E)^{n})=2$ for $E=2$, and $=2(-1)^{n}$ for $E=-2$, these are
consistent with $2T_{n}(E/2)$. $\Box$

Then, our main theorem is stated as follows:

######  Theorem 1.

The discriminant $\Delta_{L}(E)$ can be represented as

$\Delta_{L}(E)=2T_{L}(E/2)-vU_{L-1}(E/2)$ (3.3)

for $L\geq 1$.

###### Proof.

From the cyclic invariance of the matrix trace, the discriminant
$\Delta_{L}(E)$ equals the trace of the matrix given by

$\Phi_{0}(E)^{L-1}\left(\begin{array}[]{cc}E-v&-1\\\
1&0\end{array}\right)=\Phi_{0}(E)^{L}-v\Phi_{0}(E)^{L-1}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right).$ (3.4)

For $L=1,2$, by direct computation, we have

$\displaystyle\Delta_{1}(E)$ $\displaystyle=$ $\displaystyle
E-v=2T_{1}(E/2)-vU_{0}(E/2),$ $\displaystyle\Delta_{2}(E)$ $\displaystyle=$
$\displaystyle E^{2}-2-vE=2T_{2}(E/2)-vU_{1}(E/2).$

Next, assume $L\geq 3$, using (3.1), we have

$\Phi_{0}(E)^{L-1}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)=\frac{\lambda^{L-1}-\lambda^{-(L-1)}}{\lambda-\lambda^{-1}}\left(\begin{array}[]{cc}E&0\\\
0&0\end{array}\right)-\frac{\lambda^{L-2}-\lambda^{-(L-2)}}{\lambda-\lambda^{-1}}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right).$

Taking trace in both sides shows that

$\displaystyle\text{\rm
tr}\left\\{\Phi_{0}(E)^{L-1}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\right\\}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{L-1}-\lambda^{-(L-1)}}{\lambda-\lambda^{-1}}E-\frac{\lambda^{L-2}-\lambda^{-(L-2)}}{\lambda-\lambda^{-1}}$
$\displaystyle=$ $\displaystyle EU_{L-2}(E/2)-U_{L-3}(E/2)=U_{L-1}(E/2).$
(3.8)

From Lemma 1, we learn that $\text{\rm tr}(\Phi_{0}(E)^{L})=2T_{L}(E/2)$.
Substituting $\text{\rm tr}(\Phi_{0}(E)^{L})=2T_{L}(E/2)$ and (3.8) into
(3.4), we have

$\displaystyle\Delta_{L}(E)$ $\displaystyle=$ $\displaystyle\text{\rm
tr}(\Phi_{L}(E))-v\text{\rm
tr}\left\\{\Phi_{0}(E)^{L-1}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\right\\}$ $\displaystyle=$ $\displaystyle
2T_{L}(E/2)-vU_{L-1}(E/2).$

$\Box$

######  Corollary 1.

Let $\beta_{k}$ be the $k$-th root of $U_{L-1}(E/2)$ is given by

$\beta_{k}=2\cos\frac{k\pi}{L}\quad(k=1,2,\ldots,L-1).$

Then, $\\{\beta_{1},\beta_{2},\ldots,\beta_{L-1}\\}\subset\sigma(H)$.
Moreover, $2\in\sigma(H)$ if and only if $v\in[0,4/L]$, also, $-2\in\sigma(H)$
if and only if $v\in[-4/L,0]$. In particular, $\pm 2\not\in\sigma(H)$ if
$v\not\in\mathbb{R}$.

###### Proof.

Substituting $\beta_{k}$ into (3.3) yields
$\Delta_{L}(\beta_{k})=2\cdot(-1)^{k}\in[-2,2]$. Furthermore, substituting
$E=\pm 2$ into (3.3) yields $\Delta_{L}(2)=2-Lv$ and
$\Delta_{L}(-2)=(-1)^{L}(2+Lv)$. $2\in\sigma(H)$ if and only if
$\Delta_{L}(2)=2-Lv\in[-2,2]$, which means that $v\in[0,4/L]$. Similarly, it
can be shown that $-2\in\sigma(H)$ and that $v\in[-4/L,0]$ is equivalent.
$\Box$

Corollary 1 shows that the roots $\beta_{1},\beta_{2},\ldots,\beta_{L-1}$ of
$\Delta_{L}(E)$ that corresponding to one of the endpoints of each spectral
arc is concentrated at endpoints $E=\pm 2$ of the spectrum of the free
Laplacian as $L$ is large, and their density is approximately
$\displaystyle\frac{1}{\pi\sqrt{4-E^{2}}}$ on the real axis.

######  Corollary 2.

$\int_{-2}^{2}\Delta_{L}(E)dE=\left\\{\begin{array}[]{cl}-\frac{4}{L^{2}-1}&\mbox{$L$:
even}\\\ -\frac{2v}{L}&\mbox{$L$: odd}.\end{array}\right.$ (3.10)

###### Proof.

Using (3.3) and the definition of the Chebyshev polynomial, and substituting
$E=2\cos\theta$, the integral then becomes

$\displaystyle\int_{-2}^{2}\Delta_{L}(E)dE$ $\displaystyle=$
$\displaystyle\int_{-2}^{2}(2T_{L}(E/2)-vU_{L-1}(E/2))dE$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}(2\cos L\theta\sin\theta-v\sin L\theta)d\theta.$

This integral can be easily computed to obtain the desired result. $\Box$

This integral (3.10) approaches $0$ as $L$ is large, which means that
$\Delta_{L}(E)$ oscillates intensively within $[-2,2]$ as large $L$.

######  Theorem 2.

$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|\Delta_{L}(E)|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle=$ $\displaystyle 2+|v|^{2}\quad(L\geq 2)$
$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|\Delta_{1}(E)|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle=$ $\displaystyle 1+|v|^{2}$

###### Proof.

For $L=1$, direct computation of the integral yields

$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|\Delta_{1}(E)|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|E-v|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}(E^{2}-(v+\overline{v})E+|v|^{2})\sqrt{4-E^{2}}dE$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}(E^{2}+|v|^{2})\sqrt{4-E^{2}}dE=1+|v|^{2}.$

We consider the case $L\geq 2$. Theorem 1 and the third identity in
Proposition 1 together shows that

$\displaystyle\Delta_{L}(E)$ $\displaystyle=$ $\displaystyle
2T_{L}(E/2)-vU_{L-1}(E/2)$ $\displaystyle=$ $\displaystyle
U_{L}(E/2)-U_{L-2}(E/2)-vU_{L-1}(E/2).$

Using Proposition 3 and Parseval’s identity, one can show

$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|\Delta_{L}(E)|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-2}^{2}|U_{L}(E/2)|^{2}\sqrt{4-E^{2}}dE+\frac{1}{2\pi}\int_{-2}^{2}|U_{L-2}(E/2)|^{2}\sqrt{4-E^{2}}dE$
$\displaystyle+|v|^{2}\frac{1}{2\pi}\int_{-2}^{2}|U_{L-1}(E/2)|^{2}\sqrt{4-E^{2}}dE=2+|v|^{2}.$

$\Box$

The integral in Theorem 2 does not depend on $L(\geq 2)$, but only on $v$.
Theorem 2 that the value of the weight $\sqrt{4-E^{2}}$ becomes smaller near
$\pm 2$ indicates that as $L$ becomes larger, $|\Delta_{L}(E)|$ becomes larger
near $\pm 2$. From this fact and Corollary 2, we learn that $\Delta_{L}(E)$
oscillates intensively near $\pm 2$ for large $L$.

##  4 The first-order Taylor approximation of the discriminant

In this section, we give the first-order Taylor polynomials at
$\alpha_{j}~{}(j=1,2,\ldots,L)$, $\beta_{k}~{}(k=1,2,\ldots,L-1)$, and $\pm 2$
on the real axis of the discriminant for application to the analysis of the
spectrum of $H$ near $[-2,2]$.

######  Lemma 2.

The first-order Taylor polynomials at $\alpha_{j}~{}(j=1,2,\ldots,L)$,
$\beta_{k}~{}(k=1,2,\ldots,L-1)$, and $\pm 2$ are given by:

$\displaystyle\Delta_{L}(E)$ $\displaystyle=$
$\displaystyle\frac{v(-1)^{j}}{\sin\theta_{j}}+\frac{2L\sin^{2}\theta_{j}-v\cos\theta_{j}}{2\sin^{3}\theta_{j}}(E-\alpha_{j})+O((E-\alpha_{j})^{2})$
(4.1) $\displaystyle\Delta_{L}(E)$ $\displaystyle=$ $\displaystyle
2(-1)^{k}+\frac{v(-1)^{k}}{2\sin^{3}\phi_{k}}(E-\beta_{k})+O((E-\beta_{k})^{2}),$
(4.2) $\displaystyle\Delta_{L}(E)$ $\displaystyle=$ $\displaystyle
2-vL+\left\\{L^{2}-\frac{v}{4}L(L^{2}-1)\right\\}(E-2)+O((E-2)^{2})$ (4.3)
$\displaystyle\Delta_{L}(E)$ $\displaystyle=$
$\displaystyle(-1)^{L}(2+vL)+(-1)^{L}\left\\{L^{2}+\frac{v}{4}L(L^{2}-1)\right\\}(E+2)$
(4.4) $\displaystyle+O((E+2)^{2}),$

where $\alpha_{j}=2\cos\theta_{j}=2\cos\frac{(2j-1)\pi}{2L}$ and
$\beta_{k}=2\cos\phi_{k}=2\cos\frac{k\pi}{L}$.

###### Proof.

Taking derivative of (3.3) and Proposition 2 together shows that

$\displaystyle\Delta_{L}^{\prime}(E)$ $\displaystyle=$ $\displaystyle
T_{L}^{\prime}(E/2)-\frac{v}{2}U_{L-1}^{\prime}(E/2)$ (4.5) $\displaystyle=$
$\displaystyle
LU_{L-1}(E/2)-\frac{v}{2}\cdot\frac{LT_{L}(E/2)-\frac{E}{2}U_{L-1}(E/2)}{(E/2)^{2}-1}$
$\displaystyle=$ $\displaystyle
LU_{L-1}(E/2)-v\frac{2LT_{L}(E/2)-EU_{L-1}(E/2)}{E^{2}-4}.$

Since $\alpha_{j}$ satisfies $T_{L}(\alpha_{j}/2)=0$ and $\beta_{k}$ satisfies
$U_{L-1}(\beta_{k}/2)=0$, substituting $\alpha_{j}$ and $\beta_{k}$ into (4.5)
yields (4.1) and (4.2).

Substituting $E=2\cos\theta~{}(0<\theta<\pi)$ into (4.5), we have

$\displaystyle\Delta_{L}^{\prime}(2\cos\theta)$ $\displaystyle=$
$\displaystyle LU_{L-1}(\cos\theta)-v\frac{2LT_{L}(\cos\theta)-2\cos\theta
U_{L-1}(\cos\theta)}{4\cos^{2}\theta-4}$ (4.6) $\displaystyle=$ $\displaystyle
L\frac{\sin L\theta}{\sin\theta}+v\frac{2L\cos L\theta-2\cos\theta\frac{\sin
L\theta}{\sin\theta}}{4\sin^{2}\theta}$ $\displaystyle=$ $\displaystyle
L\frac{\sin L\theta}{\sin\theta}+v\frac{L\sin\theta\cos L\theta-\cos\theta\sin
L\theta}{2\sin^{3}\theta}.$

Taking limit $\theta\to 0$ of both sides of (4.6), we have

$\Delta_{L}^{\prime}(2)=L^{2}-\frac{v}{4}L(L^{2}-1).$ (4.7)

Combining (4.7) and $\Delta_{L}(2)=2-vL$ yields (4.3). Using

$\displaystyle\Delta_{L}(-E)$ $\displaystyle=$ $\displaystyle
2T_{L}(-E/2)-vU_{L-1}(-E/2)$ $\displaystyle=$ $\displaystyle
2(-1)^{L}T_{L}(E/2)-v(-1)^{L-1}U_{L-1}(E/2),$

we have

$\Delta_{L}^{\prime}(-2\cos\theta)=L(-1)^{L}\frac{\sin
L\theta}{\sin\theta}+v(-1)^{L-1}\frac{L\sin\theta\cos L\theta-\cos\theta\sin
L\theta}{2\sin^{3}\theta}$

and then, taking limit $\theta\to 0$, we have

$\Delta_{L}^{\prime}(-2)=(-1)^{L}L^{2}-(-1)^{L-1}\frac{v}{4}L(L^{2}-1).$ (4.8)

Combining (4.8) and $\Delta_{L}(-2)=(-1)^{L}(2+vL)$ yields (4.4). $\Box$

######  Remark 2.

Since (4.5) is the derivative of polynomial $\Delta_{L}(E)$, (4.5) is also a
polynomial, therefore, $2LT_{L}(E/2)-EU_{L-1}(E/2)$ becomes a divisor of
$E^{2}-4$. For example, in the case of $L=3$, it is equals to $2\cdot
3T_{3}(E/2)-EU_{2}(E/2)=2E^{3}-8E=2E(E^{2}-4)$.

######  Remark 3.

From (4.1) in Lemma 2, we have

$\Delta_{L}(\alpha_{j})=\frac{v(-1)^{j}}{\sin\frac{(2j-1)\pi}{2L}}.$

This indicates that the discriminant’s amplitude increases almost in
proportion to $Lv$ near both ends of $[-2,2]$. It is consistent with Corollary
2 and Theorem 2.

##  5 Spectrum for real $v$

In this section, we consider the case of real $v$ compared to where $v$ is not
real. $H$ is a self-adjoint operator if $v$ is real, therefore, the spectrum
$\sigma(H)\subset\mathbb{R}$. Throughout this section, interval means an
interval on the real axis.

######  Theorem 3.

(A part of Theorem 4.10 in Kato [8]) Let $T$ be self-adjoint, and $A$ bounded
symmetric. Then $S=T+A$ is self-adjoint and

$\sup_{\zeta\in\sigma(S)}\text{\rm dist}(\zeta,\sigma(T))\leq\|A\|,$

where $\text{\rm dist}(\zeta,D)=\inf\\{|\zeta-x|:x\in D\\}$.

Theorem 3 implies the roughest estimate of the location of the spectrum as
follows.

######  Proposition 5.

$\sup_{\zeta\in\sigma(H)}\text{\rm dist}(\zeta,[-2,2])\leq|v|,$

that is, $\sigma(H)\subset[-2-|v|,2+|v|]$.

Since $\Delta_{L}(E)$ is a polynomial of degree $L$, the number of extreme
values of the discriminant is at most $L-1$; therefore,
$\sigma(H)=\\{E\in\mathbb{R}:\Delta_{L}(E)\in[-2,2]\\}$ consists of at most
$L$ closed intervals. The structure of the spectrum can be further detailed,
as in the following Theorem 4.

######  Theorem 4.

If $v\in\mathbb{R}$ is nonzero, $\sigma(H)$ consists of exactly $L$ closed
intervals.

###### Proof.

Let $\beta_{k}~{}(k=1,2,\ldots,L-1)$ be defined in Lemma 2, then
$\Delta_{L}(\beta_{k})=2(-1)^{k}$. Since the first-order coefficient of (4.2),
i.e.,

$\eta_{k}=\frac{v(-1)^{k}}{2\sin^{3}\frac{k\pi}{L}}$

is never zero, therefore, $\Delta_{L}(E)$ intersects $2(-1)^{k}$ transversely
at $E=\beta_{k}~{}(k=1,2,\ldots,L-1)$. This result means that $\beta_{k}$ is
an endpoint of a spectral band. Since it can be shown the same way for $v<-2$
or $L$ is even, we will only prove the $v>2$ and odd $L$ case here.

Since $\Delta_{L}(-2)=-(2+Lv)<-2$ and $\Delta_{L}(\beta_{L-1})=2$, there
exists $\gamma_{L-1}\in(-2,\beta_{L-1})$ such that
$\Delta_{L}(\gamma_{L-1})=-2$. Therefore, $[\gamma_{L-1},\beta_{L-1}]$
contains at least one spectral band $I_{L-1}$ of $\sigma(H)$. Next, since
$\Delta_{L}^{\prime}(\beta_{L-1})=\eta_{L-1}>0$ and
$\Delta_{L}(\beta_{L-2})=-2$, there exists
$\gamma_{L-2}\in(\beta_{L-1},\beta_{L-2})$ and $[\gamma_{L-2},\beta_{L-2}]$
contains at least one spectral band $I_{L-2}$. In the same way, we show that
there exists
$\gamma_{L-3}\in(\beta_{L-2},\beta_{L-3}),\cdots,\gamma_{1}\in(\beta_{2},\beta_{1})$
such that all $[\gamma_{k},\beta_{k}]$ each contain a spectral band $I_{k}$
for every $k=1,2,\ldots,L-1$. Since $\Delta_{L}(E)$ is a monic polynomial,
there exists $\beta_{0}>\beta_{1}$ such that $\Delta_{L}(\beta_{0})=2$.
Therefore, since $\Delta_{L}(\beta_{1})=-2$,
$\Delta_{L}^{\prime}(\beta_{1})=\eta_{1}<0$, there exists
$\gamma_{0}\in(\beta_{1},\beta_{0})$ such that $\Delta_{L}(\gamma_{0})=-2$,
that is, $[\gamma_{0},\beta_{0}]$ cotains at least a spectral band $I_{0}$.
From the above, there exist at least $L$ spectral bands. Since there exist at
most $L$ spectral bands, this indicates that
$I_{k}=[\gamma_{k},\beta_{k}]~{}(k=0,1,\ldots,L-1)$. Therefore, we conclude

$\sigma(H)=\bigcup_{k=0}^{L-1}[\gamma_{k},\beta_{k}].$

Thus, we complete the proof. $\Box$

By carefully observing the proof of Theorem 4 and the value at $E=\pm 2$ of
the discriminant $\Delta_{L}(E)$, the following result is easily derived.

######  Corollary 3.

Suppose that $\sigma(H)=\bigcup_{k=0}^{L-1}I_{k}$, where
$I_{k}=[\gamma_{k},\beta_{k}]$ with
$\gamma_{L-1}<\beta_{L-1}<\cdots<\gamma_{0}<\beta_{0}$. Then,
$[-2,2]\cap[\gamma_{0},\beta_{0}]\neq\emptyset$ if and only if $0\leq v\leq
4/L$, and $[-2,2]\cap[\gamma_{L-1},\beta_{L-1}]\neq\emptyset$ if and only if
$-4/L\leq v\leq 0$.

We remark that Corollary 3 implies that for $|v|>4/L$, $[-2,2]$ contains
exactly $L-1$ spectral bands, and there exists only one band outside $[-2,2]$.
If $L$ is large enough, $|v|>4/L$ is satisfied; moreover, we have a good
approximation of the location of the spectral band outside $[-2,2]$ as
follows.

######  Corollary 4.

The spectral band $I_{*}$ outside $[-2,2]$ closes a point either
$\sqrt{4+v^{2}}$ for $v>0$, or $-\sqrt{4+v^{2}}$ as $L\to\infty$. That is,

$\sup_{E\in I_{*}}\text{\rm dist}(E,\\{\text{\rm sgn}(v)\sqrt{4+v^{2}}\\})\to
0\quad(L\to\infty),$

where sgn is the signum function.

###### Proof.

It is sufficient to show that all Floquet eigenvalues in the spectral band
outside $[-2,2]$ converge to either $\sqrt{4+v^{2}}$ for $v>0$, or
$-\sqrt{4+v^{2}}$ for $v<0$ as $L\to\infty$. By letting
$E(\kappa)=2\cosh\xi(\kappa)~{}(\xi(\kappa)>0)$ be a Floquet eigenvalue larger
than $2$, we obtain

$\displaystyle\Delta_{L}(2\cosh\xi(\kappa))$ $\displaystyle=$ $\displaystyle
2T_{L}(\cosh\xi(\kappa))-vU_{L-1}(\cosh\xi(\kappa))$ $\displaystyle=$
$\displaystyle 2\cosh L\xi(\kappa)-v\frac{\sinh
L\xi(\kappa)}{\sinh\xi(\kappa)}=2\cos\kappa.$

Thus, we have

$2\coth L\xi(\kappa)-\frac{v}{\sinh\xi(\kappa)}=\frac{2\cos\kappa}{\sinh
L\xi(\kappa)}.$ (5.1)

Taking limit $L\to\infty$ of both sides of (5.1), we learn that
$\sinh\xi(\kappa)\to v/2$ as $L\to\infty$, therefore,
$E=2\cos\xi(\kappa)\to\sqrt{4+v^{2}}$ as $L\to\infty$. In the case $v<0$,
$E\to-\sqrt{4+v^{2}}$ can be found similarly. $\Box$

##  6 Floquet spectrum for small and large $v$

In this section, we study perturbations of the Floquet spectrum for small and
large $v$. We first note that from Proposition 4, all roots of $T_{n}(x)$ and
$U_{n}(x)$ are simple.

######  Theorem 5.

(Lagrange inversion formula, 3.6.6. in Abramowitz and Stegun[1]) Suppose
$w=f(z)$ is analytic at a point $z_{0}$ and $f^{\prime}(z_{0})\neq 0$. Then
$z=g(w)$ given by a power series has a nonzero radius of convergence:

$g(w)=z_{0}+\sum_{n=1}^{\infty}\frac{g_{n}}{n!}(w-f(z_{0}))^{n},$

where

$g_{n}=\lim_{z\to
a}\frac{d^{n-1}}{dz^{n-1}}\left\\{\left(\frac{z-z_{0}}{f(z)-f(z_{0})}\right)^{n}\right\\}.$

######  Lemma 3.

Let $f(z)$ be a polynomial with $\alpha$ as a simple root, and let $g(z)$ be a
polynomial satisfying $g(\alpha)\neq 0$. Then, for $t\in\mathbb{C}$ with
sufficiently small $|t|$, $f(z)-sg(z)-t$ has a simple root $\alpha(v,t)$ near
$\alpha$ such that

$\alpha(v,t)=\alpha+\frac{g(\alpha)}{f^{\prime}(\alpha)}s+\frac{t}{f^{\prime}(\alpha)}+O((|s|+|t|)^{2})$

Sketch of the proof of Lemma 3

First, we note that $f^{\prime}(\alpha)\neq 0$ since $\alpha$ is a simple root
of $f(z)$. We consider the polynomial $h(z)=f(z)-sg(z)-t$. If $|s|$ and $|t|$
are small enough, $h(z)$ has a simple root $\alpha(s,t)$ near $\alpha$.
Theorem 5 shows

$\alpha(s,t)=\alpha(s)+\frac{t}{f^{\prime}(\alpha(s))-sg^{\prime}(\alpha(s))}+O(|t|^{2}),$
(6.1)

where $\alpha(s)$ is a root of $f(z)-sg(z)=0$. $\alpha(s)$ is close enough to
$\alpha$ for every sufficiently small $|s|$. Since $f(z)/g(z)$ is analytic at
$\alpha$ the assumption, we can use Theorem 5 again, it shows

$\displaystyle\alpha(s)$ $\displaystyle=$
$\displaystyle\alpha+\frac{s}{\left.\left(\frac{f(z)}{g(z)}\right)^{\prime}\right|_{z=\alpha}}+O(|s|^{2})$
(6.2) $\displaystyle=$
$\displaystyle\alpha+\frac{g(\alpha)}{f^{\prime}(\alpha)}s+O(|s|^{2})$

Substituting (6.2) into (6.1) and expanding the first-order term for $t$ using
the geometric series for small $|s|$, we have

$\displaystyle\alpha(s,t)$ $\displaystyle=$
$\displaystyle\alpha+\frac{g(\alpha)}{f^{\prime}(\alpha)}s+\frac{t}{f^{\prime}(\alpha(s))-sg^{\prime}(\alpha(s))}+O(|s|^{2})+O(|t|^{2})$
$\displaystyle=$
$\displaystyle\alpha+\frac{g(\alpha)}{f^{\prime}(\alpha)}s+\frac{t}{f^{\prime}(\alpha(s))}\sum_{n=0}^{\infty}\left(\frac{g^{\prime}(\alpha(s))}{f^{\prime}(\alpha(s))}\right)^{n}s^{n}+O(|s|^{2})+O(|t|^{2})$
$\displaystyle=$
$\displaystyle\alpha+\frac{g(\alpha)}{f^{\prime}(\alpha)}s+\frac{t}{f^{\prime}(\alpha(s))}+O(|st|)+O(|s|^{2})+O(|t|^{2})$

Since it follows from (6.2) that
$f^{\prime}(\alpha(s))=f^{\prime}(\alpha)+O(|s|)$, we have

$\frac{1}{f^{\prime}(\alpha(s))}=\frac{1}{f^{\prime}(\alpha)}+O(|s|).$

Summarize the above to reach the desired result. $\Box$

######  Theorem 6.

Let $\alpha_{j}$ be defined in Lemma 2, and $\alpha_{j}(v,\kappa)$ be the
element nearest $\alpha_{j}$ of the Floquet spectrum $\sigma_{\kappa}(H)$. For
$v\in\mathbb{C}$ with sufficiently small $|v|$ and $\kappa\in[0,\pi]$ with
sufficiently small $|\cos\kappa|$,

$\alpha_{j}(v,\kappa)=\alpha_{j}+\frac{v}{L}+2(-1)^{j-1}\frac{\sin\frac{(2j-1)\pi}{2L}}{L}\cos\kappa+O((|v|+|\cos\kappa|)^{2}).$

###### Proof.

Consider the equation
$\Delta_{L}(E)-2\cos\kappa=2T_{L}(E/2)-vU_{L-1}(E/2)-2\cos\kappa=0$. By
applying Lemma 3 to this equation, we have

$\alpha_{j}(v,\kappa)=\alpha_{j}+\frac{U_{L-1}(\alpha_{j}/2)}{T_{L}^{\prime}(\alpha_{j}/2)}v+\frac{2\cos\kappa}{T_{L}^{\prime}(\alpha_{j}/2)}+O(|v|^{2}+|\cos\kappa|^{2}).$
(6.3)

Proposition 2 leads

$T_{L}^{\prime}(\alpha_{j}/2)=LU_{L-1}(\alpha_{j}/2)=\frac{(-1)^{j-1}L}{\sin\frac{(2j-1)\pi}{2L}}.$

Substituting the above into (6.3) completes the proof. $\Box$

Theorem 6 implies that $\sigma(H)$ has $L$ arcs approximately parallel to the
real axis near $\alpha_{j}+v/L$ for $j=1,2,\ldots,L$. Moreover, as $L$
increases, the potential becomes sparse, therefore, the spectrum close to the
set $[-2,2]$ on the real line.

Figure 2: The spectrum for $L=5$ and $v=i/2$

######  Example 2.

Figure 2 shows the spectrum for $L=5$, $v=i/2$. In Figure 2, the dashed lines
are the curves represented by $\text{{\rm Re}}\Delta_{5}(E)=\text{{\rm
Re}}(2T_{5}(E/2))+\frac{1}{2}\text{{\rm Im}}U_{4}(E/2)=\pm 2$ and the dotted
lines are the curves represented by $\text{{\rm Im}}\Delta_{5}(E)=\text{{\rm
Im}}(2T_{5}(E/2))-\frac{1}{2}\text{{\rm Re}}U_{4}(E/2)=0$, where

$\displaystyle\text{{\rm Re}}(2T_{5}(E/2))$ $\displaystyle=$ $\displaystyle
x^{5}-10x^{3}y^{2}-5x^{3}+5xy^{4}+15xy^{2}+5x$ $\displaystyle\text{{\rm
Im}}(2T_{5}(E/2))$ $\displaystyle=$ $\displaystyle
5x^{4}y-10x^{2}y^{3}-15x^{2}y+y^{5}+5y^{3}+5y$ $\displaystyle\text{{\rm
Re}}U_{4}(E/2)$ $\displaystyle=$ $\displaystyle
x^{4}-6x^{2}y^{2}-3x^{2}+y^{4}+3y^{2}+1$ $\displaystyle\text{{\rm
Im}}U_{4}(E/2)$ $\displaystyle=$ $\displaystyle 4x^{3}y-4xy^{3}-6xy$

for $E=x+iy~{}((x,y)\in\mathbb{R}^{2})$. The solid lines are the spectrum
(spectral arcs), and the filled circles on the real axis are
$\alpha_{1},\alpha_{2},\ldots,\alpha_{5}$ in order from right to left. The
figure shows that the spectral arcs are almost parallel to the real axis in
the neighborhood of each $\alpha_{j}+v/L=\alpha_{j}+i/10(j=1,2,\ldots,5)$.

We next consider the case of large $|v|$.

######  Theorem 7.

Let $\beta_{k}$ be defined in Lemma 2, and $\beta_{k}(v,\kappa)$ be the
element nearest $\beta_{k}$ of the Floquet spectrum $\sigma_{\kappa}(H)$. For
$v$ with sufficiently large $|v|$ and $\kappa\in[0,\pi]$ with sufficiently
small $|\cos\kappa/v|$,

$\beta_{k}(v,t)=\beta_{k}-\frac{2\sin^{2}\frac{k\pi}{L}}{L}\frac{1}{v}-(-1)^{k}\frac{2\sin^{2}\frac{k\pi}{L}}{L}\frac{\cos\kappa}{v}+O((|1/v|+|\cos\kappa/v|)^{2}),$

where $\beta_{k}$ is defined in Lemma 2.

###### Proof.

The equation $\Delta_{L}(E)-2\cos\kappa=0$ can be rewritten as

$U_{L-1}(E/2)-\frac{2}{v}T_{L}(E/2)+\frac{2\cos\kappa}{v}=0.$ (6.4)

We recall

$T_{L}(\beta_{k}/2)=(-1)^{k},\quad
U_{L-1}(\beta_{k}/2)=-\frac{L(-1)^{k}}{\sin^{2}\frac{k\pi}{L}},$

and Lemma 3, hence we have

$\displaystyle\beta_{k}(v,\kappa)$ $\displaystyle=$
$\displaystyle\beta_{k}+\frac{2T_{L}(\beta_{k}/2)}{U_{L-1}^{\prime}(\beta_{k}/2)}\frac{1}{v}-\frac{2\cos\kappa}{U_{L-1}^{\prime}(\beta_{k}/2)}\frac{1}{v}+O((|1/v|+|\cos\kappa/v|)^{2})$
$\displaystyle=$
$\displaystyle\beta_{k}-\frac{2\sin^{2}\frac{k\pi}{L}}{L}\frac{1}{v}-(-1)^{k}\frac{2\sin^{2}\frac{k\pi}{L}}{L}\frac{\cos\kappa}{v}+O((|1/v|+|\cos\kappa/v|)^{2}).$

$\Box$

Figure 3: The spectrum for $L=5$ and $v=5i$

######  Example 3.

Figure 3 shows the spectrum for $L=5$, $v=5i$. The discriminant is the same as
in Example 2 except for the value of $v$. In Figure 3, the dashed lines are
the curves represented by $\text{{\rm Re}}\Delta_{5}(E)=\text{{\rm
Re}}(2T_{5}(E/2))+5\text{{\rm Im}}U_{4}(E/2)=\pm 2$ and the dotted lines are
the curves represented by $\text{{\rm Im}}\Delta_{5}(E)=\text{{\rm
Im}}(2T_{5}(E/2))-5\text{{\rm Re}}U_{4}(E/2)=0$ for
$E=x+iy~{}((x,y)\in\mathbb{R}^{2})$. The solid lines are the spectral arcs,
and the filled circles on the real axis are $\beta_{1},\beta_{2},\beta_{3}$,
and $\beta_{4}$ in order from right to left. Four curves extend from
$\beta_{1}$, $\beta_{2}$, $\beta_{3}$, and $\beta_{4}$ almost in the direction
of the imaginary axis, i.e., in the direction of $-1/v$. In the case of small
$v$, five connected components of the spectrum exist, as in Figure 2, but as
$v$ increases, there are four.

Acknowledgements
The author would like to thank Dr. Yu Morishima for helpful suggestions on
drawing figures in Python.

## References

  * [1] M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1972.
  * [2] C. M. Bender, Introduction to $\mathcal{PT}$-symmetric quantum theory, Contemporary physics 46 (4), 277-292.
  * [3] G. Borg, Eine Umkerhrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math. 78 (1946), 1-96.
  * [4] M. G. Gasimov, Spectral analysis of a class of second-order non-self-adjoint differential operators, Functional Anal., Appl. 14(1980), 11-15.
  * [5] I. M. Gel’fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad Nauk SSSR 73 (1950), 1117-1120. (in Russian)
  * [6] V. Guillemin, A. Uribe, Spectral properties of a certain class of complex potentials, Trans. of Americal Mathematical Society 279 (1983), No. 2, 759-771.
  * [7] H. Hochstadt, On the theory of Hill’s matrices and related inverse spectral problems. Linear Algebra Appl. 11 (1975), 41–52.
  * [8] T. Kato, Perturbation Theory for Linear Operators(2nd ed.), Springer-Verlag Berlin Heiderberg NewYork, 1995.
  * [9] C.Kittel, Introduction to Solid State Physics (Seventh ed.), New York: Wiley, 1996.
  * [10] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, 2002.
  * [11] V. G. Papanicolaou, Periodic Jacobi operators with complex coefficients, J. Spectr. Theory 11 (2021), no. 2, 781–819.
  * [12] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.
  * [13] F. S. Rofe-Beketov, On the spectrum of non-selfadjoint differential operators with periodic coefficients, Dokl. Akad. Nauk SSSR 152 (1963), No. 6, 1312–1315. (in Russian)
  * [14] O. Valiev, Non-self-adjoint Schrödinger Operator with a Periodic Potential, Springer, 2021.

Department of Information Technology

Faculty of Engineering

Tohoku Gakuin University

1-13-1 Chuo, Tagajo, Miyagi 985-0873, Japan


\funint{\bar{\psi},\psi,\Sigma}\exp\left\{i\int\left[\bar{\psi}\left(i\widehat{\partial} + \widehat{\Sigma}_{11}
- \lambda_{21\bar{i}}\tau_{\bar{i}\bar{\alpha}}\tau_{\bar{i}}^{\bar{\alpha}}\right)\psi
+ \bar{J}_1\psi +\bar{\psi}J_1
- \frac{\Sigma_{11\bar{i}\bar{\alpha}}\Sigma_{11\bar{i}}^{\bar{\alpha}}}{4\lambda_{11\bar{i}}}
+ J_{11\bar{i}\bar{\alpha}}\Sigma_{11\bar{i}}^{\bar{\alpha}} - \right.\right.\\
+ \bar{\Sigma}_{21\bar{i}\bar{\alpha}}\left(\tau_{\bar{i}}^{\bar{\alpha}}\psi + g_{21\bar{i}}\psi\bar{\psi}\tau_{\bar{i}}^{\bar{\alpha}}\psi\right)
+ \left(\bar{\psi}\tau_{\bar{i}}^{\bar{\alpha}} + g_{21\bar{i}}\bar{\psi}\tau_{\bar{i}}^{\bar{\alpha}}\psi\bar{\psi}\right)\Sigma_{21\bar{i}\bar{\alpha}} +\right.\right.\\
&\left.\left.+ \bar{\Sigma}_{21\bar{i}\bar{\alpha}}J_{21\bar{i}}^{\bar{\alpha}}
+ \bar{J}_{21\bar{i}\bar{\alpha}}\Sigma_{21\bar{i}}^{\bar{\alpha}}
\right]d^4x\right\}\\
g_{21\bar{i}} &= \sqrt{\frac{\lambda_{21\bar{i}}}{\varkappa_{21\bar{i}}}}
\end{split} \label{trifermion/compfields}
\end{equation}

The $\varkappa$ and $\lambda$ coefficients are assumed to be positive real numbers here.
The composite fields $\Sigma_{21\bar{i}}^{\bar{\alpha}}$ have various half-integer spins.
These fields can couple to both linear and cubic combinations of $\psi$ and $\bar{\psi}$.
Proper scaling of the fields and their sources reduces the number of related coupling constants
to two families $\varkappa_{21\bar{i}}$ and $\lambda_{21\bar{i}}$.

Fourth- and sixth-order interaction terms must cancel independently. This imposes two sets of relations on the constants.
However, only a small subset of sixth-order Fierz identities is involved: the ones for the terms of type

\begin{equation}
\Lgr_{6\bar{i}} = \bar{\psi}\psi\bar{\psi}\tau_{\bar{i}\bar{\alpha}}\psi\bar{\psi}\tau_{\bar{i}}^{\bar{\alpha}}\psi
\end{equation}

(summation over $\bar{i}$ not implied) which are essentially the same as fourth-order Fierz identities.
Thus, the complete set of cancellation conditions is

\begin{equation}
\begin{split}
\lambda_{21\,10}\\\lambda_{21\,20}\\\lambda_{21\,30}\\\lambda_{21\,40}\\\lambda_{21\,50}
\end{bmatrix} +
\begin{bmatrix}
-\frac{N + 4}{8N}&-\frac{1}{2}&-\frac{3}{2}&\frac{1}{2}&-\frac{1}{8}\\
-\frac{1}{2}&\frac{N - 2}{4N}&0&\frac{1}{4}&\frac{1}{8}\\
-\frac{1}{16}&0&\frac{N - 2}{4N}&0&-\frac{1}{16}\\
\frac{1}{8}&\frac{1}{4}&0&\frac{N - 2}{4N}&-\frac{1}{8}\\
-\frac{1}{8}&\frac{1}{2}&-\frac{3}{2}&-\frac{1}{2}&-\frac{N + 4}{8N}
\end{bmatrix}
\begin{bmatrix}
\lambda_{21\,11}\\\lambda_{21\,21}\\\lambda_{21\,31}\\\lambda_{21\,41}\\\lambda_{21\,51}
\end{bmatrix} = 0\\
\theta_{10}\\\theta_{20}\\\theta_{30}\\\theta_{40}\\\theta_{50}
\end{bmatrix} +
\begin{bmatrix}
-\frac{N + 4}{8N}&-\frac{1}{2}&-\frac{3}{2}&\frac{1}{2}&-\frac{1}{8}\\
-\frac{1}{2}&\frac{N - 2}{4N}&0&\frac{1}{4}&\frac{1}{8}\\
-\frac{1}{16}&0&\frac{N - 2}{4N}&0&-\frac{1}{16}\\
\frac{1}{8}&\frac{1}{4}&0&\frac{N - 2}{4N}&-\frac{1}{8}\\
-\frac{1}{8}&\frac{1}{2}&-\frac{3}{2}&-\frac{1}{2}&-\frac{N + 4}{8N}
\end{bmatrix}
\begin{bmatrix}
\theta_{11}\\\theta_{21}\\\theta_{31}\\\theta_{41}\\\theta_{51}
\end{bmatrix} =
\begin{bmatrix}
\end{bmatrix}\\
&\theta_{\bar{i}} = \lambda_{11\,\bar{i}} + \sqrt{\varkappa_{21\,\bar{i}}\lambda_{21\,\bar{i}}}
\end{split} \label{trifermion/cancellation}
\end{equation}

It is easy to see that addition of three-fermion fields makes $\lambda_{11\bar{i}}$ essentially free parameters.

It is also possible to introduce 3-0 composite fields dual to

\begin{equation}
\mu_{30\,ijk} = \psi_i\otimes\psi_j\otimes\psi_k
\end{equation}

Such tensor products form a reducible representation of the model's symmetry group and its decomposition
into irreducible components depends on color number $N$. The $N=3$ case is of special interest in the QCD context.

It is also worth noting that one-loop approximation of the effective action will not contain a proper kinetic term for three-fermion fields:
if all fields are split into background and deviation parts, the quadratic in deviations terms of the composite fields lagrangian will contain
the background fields $\bar{\Sigma}_{21\bar{i}\bar{\alpha}}$ and $\Sigma_{21\bar{i}\bar{\alpha}}$ coupled to $\Psi$, $\bar{\Psi}$ only.

Three-fermion fields will acquire kinetic term from renormalization of higher-loop diagrams, the first one being sunset

\begin{equation}
\begin{tikzpicture}[baseline=(a)]
\begin{feynman}
\vertex(a);
\vertex [right=1cm of a] (b);
\vertex [left=0.5cm of a] (o) {\(\bar{\Sigma}_{21}\)};
\vertex [right=0.5cm of b] (i) {\(\Sigma_{21}\)};
\diagram* {
(a) --[fermion, very thick] (o),
(b) --[fermion, half right] (a),
(a) --[fermion] (b),
(a) --[fermion, half right] (b),
(i) --[fermion, very thick] (b),
\end{feynman}
\end{tikzpicture}
\end{equation}

Constructing effective action for QCD with two- and three-fermion fields up to two loops is a task for further research.

§ CONCLUSIONS

The conclusions that can be drawn from the study above fall into two broad groups.

The first is relevant to NJL and related models.
Non-uniqueness of bosonization due to Fierz identities implies that
particularities of fermion interaction in such models do not manifest themselves
at one-loop level. This also sheds some light on the relation between NJL and QCD:
the two theories produce similar meson dynamics up to one loop.

Interestingly, if three-fermion composite fields are considered,
one-loop approximation is not enough to give them physical dynamics
and it is necessary to build effective action up to two loops at least, where the interaction would matter.

Non-uniqueness of bosonization is also relevant to the subquark model of fundamental interactions
proposed by Terazawa, Chikashige, and Akama [8].
The one-loop finite renormalization Lagrangian (<ref>) has vector boson and higgsoid sectors
and, as discussed above, it does so regardless of the subtleties of underlying subquark interactions.

The biggest obstacle to such subquark theories is, of course, the need to produce emergent gauge symmetry.
The formalism developed here naturally produces almost gauge invariant effective actions, where the only symmetry breaking terms are the vector boson mass terms.
Interestingly, if symmetry conditions are imposed, these terms cannot be cancelled exactly by renormalization.
Nevertheless, there is enough freedom to make observable vector boson masses arbitrarily small.

What (<ref>) also predicts is plenty of interesting nonlinear phenomena.
In particular, the effective field equations should have 't Hooft-Polyakov monopole-like solutions.
However, if vector field has mass, albeit small, the monopole energy becomes infinite.
This would neatly explain why monopoles are not observed.

Another type of nonlinear field equation solutions are kinks.
Their existence is determined by $\chi$-symmetry (<ref>) under which scalar composite fields change signs.
This means that if

\begin{equation}
\begin{split}
\Sigma_{10} = M = \text{const}\\
\Sigma_{11}^a = M^a = \text{const}
\end{split}
\end{equation}

is a solution of the effective field equations, there must also be an opposite-sign solution, and there must also be a solution that
approaches $M^{(a)}$ and $-M^{(a)}$ asymptotically in the opposite spatial directions.
This is another evidence that Wightman's vacuum homogeneity assumption is incompatible with effective action formalism
since the latter naturally produces inhomogeneous ground states.

The second group of conclusions comprises the considerations relevant to the general formalism of quantum field theory.
Application of R-operation to effective action formalism is obviously one such result.

Applicability of composite fields formalism to any field theory raises the question whether it preserves renormalizability.
In the traditional R-operation formalism, renormalizability arises from extra requirements, such as gauge invariance, being imposed on renormalization constants
and making the number of independent parameters finite.
The question is whether this persists when composite fields are added.
If not, this would mean blurring of the line between renormalizable and non-renormalizable theories.
In the latter case, renormalizablity would be the property of some sectors of the theory only.

But the main result here is the relation between bosonization and quantum BBGKY hierarchy. Bosonization, far from being an ad hoc recipe, is demonstrated to be a special case of the latter.
Physically, it means that nontrivial quantum states and processes involving them are under consideration. Successful application of bosonization to NJL model suggests that in some theories such nontrivial quantum states may comprise the actual low-energy sector. Interestingly, presence of various condensates suggests that QCD also belongs to this class (see e.g. [33]),
which makes studying composite fields dynamics in QCD a promising research program.

§ COMPUTATION OF 6-ORDER FIERZ IDENTITIES

Reduction of a generic term (<ref>) to ta sum of canonical terms (<ref>) is a two-stage process.
In the first stage, the formulas (<ref>) are used to express it in color indices. This results in
two types of nontrivial terms

\begin{equation}
\begin{split}
\Lgr_{1ijk}^{\bar{\alpha}\bar{\beta}\bar{\gamma}} =
\bar{\psi}_{l_1}\Gamma^{\bar{\alpha}}_i\psi_{l_1}
\bar{\psi}_{l_2}\Gamma^{\bar{\beta}}_j\psi_{l_3}
\bar{\psi}_{l_3}\Gamma^{\bar{\gamma}}_k\psi_{l_2}\\
\Lgr_{2ijk}^{\bar{\alpha}\bar{\beta}\bar{\gamma}} =
\bar{\psi}_{l_1}\Gamma^{\bar{\alpha}}_i\psi_{l_2}
\bar{\psi}_{l_2}\Gamma^{\bar{\beta}}_j\psi_{l_3}
\bar{\psi}_{l_3}\Gamma^{\bar{\gamma}}_k\psi_{l_1}\\
\end{split}
\end{equation}

Then (<ref>) is used to convert $\Lgr_2$ to $\Lgr_1$ and $\Lgr_1$s to canonical terms.
All that remains is to simplify products of $\gamma$-matrices. This was done with the help of dirac software
package [34]. The resulting identities are listed below.

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_3}\bar{\psi}_{i_3}\psi_{i_2} = \bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\psi_{i_1}\bar{\psi}_{i_3}\psi_{i_3} =
-\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} +\\
&+ \frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\psi_{i_3}\bar{\psi}_{i_3}\psi_{i_1} =
\frac{1}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
+\frac{3}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} -\\
&- \frac{i}{32}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \frac{3}{16}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}  -\\
- \frac{3}{32}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_3}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_2} =
-\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3} +
\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} +\\
+ \frac{1}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_1}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3} =
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_3}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_1} =
\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
-\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} - \\
& -\frac{3}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3}
+ \frac{i}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  -\\
+ \frac{1}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\lambda\mu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\lambda\mu}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_2} = &
-3\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3} -\\
+ \frac{1}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\lambda\mu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\sigma_{\sigma_{1}\sigma_{2}}\psi_{i_1}\bar{\psi}_{i_3}\sigma^{\sigma_{1}\sigma_{2}}\psi_{i_3} =
\frac{i}{4}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} - \\
- \frac{i}{2}\bar{\psi}_{i_1}{\sigma_\kappa}^\lambda\psi_{i_1}\bar{\psi}_{i_2}{\sigma_\lambda}^\mu\psi_{i_2}\bar{\psi}_{i_3}{\sigma_\mu}^\kappa\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\sigma_{\sigma_{1}\sigma_{2}}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\sigma_{1}\sigma_{2}}\psi_{i_1} =\frac{3}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
+ \frac{9}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} + \\
& + \frac{3}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3}
- \frac{i}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} -\frac{9}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} +\\
&+ \frac{1}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_2} =
\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
-\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} + \\
& +\frac{1}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3}
+ \frac{1}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_1}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} =
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_1} =
+ \frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}  -\\
&  -\frac{3}{8}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
- \frac{i}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  -\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\psi_{i_2} =
-\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} -\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\psi_{i_1}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} =
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\psi_{i_1} =
\frac{1}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
+\frac{3}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} - \\
&- \frac{I}{32}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
-\frac{7}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3} -\\
+\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} +\\
&+ \frac{1}{32}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \frac{13}{32}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_{\kappa}\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^{\kappa}\psi_{i_2} =
\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\psi_{i_2}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_1}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} =
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\psi_{i_2}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_1} =
+ \frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}  -\\
& - \frac{i}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
-\frac{1}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  -\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma_{\kappa}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^{\kappa}\psi_{i_2} =
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma_{\kappa}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\psi_{i_3}\bar{\psi}_{i_3}\gamma^5\gamma^{\kappa}\psi_{i_1} =
\frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
+\frac{1}{4}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} -\\
+ \frac{1}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} + \\
+ \frac{1}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \frac{i}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma_{\lambda}\psi_{i_1}\bar{\psi}_{i_2}\gamma_{\mu}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_2} =  - \frac{3i}{2}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma_{\lambda}\psi_{i_2}\bar{\psi}_{i_2}\gamma_{\mu}\psi_{i_1}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_3} = 0
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma_{\lambda}\psi_{i_2}\bar{\psi}_{i_2}\gamma_{\mu}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_1} =
- \frac{3i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3} + \\
+ \frac{3i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}  -\\
& - \frac{9i}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3}
- \frac{i}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} -\\
& - \frac{9i}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
+ \frac{i}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
- \frac{1}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\gamma_{\lambda}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_{\mu}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_2} =  - \frac{3i}{2}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\gamma_{\lambda}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma_{\mu}\psi_{i_1}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_3} =  0
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}\gamma^5\gamma_{\lambda}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma_{\mu}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\lambda\mu}\psi_{i_1} =
\frac{3i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}  -\\
& - \frac{3i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}
+\frac{1}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} + \\
+ \frac{i}{16}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} + \\
- \frac{i}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \frac{3i}{16}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^{\kappa}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^{\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_2} =
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^{\kappa}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma^{\lambda}\psi_{i_1}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} =
\frac{i}{2}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^{\kappa}\psi_{i_2}\bar{\psi}_{i_2}\gamma^5\gamma^{\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_1} =
-\frac{3}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3} + \\
+ \frac{9}{4}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3}  -\\
+ \frac{1}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
-\frac{15}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} + \\
&+\frac{1}{4}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} +
\frac{i}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\gamma^{\kappa}\psi_{i_1}\bar{\psi}_{i_2}\gamma^{\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_2} =
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\gamma^{\kappa}\psi_{i_2}\bar{\psi}_{i_2}\gamma^{\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_1} =
\frac{3}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
-\frac{3}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} -\\
-\frac{1}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} - \\
& - \frac{i}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
-\frac{9}{8}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\gamma^\mu\psi_{i_3} + \\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_2} =  - 12i\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_2}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_1}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} =
-\frac{1}{2}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3} + \\
- i\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^\kappa\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\gamma^\lambda\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \bar{\psi}_{i_1}{\sigma_\kappa}^\lambda\psi_{i_1}\bar{\psi}_{i_2}{\sigma_\lambda}^\mu\psi_{i_2}\bar{\psi}_{i_3}{\sigma_\mu}^\kappa\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_2}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_3}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_1} =
\frac{9i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3} + \\
+ \frac{3i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
+ i\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  -\\
- \frac{3i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma_\mu\psi_{i_2}\bar{\psi}_{i_3}\gamma^\mu\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}{\sigma_{\kappa}}^{\lambda}\psi_{i_1}\bar{\psi}_{i_2}{\sigma_{\lambda}}^{\mu}\psi_{i_3}\bar{\psi}_{i_3}{\sigma_{\mu}}^{\kappa}\psi_{i_2} =
\bar{\psi}_{i_1}{\sigma_{\kappa}}^{\lambda}\psi_{i_2}\bar{\psi}_{i_2}{\sigma_{\lambda}}^{\mu}\psi_{i_1}\bar{\psi}_{i_3}{\sigma_{\mu}}^{\kappa}\psi_{i_3} =
\bar{\psi}_{i_1}{\sigma_{\kappa}}^{\lambda}\psi_{i_3}\bar{\psi}_{i_2}{\sigma_{\lambda}}^{\mu}\psi_{i_3}\bar{\psi}_{i_3}{\sigma_{\mu}}^{\kappa}\psi_{i_1} = 0
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}{\sigma_{\kappa}}^{\lambda}\psi_{i_2}\bar{\psi}_{i_2}{\sigma_{\lambda}}^{\mu}\psi_{i_3}\bar{\psi}_{i_3}{\sigma_{\mu}}^{\kappa}\psi_{i_1} =
\frac{i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
+ \frac{1}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  -\\
& - \frac{9i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}
- \frac{3i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\bar{\psi}_{i_1}{\sigma_{\lambda}}^{\mu}\psi_{i_2}\bar{\psi}_{i_2}{\sigma_{\kappa}}^{\lambda}\psi_{i_3}\bar{\psi}_{i_3}{\sigma_{\mu}}^{\kappa}\psi_{i_1} =
\frac{i}{4}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\sigma_{\mu\nu}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}
- \frac{1}{8}\epsilon_{\kappa\lambda\mu\nu}\bar{\psi}_{i_1}\gamma^5\psi_{i_1}\bar{\psi}_{i_2}\sigma^{\kappa\lambda}\psi_{i_2}\bar{\psi}_{i_3}\sigma^{\mu\nu}\psi_{i_3}  +\\
& + \frac{9i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\gamma^5\psi_{i_2}\bar{\psi}_{i_3}\gamma^5\psi_{i_3}
+ \frac{3i}{2}\bar{\psi}_{i_1}\psi_{i_1}\bar{\psi}_{i_2}\psi_{i_2}\bar{\psi}_{i_3}\psi_{i_3}
\end{split}
\end{equation}

[1]
A. Svidzinskyi, Spatially inhomogeneous propblems in the theory of
Moscow: Nauka, 1982.
(in Russian).

[2]
S. Hollands and R. M. Wald, “Axiomatic quantum field theory in curved
spacetime,” Communications in Mathematical Physics, vol. 293,
pp. 85–125, sep 2009.

[3]
Y. Nambu and G. Jona-Lasinio, “Dynamical model of elementary particles based
on an analogy with superconductivity. I,” Phys. Rev., vol. 122,
pp. 345–358, Apr 1961.

[4]
K. Kikkawa, “Quantum corrections in superconductor models,” Progress of
Theoretical Physics, vol. 56, pp. 947–955, 09 1976.

[5]
D. Ebert and H. Reinhardt, “Functional approach to nuclear field theory: A
schematic model with pairing and particle-hole forces,” Nuclear Physics
A, vol. 298, no. 1, pp. 60–76, 1978.

[6]
H. Reinhardt, “Nuclear field theory,” Nuclear Physics A, vol. 298,
no. 1, pp. 77–92, 1978.

[7]
M. K. Volkov, “Meson lagrangians in a superconductor quark model,” Annals Phys., vol. 157, pp. 282–303, 1984.

[8]
H. Terazawa, Y. Chikashige, and K. Akama, “Unified model of the
Nambu-Jona-Lasinio type for all elementary-particle forces,” Phys. Rev. D, vol. 15, pp. 480–487, Jan 1977.

[9]
R. Jackiw, “Functional evaluation of the effective potential,” Phys.
Rev. D, vol. 9, p. 1686, 1974.

[10]
J. M. Cornwall, R. Jackiw, and E. Tomboulis, “Effective action for composite
operators,” Phys. Rev. D, vol. 10, pp. 2428–2445, Oct 1974.

[11]
V. Gusynin and V. Miransky, “Nonperturbative scale anomaly in gauge
theories,” Zh. Eksp. Teor. Fiz, vol. 95, pp. 410–427, 1989.

[12]
V. P. Gusynin, V. A. Kushnir, and V. A. Miransky, “Green's functions of
composite operators and bound states in gauge theories,” Phys. Rev. D,
vol. 39, pp. 2355–2367, Apr 1989.

[13]
V. Gusynin and V. Miransky, “On the effective action in field theories with
dynamical symmetry breaking,” Modern Physics Letters A, vol. 06, 11

[14]
V. Gusynin and V. Miransky, “Effective action in the nambu-jona-lasinio gauge
model,” Zh. Eksp. Teor. Fiz, vol. 101, pp. 414–430, 1992.

[15]
E. Calzetta and B. L. Hu, “Nonequilibrium quantum fields: Closed-time-path
effective action, Wigner function, and Boltzmann equation,” Phys.
Rev. D, vol. 37, pp. 2878–2900, May 1988.

[16]
C. Anastopoulos, B.-L. Hu, and K. Savvidou, “Towards a field-theory based
relativistic quantum information,” Journal of Physics: Conference
Series, vol. 2533, p. 012004, jun 2023.

[17]
M. Fierz, “Zur fermischen theorie des $\beta$-zerfalls,” Zeitschrift
für Physik, vol. 104, pp. 553–565, 1937.

[18]
C. Itzykson and J. Zuber, Quantum Field Theory.
Dover Books on Physics, Dover Publications, 2012.

[19]
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications.
Cambridge University Press, 8 2013.

[20]
O. I. Zavyalov, Renormalized quantum field theory.

[21]
R. F. Streater, A. S. Wightman, PCT, spin and statistics, and all that.
W.A. Benjamin, New York, 1964.

[22]
R. Haag, “On quantum field theories,” Kong. Dan. Vid. Sel. Mat. Fys.
Med., vol. 29N12, pp. 1–37, 1955.

[23]
K. Hepp, “Proof of the Bogoliubov-Parasiuk theorem on renormalization,”
Communications in Mathematical Physics, vol. 2, no. 4, pp. 301 – 326,

[24]
W. Zimmermann, “Convergence of Bogoliubov's method of renormalization in
momentum space,” Communications in Mathematical Physics, vol. 15,
no. 3, pp. 208 – 234, 1969.

[25]
H. Weyl, The classical groups: their invariants and representations.
No. 1, Princeton university press, 1946.

[26]
H. Kraft and C. Procesi, A Primer of Classical Invariant Theory:
Preliminary Version.

[27]
H. Haber, “Useful relations among the generators in the defining and adjoint
representations of SU(n),” SciPost Physics Lecture Notes, jan

[28]
P. B. Pal, “Representation-independent manipulations with dirac matrices and
spinors,” 2007.

[29]
J. Lowenstein and W. Zimmermann, “On the formulation of theories with
zero-mass propagators,” Nuclear Physics B, vol. 86, no. 1,
pp. 77–103, 1975.

[30]
J. H. Lowenstein, “Convergence Theorems for Renormalized Feynman Integrals
with Zero-Mass Propagators,” Commun. Math. Phys., vol. 47,
pp. 53–68, 1976.

[31]
D. N. Blaschke, F. Gieres, F. Heindl, M. Schweda, and M. Wohlgenannt, “BPHZ
renormalization and its application to non-commutative field theory,” The European Physical Journal C, vol. 73, sep 2013.

[32]
K. I. Kondo, “Vacuum condensates, effective gluon mass and color confinement
in a new reformulation of qcd,” 2003.

[33]
B. L. Ioffe, “Condensates in quantum chromodynamics,” Phys. Atom.
Nucl., vol. 66, pp. 30–43, 2003.

[34]
S. Kutnii, “Dirac: a command-line $\gamma$-matrix calculator,” Computer
Physics Communications, vol. 294, to be published in 2024.