# A Fixed-Point Theorem For Mapping Satisfying a General Contractive Condition
Of Integral Type Depended an Another Function 1112000 Mathematics Subject
Classification: Primary 46J10, 46J15, 47H10.

S. Moradi222First author and A. Beiranvand

Faculty of Science, Department of Mathematics
Arak University, Arak, Iran

###### Abstract

In this paper, we study the existence of fixed points for mappings defined on
complete metric space ($X,d$) satisfying a general contractive inequality of
integral type depended on another function. This conditions is analogous of
Banach conditions and Branciari Theorem.

Keywords: Fixed point, contraction mapping, contractive mapping, sequently
convergent, subsequently convergent, integral type.

##  1 Introduction

The first important result on fixed points for contractive-type mapping was
the well-known Banach’s Contraction Principle appeared in explicit form in
Banach’s thesis in 1922, where it was used to establish the existence of a
solution for an integral equation [1]. In the general setting of complete
metric space this theorem runs as follows(see[5,Theorem 2.1]
or[10,Theorem1.2.2]).

######  Theorem 1.1.

$($Banach’s Contraction Principle$)$ Let $(X,d)$ be a complete metric space
and $f:X\longrightarrow X$ be a contraction $($there exists $k\in(0,1)$ such
that for each $x,y\in X$; $d(fx,fy)\leq kd(x,y)$$)$. Then $f$ has a unique
fixed point in $X$, and for each $x_{0}\in X$ the sequence of iterates
$\\{f^{n}x_{0}\\}$ converges to this fixed point.

After this classical result Kannan in [4] analyzed a substantially new type of
contractive condition. Since then there have been many theorems dealing with
mappings satisfying various types of contractive inequalities. Such conditions
involve linear and nonlinear expressions (rational, irrational, and of general
type). The intrested reader who wants to know more about this matter is
recommended to go deep into the survey articles by Rhoades [7,8,9] and
Meszaros [6], and into the references therein. Another result on fixed points
for contractive-type mapping is generally attributed to Edelstein (1962) who
actually obtained slightly more general versions. In the general setting of
compact metric spaces this result runs as followes (see [5, Theorem 2.2]).

######  Theorem 1.2.

Let $(X,d)$ be a compact metric space and $f:X\longrightarrow X$ be a
contractive $($for every $x,y\in X$ such that $x\neq y$; $d(fx,fy)<d(x,y)$$)$.
Then $f$ has a unique fixed point in $X$, and for any $x_{0}\in X$ the
sequence of iterates $\\{f^{n}x_{0}\\}$ converges to this fixed point.

Also in 2002 in [3] A. Branciari analyzed the existence of fixed point for
mapping $f$ defined on a complete metric space $(X,d)$ satisfying a
contractive condition of integral type.(see the following theorem).

######  Theorem 1.3.

Let $(X,d)$ be a complete metric space, $\alpha\in(0,1)$ and
$f:X\longrightarrow X$ be a mapping such that for each $x,y\in X$,
$\int_{0}^{d(fx,fy)}\phi(t)dt\leq\alpha{\int_{0}^{d(x,y)}\phi(t)dt}$, where
$\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping
which is summable (i.e., with finite integral) on each compact subset of
$[0,+\infty)$, nonnegative, and such that for each
$\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$; then $f$ has a unique fixed point
$a\in X$ such that for each $x\in X$,
$\underset{n\rightarrow\infty}{\lim}f^{n}x=a$.

The aim of this paper is to study the existence of fixed point for mapping $f$
defined on a compact metric space$(X,d)$ such that is
$T_{\int\phi}-contraction$. In particular, we extend the main theorem due to
A. Branciari [3] (Theorem 1.3) and the main theorem in [2] (2008). First we
introduce the $T_{\int\phi}-contraction$ function and then extended the
A.Branciari Theorem and the main theorem in [2] and Banach-contraction
principle, by the same metod for proof of the A. Branciari Theorem. At the end
of paper some examples and applications concerning this kind of contractions.
In [3] A. Branciari gave an example (Example 3.6) such that we can conclude
this example by theorem 1.2. (because
$X=\\{1/n:n\in\mathbb{N}\\}\bigcup\\{0\\}$, with metric induced by
$\mathbb{R}$, $d(x,y)=|x-y|$, is a compact metric space and $f$ is a
contractive mapping). In the end of this paper we give an example (Example
3.5) such that we can not conclude this example by Theorem 1.1, Theorem 1.2.
Branciari Theorem and the main theorem in [2], but we can conclude this
example by the main theorem (Theorem 2.5 ) in this paper. In the sequel,
$\mathbb{N}$ will represent the set of natural numbers, $\mathbb{R}$ the set
of real number and $\mathbb{R}^{+}$ the set of nonnegative real number.

##  2 Definitions and Main Result

The following theorem (Theorem 2.5) is the main result of this paper. In the
first, we define some new definitions.

######  Definition 2.1.

Let $(X,d)$ be a metric space and $f,T:X\longrightarrow X$ be two functions
and $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ be a Lebesgue-integrable
mapping. A mapping $f$ is said to be a $T_{\int\phi}-contraction$ if there
exists $\alpha\in(0,1)$ such that for all $x,y\in X$

$\int_{0}^{d(Tfx,Tfy)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}$

######  Remark 2.2.

By taking $Tx=x$ and $\phi=1$, $T_{\int\phi}-contraction$ and contraction are
equivalent. Also by taking $Tx=x$ we can define $\int\phi-contraction$.

######  Example 2.3.

Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}$: $d(x,y)=|x-y|$. We
consider two mappings $T,f:X\longrightarrow X$ by $Tx=\frac{1}{x}+1$ and
$fx=2x$. Obviously $f$ is not contraction but $f$ is
$T_{\int{1}}-contraction$.

######  Definition 2.4.

$[2]$ Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said
sequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if
$\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ also is convergence. $T$ is
said subsequentially convergent if we have, for every sequence $\\{y_{n}\\}$,
if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ has a convergent
subsequence.

######  Theorem 2.5.

$[$Main theorem $]$ Let $(X,d)$ be a complete metric space, $\alpha\in(0,1)$,
$T,f:X\longrightarrow X$ be mapping such that $T$ is continuous, one-to-one
and subsequentially convergent and $f$ is $T_{\int\phi}-contraction$ where
$\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping
which is summable on each compact subset of $[0,+\infty)$, nonnegative and
such that for each $\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$; then $f$ has a
unique fixed point $a\in X$. Also if $T$ is sequentially convergent, then for
each $x_{0}\in X$, the sequence of iterates $\\{f^{n}x_{0}\\}$ converges to
this fixed point.

###### Proof.

STEP 1. Let $\alpha\in(0,1)$ such that for all $x,y\in X$

$\int_{0}^{d(Tfx,Tfy)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}.\hskip
56.9055pt(2.1)$

So if for $a,b>0$, $\int_{0}^{a}\phi(t)dt\leq\alpha{\int_{0}^{b}\phi(t)dt}$
then $a<b.$

STEP 2. We show that $f$ is a continuouse mapping.
If $\underset{n\rightarrow\infty}{\lim}x_{n}=x$ then by
$\int_{0}^{d(Tfx_{n},Tfx)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx_{n},Tx)}\phi(t)dt}$
and $\underset{n\rightarrow\infty}{\lim}d(Tx_{n},Tx)=0$, we conclude that:

$\lim_{n\rightarrow\infty}d(Tfx_{n},Tfx)=0.$

Since $T$ is subsequentially convergent, $\\{fx_{n}\\}$ has a subsequence such
$\\{{fx_{n}}_{k}\\}_{k=1}^{\infty}$ converge to a $y\in X$. So $d(Ty,Tfx)=0$.
Since $T$ is one-to-one, $y=fx$. Hence, $\\{fx_{n}\\}$ has a subsequence
converge to $fx$.
Therefore for every sequence $\\{x_{n}\\}$ converge to $x$, the sequence
$\\{fx_{n}\\}$ has a subsequence converge to $fx$. This shows that $f$ is
continuouse at $x$.

STEP 3. Since (2.1) is holds, for all $n\in\mathbb{N}:$

$\int_{0}^{d(Tf^{n+1}x,Tf^{n}x)}\phi(t)dt\leq\alpha^{n}{\int_{0}^{d(Tfx,Tx)}\phi(t)dt}\qquad\forall
x\in X.$

As a consequence, since $\alpha\in(0,1)$, we further have

$\int_{0}^{d(Tf^{n+1}x,Tf^{n}x)}\phi(t)dt\rightarrow 0^{+}\hskip
14.22636ptas\hskip 14.22636ptn\rightarrow\infty\hskip 56.9055pt(2.2)$

Since

$\hskip 14.22636pt\int_{0}^{\epsilon}\phi(t)dt>0,\hskip
14.22636pt\forall\epsilon>0\hskip 56.9055pt(2.3)$

is holds we conclude that

$\lim_{n\rightarrow\infty}d(Tf^{n+1}x,Tf^{n}x)=0\hskip 85.35826pt(2.4)$

Step 4. $\\{Tf^{n}x\\}$ is a bounded sequence.
If $\\{Tf^{n}x\\}_{n=1}^{\infty}$is not a bounded sequence then, we choose the
sequence $\\{n_{k}\\}_{k=1}^{\infty}$ such that $n_{1}=1$ and for each
$k\in\mathbb{N}$, $n_{k+1}$ is ”minimal” in the sense that

$d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)>1.$

So,

$\displaystyle 1$ $\displaystyle<$ $\displaystyle
d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)$ $\displaystyle\leq$ $\displaystyle
d(Tf^{n_{k+1}}x,Tf^{n_{k+1}-1}x)+d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)$
$\displaystyle\leq$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k+1}-1}x)+1.\hskip
71.13188pt(2.5)$

Hence, by (2.4) and (2.5) we conclude that

$d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)\rightarrow 1\hskip 14.22636ptas\hskip
14.22636ptk\rightarrow\infty\hskip 56.9055pt(2.6)$

Also by step 1,

$d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)\leq d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x).$

Therefore,

$\displaystyle 1-d(Tf^{n_{k}+1}x,Tf^{n_{k}}x)$ $\displaystyle<$ $\displaystyle
d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)-d(Tf^{n_{k}+1}x,Tf^{n_{k}}x)$ $\displaystyle\leq$
$\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)$ $\displaystyle\leq$
$\displaystyle d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)$ $\displaystyle\leq$
$\displaystyle 1.$

Hence, by (2.4),

$d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)\rightarrow 1\hskip 14.22636ptas\hskip
14.22636ptk\rightarrow\infty.\hskip 56.9055pt(2.7)$

Therefore,

$\displaystyle\int_{0}^{d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)}\phi(t)dt$
$\displaystyle\leq$
$\displaystyle\alpha{\int_{0}^{d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)}\phi(t)dt}$
$\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{1}\phi(t)dt}.\hskip
56.9055pt(2.8)$

By (2.7) and (2.8) we conclude that

$\displaystyle\int_{0}^{1}\phi(t)dt$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\int_{0}^{d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)}\phi(t)dt$
$\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{1}\phi(t)dt}.$

So $\int_{0}^{1}\phi(t)dt=0$ and this is contradiction.

STEP 5. By (2.1) for every $m,n\in\mathbb{N}(m>n)$,

$\int_{0}^{d(Tf^{m}x,Tf^{n}x)}\phi(t)dt\leq\alpha^{n}{\int_{0}^{d(Tf^{m-n}x,Tx)}\phi(t)dt}.\hskip
56.9055pt(2.9)$

By step 4, (2.9) and $\alpha\in(0,1),$

$\underset{m,n\rightarrow\infty}{\lim}\int_{0}^{d(Tf^{m}x,Tf^{n}x)}=0\hskip
56.9055pt(2.10)$

Since (2.3) is hold
$\underset{m,n\rightarrow\infty}{\lim}d(Tf^{m}x,Tf^{n}x)=0$, and this shows
that $\\{Tf^{n}x\\}_{n=1}^{\infty}$ is a Cauchy sequence. Hence there exists
$a\in X$ such that

$\underset{n\rightarrow\infty}{\lim}Tf^{n}x=a\hskip 56.9055pt(2.11)$

STEP 6. Since $T$ is a subsequentially convergent, $\\{f^{n}x\\}$ has a
convergent subsequence.
So there exists $b\in X$ and $\\{n_{k}\\}_{k=1}^{\infty}$ such that
$\underset{k\rightarrow\infty}{\lim}f^{n_{k}}x=b$. Since $T$ is continuouse
$\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}}x=Tb$, and by (2.11) we conclude
that

$Tb=a.\hskip 56.9055pt(2.12)$

Since $f$ is continuouse (step 2) and
$\underset{k\rightarrow\infty}{\lim}f^{n_{k}}x=b,\underset{k\rightarrow\infty}{\lim}f^{n_{k}+1}x=fb$
and so $\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}+1}x=Tfb.$
Again by (2.11) we have

$\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}+1}x=a$

and therefore, $Tfb=a.$ So by (2.12), $Tfb=Tb.$ Since $T$ is one-to one,
$fb=b.$ Therefore $f$ has a fixed point.
STEP 7. Since $T$ is one-to-one and $f$ is $T_{\int\phi}-contraction$, $f$ has
a unique fixed point. ∎

##  3 Examples and Applications

In this section, we give some applications and some examples concerning these
contractive mapping of integral type, which clarify the connection between our
result and the classical ones.

######  Remark 3.1.

Theorem 2.5 is a generalization of the Banach’s contraction principle (Theorem
1.1), letting $\phi(t)=1$ for each $t\geq 0$ and $Tx=x$ for each $x\in X$ in
Theorem 2.5, we have

$\displaystyle\int_{0}^{d(Tfx,Tfy)}\phi(t)dt$ $\displaystyle=$ $\displaystyle
d(fx,fy)$ $\displaystyle\leq$ $\displaystyle\alpha d(x,y)$ $\displaystyle=$
$\displaystyle\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}$

######  Remark 3.2.

Theorem 2.5 is a generalization of the A. Branciari theorem (Theorem 1.3),
letting $Tx=x$ for each $x\in X$ in Theorem 2.5, so

$\displaystyle\int_{0}^{d(Tfx,Tfy)}\phi(t)dt$ $\displaystyle=$
$\displaystyle\int_{0}^{d(fx,fy)}\phi(t)dt$ $\displaystyle\leq$
$\displaystyle\alpha{\int_{0}^{d(x,y)}\phi(t)dt}$ $\displaystyle=$
$\displaystyle\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}.$

We can conclude the following theorem ( the main Theorem in [2]) by Theorem
2.5.

######  Theorem 3.3.

Let $(X,d)$ be a complete metric space and $T:X\longrightarrow X$ be a one-to-
one, continuouse and subsequentially convergent mapping. Then for every
$T-contraction$ function $f:X\longrightarrow X$, $f$ has a unique fixed point.
Also if $T$ is sequentially convergent, then for each $x_{0}\in X$, the
sequence of iterates $\\{f^{n}x\\}$ converges to this fixed point.
($f:X\longrightarrow X$ is $T-contraction$ if there exist $\alpha\in(0,1)$
such that for all $x,y\in X$

$d(Tfx,Tfy)\leq\alpha{d(Tx,Ty)}.)$

###### Proof.

By taking $\phi(t)=1$ for each $t\in[0,+\infty)$ in Theorem 2.5 we can
conclude this theorem.
∎

######  Example 3.4.

Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}:d(x,y)=|x-y|,$ thus,
since $X$ is a closed subset of $\mathbb{R},$ it is a complete metric space.
we define $T,f:X\longrightarrow X$ by $Tx=\ln{x}+1$ and $fx=k\sqrt{x}$ such
that $k\geq 1$ be a fixed element of $\mathbb{R}.$ Obviousely $f$ is not
contraction, but $f$ is $T_{\int 1}-contraction$ and $T$ is one-to-one,
continuouse and sequentially convergent. So $f$ has a unique fixed point by
Theorem 2.5.

The following example is the main example of this paper. In the following we
show that, we can not conclude this example by Theorem 1.1, Theorem 1.2,
Theorem 1.3 (Branciari Theorem) and Theorem 3.3.

######  Example 3.5.

Let $X:={\\{\frac{1}{n}\ \ |\ \ n\in\mathbb{N}\\}}\bigcup{\\{0}\\}$ with
metric induced by $\mathbb{R}:d(x,y):=|x-y|$, thus, since $X$ is a closed
subset of $\mathbb{R}$, it is a complete metric space. We consider a mapping
$f:X\longrightarrow X$ defined by

$fx=\left\\{\begin{array}[]{ll}\frac{1}{n+3}&;x=\frac{1}{n},\>n\>is\>odd\\\
0&;x=0\\\ \frac{1}{n-1}&;x=\frac{1}{n},\>n\>is\>even\end{array}\right.$

and defined $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ by

$\phi(t)=\left\\{\begin{array}[]{ll}t^{{\frac{1}{t}}-2}[1-\log{t}]&;t>0\\\
0&;t=0\end{array}\right.$

we have $\int_{0}^{\tau}\phi(t)dt=\tau^{\frac{1}{\tau}}.$
By taking $n=2$ and $m=4$, $|f(1/m)-f(1/n)|>|1/m-1/n|$, so $f$ is not
contraction and contractive. Hence, we can not conclude that, $f$ has a fixed
point by Theorem 1.1 and Theorem 1.2.
Now we show that we can not use Branciari Theorem for this example. For
$x=1/m$, $y=1/n$ where $m$ and $n$ are even if

$\int_{0}^{|fx-fy|}\phi(t)dt\leq\alpha{\int_{0}^{|x-y|}\phi(t)dt}$

then

$|\frac{1}{m-1}-\frac{1}{n-1}|^{\frac{1}{|\frac{1}{m-1}-\frac{1}{n-1}|}}\leq\alpha{|\frac{1}{m}-\frac{1}{n}|^{\frac{1}{|\frac{1}{m}-\frac{1}{n}|}}}$

$\Rightarrow\hskip
28.45274pt|\frac{m-n}{(m-1)(n-1)}|^{|\frac{(m-1)(n-1)}{m-n}|}\leq\alpha{|\frac{m-n}{mn}|^{|\frac{mn}{m-n}|}}$

For $m=4$ and $n=2$ we conclude that $1<\alpha.$ So we can not use Branciari
Theorem.
Now we defined $T:X\longrightarrow X$ by

$Tx=\left\\{\begin{array}[]{ll}\frac{1}{n-1}&;x=\frac{1}{n},\>n\>is\>even\\\
0&;x=0\\\ \frac{1}{n+1}&;x=\frac{1}{n},\>n\>is\>odd\end{array}\right.$

Obviously $T$ is one-to-one and sequentially convergent and continuouse.
we have

$Tfx=\left\\{\begin{array}[]{ll}\frac{1}{n+2}&;x=\frac{1}{n},\>n\>is\>odd\\\
0&;x=0\\\ \frac{1}{n}&;x=\frac{1}{n},\>n\>is\>even\end{array}\right.$

Since $\sup{\frac{|Tfx-Tfy|}{|Tx-Ty}|}=1$, $f$ is not $T-contraction$, and so
we can not use Theorem 3.3 for this example. Now we show that the condition of
Theorem 2.5 are holds. We show that $f$ is $T_{\int\phi}-contraction$ and

$\int_{0}^{|Tfx-Tfy|}\phi(t)dt\leq{\frac{1}{2}\int_{0}^{|Tx-
Ty|}\phi(t)dt}\hskip 14.22636ptfor\ \ all\ \ x,y\in X.\hskip 56.9055pt(2.13)$

Case 1. Let $x=\frac{1}{m},y=\frac{1}{n}$ and $m$ and $n$ are even. Then

$\displaystyle\qquad\qquad\int_{0}^{|Tfx-
Tfy|}\phi(t)dt\leq{\frac{1}{2}\int_{0}^{|Tx-Ty|}\phi(t)dt}$
$\displaystyle\Leftrightarrow\hskip
28.45274pt|\frac{1}{m}-\frac{1}{n}|^{\frac{1}{|\frac{1}{m}-\frac{1}{n}|}}\leq\frac{1}{2}{|\frac{1}{m-1}-\frac{1}{n-1}|^{\frac{1}{|\frac{1}{m-1}-\frac{1}{n-1}|}}}$
$\displaystyle\Leftrightarrow\hskip
28.45274pt|\frac{m-n}{mn}|^{|\frac{mn}{m-n}|}.|\frac{(m-1)(n-1)}{m-n}|^{|\frac{(m-1)(n-1)}{m-n}|}\leq\frac{1}{2}$
$\displaystyle\Leftrightarrow\hskip
28.45274pt|\frac{(m-1)(n-1)}{mn}|^{|\frac{(m-1)(n-1)}{m-n}|}.|\frac{(m-n)}{mn}|^{|\frac{(m+n-1)}{m-n}|}\leq\frac{1}{2}$

Obviously the last inequality is holds, because

$|\frac{(m-1)(n-1)}{mn}|\leq 1\>\>and\>\>|\frac{(m-1)(n-1)}{m-n}|\geq 1$

and so

$|\frac{(m-1)(n-1)}{mn}|^{|\frac{(m-1)(n-1)}{m-n}|}\leq 1,$

and

$|\frac{m-n}{mn}|^{|\frac{m+n-1}{m-n}|}\leq\frac{1}{2}.$

Therefore for this case (2.13) is holds.

Case 2. Let $x=\frac{1}{m},y=\frac{1}{n}$ and $m$ and $n$ are odd.

Case 3. Let $x=\frac{1}{m},y=\frac{1}{n}$, $m$ is even and $n$ is odd.
By the same argument in case 1 we conclude that (2.13) for case 2 and case 3
is holds.

Case 4. Let $x=0,y=\frac{1}{n}$ such that $n$ is even. Then

$\displaystyle\int_{0}^{|Tfx-Tfy|}\phi(t)dt\leq\frac{1}{2}{\int_{0}^{|Tx-
Ty|}\phi(t)dt}$
$\displaystyle\Leftrightarrow(\frac{1}{n})^{n}\leq\frac{1}{2}{(\frac{1}{n-1})^{n-1}}$
$\displaystyle\Leftrightarrow(\frac{1}{n})^{n}(n-1)^{n-1}\leq\frac{1}{2}$
$\displaystyle\Leftrightarrow(\frac{n-1}{n})^{n-1}.\frac{1}{n}\leq\frac{1}{2}.$

The last inequality is holds, because,

$(\frac{n-1}{n})^{n-1}\leq 1\qquad and\qquad\frac{1}{n}\leq\frac{1}{2}.$

Therefore (2.13) is true for this case.
Case 5. Let $x=0\ \ \ y=\frac{1}{n}$ such that $n$ is odd. By the same
argument in case 4 we conclude that, (2.13) is holds for this case.
Hence, (2.13) is holds for all $x,y\in X.$ Therefore the condition of Theorem
2.5 are hold and so $f$ has a unique fixed point.

## References

  * [1] S. Banach, Sur Les Operations Dans Les Ensembles Abstraits et Leur Application Aux E’quations Inte’grales, Fund. Math. 3(1922), 133-181(French).
  * [2] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh , Two Fixed-Point Theorem For Special Mapping, to appear.

  * [3] A. Branciari, A fixed point theorem for mapping satisfying a general contractive condition of integral type Int. J. M. and M. since, 29:9 (2002), 531-536.

  * [4] R. Kannan, Some results on fixed points, Bull.Calcutta Math. Soc. 60(1968),71-76.
  * [5] Kazimierz Goebel and W. A. Kirk, Topiqs in Metric Fixed Point Theory, Combridge University Press, New York, 1990.
  * [6] J. Meszaros, A Comparison of Various Definitions of Contractive Type Mappings, Bull. Calcutta Math. Soc. 84(1992), no. 2, 167-194.
  * [7] B. E. Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc. 226(1977), 257-290.
  * [8] B. E. Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis (Toronto, Ont.,1982), Contemp. Math., Vol. 21, American Mathematical Society, Rhode Island, 1983, pp. 189-203.
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  * [10] O. R. Smart, Fixed Point Theorems, Cambridge University Press, London, 1974.

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# Block-diagonalizable two-dimensional generalized Ising systems (BD2DGIS):
the eigenvalues and eigenvectors

Vadym Sakhno &Mykola Sakhno

###### Abstract

This paper is a continuation of [1] and [2], where the block-diagonalizable
two-dimensional generalized Ising systems (BD2DGIS) were introduced. In this
paper, their eigenvalues, eigenvectors and Jordan normal form are analyzed in
detail using the simplest quantum field model.

##  1 Introduction

Since this paper is a continuation of [1] and [2], the Introduction provides
detailed summaries of [1] and [2].

###  1.1 Summary of [1]

In [1], the Ising model was generalized to a system of cells interacting
exclusively by presence of shared spins. Within the cells there were
interactions of any complexity, the simplest intracell interactions came down
to the Ising model. The approach was developed to constructing the exact
matrix model for any considered system in the simplest way. Using the
approach, the exact matrix model for a two-dimensional generalized Ising model
was constructed. The 2D system under consideration is shown in Figure 1 (see
Figure 1 of [1]).

Figure 1: The 2D system under consideration.

The 2D system of Figure 1 consists of $(N+2)$ cells having spins with two
values $\\{-\frac{1}{2},\frac{1}{2}\\}$: $N$ internal cells numbered from $1$
to $N$, start cell numbered $0$, and finish cell numbered $(N+1)$. $N$
internal cells form $R$ rows. The lowest spin of a column continues into the
highest spin of the next column (see $x_{\left(R+1\right)0}$ in Figure 1).
Thus, cells are placed along a helix. Let the first column be completed, and
the last column may be uncompleted. Each internal cell $n$ has four spins, its
energy is a given function $E_{n}\left(x_{n0},\
x_{n1},x_{(n+1)0},x_{(n+R)1}\right)$. Substituting it into (3) of [1], the
internal cell function $Z_{n}\left(x_{n0},\
x_{n1},x_{(n+1)0},x_{(n+R)1}\right)$ is

$Z_{n}\left(x_{n0},\
x_{n1},x_{(n+1)0},x_{(n+R)1}\right)=\exp{\left(-\frac{E_{n}\left(x_{n0},\
x_{n1},x_{(n+1)0},x_{(n+R)1}\right)}{k_{B}T}\right)}>0.$ (1)

In [1], for each spin $x_{n\nu}\in\\{-\frac{1}{2},\ +\frac{1}{2}\\}$, the
substituting spin-number $i_{n\nu}\in\\{0,1\\}$ was introduced according to
(6) of [1]

$x_{n\nu}=i_{n\nu}-\frac{1}{2}.$ (2)

In the internal cell function (1), the substitution of each spin with its
spin-number yielded the internal cell frame, which was a set of 16 values,
numbered with a compound number of spin-numbers. The internal cells in [1] may
vary, but in this paper they are similar. Therefore, it suffices to consider
the first cell $n=1$. Its frame is (17) of [1]

$Z_{1i_{10}i_{11}i_{20}i_{\left(R+1\right)1}}=Z_{1}\left(i_{10}-\frac{1}{2},\
i_{11}-\frac{1}{2},\
i_{20}-\frac{1}{2},i_{\left(R+1\right)1}-\frac{1}{2}\right).$ (3)

In [1], four $2^{R}\times 2^{R}$ block-diagonal matrices were constructed from
the frame, along the diagonal of which there were identical $2\times 2$
blocks. Their elements with compound row number $i_{1}i_{2}\ldots i_{R}$ were
non-zero only if all row sub-numbers except the last sub-number $R$ were equal
to the corresponding column sub-numbers. To emphasize this, these $2^{R}\times
2^{R}$ block-diagonal matrices were denoted as $2\times 2$ matrices with the
number $[R]$ (see (25) of [1]), for example

$\displaystyle\begin{pmatrix}Z_{10000}&Z_{10100}\\\
Z_{10001}&Z_{10101}\end{pmatrix}_{[R]}=\begin{pmatrix}Z_{10000}&Z_{10100}&0&0&\ldots&0&0\\\
Z_{10001}&Z_{10101}&0&0&\ldots&0&0\\\ 0&0&Z_{10000}&Z_{10100}&\ldots&0&0\\\
0&0&Z_{10001}&Z_{10101}&\ldots&0&0\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
0&0&0&0&\ldots&Z_{10000}&Z_{10100}\\\
0&0&0&0&\ldots&Z_{10001}&Z_{10101}\end{pmatrix}.$ (4)

Also $2^{R}\times 2^{R}$ cyclic shift matrix was introduced in (23) of [1]

$\displaystyle\overleftrightarrow{P_{r}}$
$\displaystyle=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}1&0&\ldots&0&0&\ldots&0&0&\ldots&0&0&\ldots\\\
0&0&\ldots&0&0&\ldots&1&0&\ldots&0&0&\ldots\\\
0&1&\ldots&0&0&\ldots&0&0&\ldots&0&0&\ldots\\\
0&0&\ldots&0&0&\ldots&0&1&\ldots&0&0&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
0&0&\ldots&1&0&\ldots&0&0&\ldots&0&0&\ldots\\\
0&0&\ldots&0&0&\ldots&0&0&\ldots&1&0&\ldots\\\
0&0&\ldots&0&1&\ldots&0&0&\ldots&0&0&\ldots\\\
0&0&\ldots&0&0&\ldots&0&0&\ldots&0&1&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\end{pmatrix}.$
(5)

Then the sought-for $2^{R+1}\times 2^{R+1}$ internal cell matrix was
represented in (24) of [1] as

$\overleftrightarrow{Z_{1}}=\begin{pmatrix}\begin{pmatrix}Z_{10000}&Z_{10100}\\\
Z_{10001}&Z_{10101}\end{pmatrix}_{[R]}&\begin{pmatrix}Z_{11000}&Z_{11100}\\\
Z_{11001}&Z_{11101}\end{pmatrix}_{[R]}\\\
\begin{pmatrix}Z_{10010}&Z_{10110}\\\
Z_{10011}&Z_{10111}\end{pmatrix}_{[R]}&\begin{pmatrix}Z_{11010}&Z_{11110}\\\
Z_{11011}&Z_{11111}\end{pmatrix}_{[R]}\end{pmatrix}*\begin{pmatrix}\overleftrightarrow{P_{r}}&0\\\
0&\overleftrightarrow{P_{r}}\end{pmatrix},$ (6)

As shown in Figure 1, the start cell energy is $E_{0}\left(x_{10},\ x_{11},\
x_{21},\ldots,x_{R1}\right)$ and the finish cell energy is
$E_{N+1}\left(x_{(N+1)0},x_{(N+1)1},x_{(N+1)2},\ldots,x_{(N+1)R}\right)$,
which can be any given functions of $(R+1)$ spins. Substituting them into (3)
of [1] gives the start cell function and the finish cell function (see (1) for
an internal cell). Then substituting each spin with its spin-number according
to (2) gives the start cell frame $Z_{0i_{10}i_{11}i_{21}\ldots i_{R1}}$ and
the finish cell frame $Z_{(N+1)i_{(N+1)0}\ldots i_{(N+1)(R-1)}i_{(N+1)R}}$ of
$2^{R+1}$ values each. The start cell vector $\overleftarrow{Z_{0}}$ is
$2^{R+1}$ column vector of the start cell frame values and the finish cell
vector $\overrightarrow{Z_{N+1}}$ is $2^{R+1}$ row vector of the finish cell
frame values, let them be called boundary conditions

$\overleftarrow{Z_{0}}=\overleftarrow{\|Z_{0i_{10}i_{11}i_{21}\ldots
i_{R1}}\|}\
,\quad\overrightarrow{Z_{N+1}}=\overrightarrow{\|Z_{(N+1)i_{(N+1)0}\ldots
i_{(N+1)(R-1)}i_{(N+1)R}}\|}\ .$ (7)

Then the resulting exact partition function $Z$ for $N$ identical internal
cell matrices $\overleftrightarrow{Z_{1}}$ was (9) in [1]

$Z=\overrightarrow{Z_{N+1}}*{\overleftrightarrow{Z_{1}}}^{N}*\overleftarrow{Z_{0}}.$
(8)

###  1.2 Summary of [2]

[2] is a continuation of [1].

The properties of internal cell matrix (6) for light block-diagonalization
were specified in (9) of [2]

$\displaystyle\begin{pmatrix}Z_{10000}&Z_{11000}\\\
Z_{10010}&Z_{11010}\end{pmatrix},\quad\begin{pmatrix}Z_{10100}&Z_{11100}\\\
Z_{10110}&Z_{11110}\end{pmatrix},\quad\begin{pmatrix}Z_{10001}&Z_{11001}\\\
Z_{10011}&Z_{11011}\end{pmatrix},\quad\begin{pmatrix}Z_{10101}&Z_{11101}\\\
Z_{10111}&Z_{11111}\end{pmatrix}$ (9) $\displaystyle are\quad
diagonalized\quad by\quad the\quad same\quad similarity\quad transformation.$

.

And an example of BD2DGIS was given in Table 1 and Figure 2 of [2].

The exact internal cell matrix (6) for the example was obtained in (11) of [2]

$\overleftrightarrow{Z_{1}}=\zeta*\begin{pmatrix}\begin{pmatrix}13&69\\\
68&64\end{pmatrix}_{[R]}&\begin{pmatrix}4&22\\\ 24&22\end{pmatrix}_{[R]}\\\
\begin{pmatrix}4&22\\\ 24&22\end{pmatrix}_{[R]}&\begin{pmatrix}7&36\\\
32&31\end{pmatrix}_{[R]}\end{pmatrix}*\begin{pmatrix}\overleftrightarrow{P_{r}}&0\\\
0&\overleftrightarrow{P_{r}}\end{pmatrix},$ (10)

where

$\zeta=\cfrac{1}{2*\sqrt[4]{11}*\sqrt[16]{2^{13}*3^{5}*1103011}}\approx
0.0465.$ (11)

Then $2^{R+1}\times 2^{R+1}$ matrix $\overleftrightarrow{S_{0}}$ was
introduced in (19) of [2]

$\overleftrightarrow{S_{0}}=\begin{pmatrix}2*\overleftrightarrow{1}&-\overleftrightarrow{1}\\\
\overleftrightarrow{1}&2*\overleftrightarrow{1}\end{pmatrix},$ (12)

where $\overleftrightarrow{1}$ is $2^{R}\times 2^{R}$ identity matrix.

And a similarity transformation with matrix $\overleftrightarrow{S_{0}}$ of
(12) was performed over the internal cell matrix $\overleftrightarrow{Z_{1}}$
of (10) in (20) of [2]

$\overleftrightarrow{Z_{1S_{0}}}={\overleftrightarrow{S_{0}}}^{-1}*\overleftrightarrow{Z_{1}}*\overleftrightarrow{S_{0}}=\begin{pmatrix}\overleftrightarrow{B_{0}}_{[R]}&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\overleftrightarrow{B_{1}}_{[R]}\end{pmatrix}*\begin{pmatrix}\overleftrightarrow{P_{r}}&0\\\
0&\overleftrightarrow{P_{r}}\end{pmatrix},$ (13)

where

$\overleftrightarrow{B_{0}}_{[R]}=\zeta*\begin{pmatrix}15&80\\\
80&75\end{pmatrix}_{[R]},\quad\overleftrightarrow{B_{1}}_{[R]}=\zeta*\begin{pmatrix}5&25\\\
20&20\end{pmatrix}_{[R]}.$ (14)

The resulting matrix $\overleftrightarrow{Z_{1S_{0}}}$ is the product of two
matrices, each of which is block-diagonal of two $2^{R}\times 2^{R}$ blocks.

Then $2^{R}\times 2^{R}$ matrices were introduced for any $\rho\in[1,R]$ in
(30) of [2]

$\overleftrightarrow{S_{1}}_{[\rho]}=\begin{pmatrix}-3+\sqrt{73}&-3-\sqrt{73}\\\
8&8\end{pmatrix}_{[\rho]},\quad\overleftrightarrow{S_{2}}_{[\rho]}=\begin{pmatrix}-3+\sqrt{89}&-3-\sqrt{89}\\\
8&8\end{pmatrix}_{[\rho]}.$ (15)

Similarity transformation with matrices $\overleftrightarrow{S_{1}}_{[\rho]}$
and $\overleftrightarrow{S_{2}}_{[\rho]}$ of (15) over matrices
$\overleftrightarrow{B_{0}}_{[\rho]}$ and
$\overleftrightarrow{B_{1}}_{[\rho]}$ of (14) diagonalized the latter in (31)
of [2]

$\displaystyle\overleftrightarrow{B_{0}}_{[\rho]}=$
$\displaystyle\overleftrightarrow{S_{1}}_{[\rho]}*\left({\overleftrightarrow{S_{1}}_{[\rho]}}^{-1}*\overleftrightarrow{B_{0}}_{[\rho]}*\overleftrightarrow{S_{1}}_{[\rho]}\right)*{\overleftrightarrow{S_{1}}_{[\rho]}}^{-1}=\lambda_{1}*\overleftrightarrow{S_{1}}_{[\rho]}*\begin{pmatrix}1&0\\\
0&\lambda_{21}\end{pmatrix}_{[\rho]}*{\overleftrightarrow{S_{1}}_{[\rho]}}^{-1},$
(16) $\displaystyle\overleftrightarrow{B_{1}}_{[\rho]}=$
$\displaystyle\overleftrightarrow{S_{2}}_{[\rho]}*\left({\overleftrightarrow{S_{2}}_{[\rho]}}^{-1}*\overleftrightarrow{B_{1}}_{[\rho]}*\overleftrightarrow{S_{2}}_{[\rho]}\right)*{\overleftrightarrow{S_{2}}_{[\rho]}}^{-1}=\lambda_{3}*\overleftrightarrow{S_{2}}_{[\rho]}*\begin{pmatrix}1&0\\\
0&\lambda_{43}\end{pmatrix}_{[\rho]}*{\overleftrightarrow{S_{2}}_{[\rho]}}^{-1},$

where taking into account (11)

$\displaystyle\lambda_{1}=$
$\displaystyle\zeta*\left(45+10\sqrt{73}\right)\approx
6.063,\quad\lambda_{21}=\cfrac{9-2\sqrt{73}}{9+2\sqrt{73}}\approx-0.3100,$
(17) $\displaystyle\lambda_{3}=$
$\displaystyle\zeta*\cfrac{25+5\sqrt{89}}{2}\approx
1.677,\quad\lambda_{43}=\cfrac{5-\sqrt{89}}{5+\sqrt{89}}\approx-0.3072.$

##  2 Constructing the model

###  2.1 Quasi-diagonalization of the internal cell matrix

Let $2^{R}\times 2^{R}$ matrices $\overleftrightarrow{S_{3}}$ and
$\overleftrightarrow{S_{4}}$ be constructed of matrices (15) as follows

$\overleftrightarrow{S_{3}}=\prod_{\rho=1}^{R}\overleftrightarrow{S_{1}}_{[\rho]},\quad\overleftrightarrow{S_{4}}=\prod_{\rho=1}^{R}\overleftrightarrow{S_{2}}_{[\rho]}.$
(18)

Matrices $\overleftrightarrow{S_{3}}$ and $\overleftrightarrow{S_{4}}$ commute
with matrix $P_{r}$ of (5). And similarity transformation with them
diagonalizes respectively matrices $\overleftrightarrow{B_{0}}_{[R]}$ and
$\overleftrightarrow{B_{1}}_{[R]}$ of (14).

Let $2^{R+1}\times 2^{R+1}$ matrix $\overleftrightarrow{S_{5}}$ be constructed
of matrices (18) as follows

$\overleftrightarrow{S_{5}}=\begin{pmatrix}\overleftrightarrow{S_{3}}&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\overleftrightarrow{S_{4}}\end{pmatrix}.$ (19)

The similarity transformation with matrix $\overleftrightarrow{S_{5}}$ of (19)
over the block-diagonalized internal cell matrix
$\overleftrightarrow{Z_{1S_{0}}}$ of (13) taking into account (16) gives the
quasi-diagonal matrix

$\overleftrightarrow{Z_{1S_{0}S_{5}}}={\overleftrightarrow{S_{5}}}^{-1}*\overleftrightarrow{Z_{1S_{0}}}*\overleftrightarrow{S_{5}}=\begin{pmatrix}\lambda_{1}*{\overleftrightarrow{\lambda}}_{[R]}\left(\lambda_{21}\right)&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\lambda_{3}*{\overleftrightarrow{\lambda}}_{[R]}\left(\lambda_{43}\right)\end{pmatrix}*\begin{pmatrix}\overleftrightarrow{P_{r}}&0\\\
0&\overleftrightarrow{P_{r}}\end{pmatrix}.$ (20)

where

${\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)=\begin{pmatrix}1&0\\\
0&\lambda\end{pmatrix}_{[R]},\quad|\lambda|<1.$ (21)

###  2.2 The simplest quantum field model

The initial task is to diagonalize the transformed internal cell
$2^{R+1}\times 2^{R+1}$ matrix $\overleftrightarrow{Z_{1S_{0}S_{5}}}$ of (20),
which boils down to diagonalization of the following $2^{R}\times 2^{R}$
matrix

$\overleftrightarrow{Z}\left(\lambda\right)={\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)*\overleftrightarrow{P_{r}}.$
(22)

Let $2^{R}\times 2^{R}$ matrix $\overleftrightarrow{Z}$ of (22) acts on
$2^{R}$ column vector $\overleftarrow{V}$, similar to $\overleftarrow{Z_{0}}$
of (7)

$\overleftarrow{V}=\overleftarrow{\|v_{\overline{\theta}}\|}=\overleftarrow{\|v_{\theta_{1}\theta_{2}\ldots\theta_{R}}\|},\quad\theta_{1}\in[0,1],\
\theta_{2}\in[0,1],\ \ldots,\ \theta_{R}\in[0,1],$ (23)

where $\overline{\theta}=\theta_{1}\theta_{2}\ldots\theta_{R}\in[0,2^{R}-1]$
\- the compound number of a vector element, consisting of subnumbers
$\theta_{1},\theta_{2},\ldots\theta_{R}$.

Let $2^{R}$ basis $W_{R}$ of $2^{R}$ basis column vectors
$\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}=\overleftarrow{W_{\overline{q}}}$
be introduced, in which each basis vector is numbered with a compound number
$\overline{q}\in[0,2^{R}-1]$, consisting of subnumbers $q_{1}\in[0,1],\
q_{2}\in[0,1],\ \ldots,\ q_{R}\in[0,1]$

$\overleftarrow{W_{\overline{q}}}=\overleftarrow{\|w_{\overline{q},\overline{\theta}}\|},\quad\overline{\theta}\in[0,2^{R}-1]\text{
- the compound number of a vector element},$ (24)

where

$w_{\overline{q},\overline{\theta}}=\begin{cases}1&\text{if
$\overline{\theta}=\overline{q}$},\\\ 0&\text{if
$\overline{\theta}\neq\overline{q}$}.\end{cases}$

For example, for $R=2$ the basis $W_{2}$ consists of $2^{2}=4$ basis column
4-vectors

$W_{00}=\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix},W_{01}=\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix},W_{10}=\begin{pmatrix}0\\\ 0\\\ 1\\\
0\end{pmatrix},W_{11}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 1\end{pmatrix}.$

Then the vector $\overleftarrow{V}$ of (23) may be represented by coordinates
$v$ with respect to the basis $W_{R}$

$\overleftarrow{V}=\sum_{q_{1},q_{2},\ldots,q_{R}\in[0,1]}v_{q_{1}q_{2}\ldots
q_{R}}*\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$ (25)

For equations (22) $\ldots$ (25), a quantum field model may be constructed
that greatly simplifies realization.

Let the space $u$ be introduced being a circle with $R$ points $u_{1},\
u_{2},\ u_{3},\ \ldots,\ u_{r},\ \ldots,\ u_{R-1},\ u_{R}$. Let the distance
between the nearest points $\Delta u$ of the circle-space $u$ be called the
space step. Let the coordinate of point $u_{1}$ be $0$, then coordinates of
points $u_{R},\ u_{R-1},\ \ldots,\ u_{2},\ u_{1}$ at counterclockwise
traversing are $\Delta u,\ 2*\Delta u,\ \ldots,\ \left(R-2\right)*\Delta u,\
\left(R-1\right)*\Delta u$ respectively (see Figure 2).

Figure 2: The circle-space $u$.

Let two basis quantum states be possible at each point $u_{r}$: a quantum is
absent or a quantum is present, which is described by a two-valued quantum
variable $q_{r}\in[0,1]$. Then $2^{R}$ basis quantum states for the quantum
field of the entire circle-space $\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$
may be introduced according to (24). And any state of the quantum field
$\overleftarrow{V}$ (see (23)) is their superposition (see (25)), let the set
of these states be called the state-space.

The state $\overleftarrow{V}$ is acted $N$ times (see (8)) by the operator
$\overleftrightarrow{Z}$ of (22). Let one action of operator
$\overleftrightarrow{Z}$ be called an elementary action. Let a change in the
state of the quantum field due to the elementary action be called motion per
time step $\Delta t$.

The task of this paper is to find and analyze eigenvalues and eigenvectors of
the elementary action $\overleftrightarrow{Z}$ of (22).

##  3 The eigenvalues of the elementary action

###  3.1 The general analysis

The elementary action $\overleftrightarrow{Z}$ of (22) is the product of two
matrices. The cyclic shift matrix $\overleftrightarrow{P_{r}}$ acts on any
basis vector $\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$ of (24), shifting
its sub-numbers cyclically as follows

$\overleftrightarrow{P_{r}}*\overleftarrow{W_{q_{1}q_{2}\ldots
q_{R}}}=\overleftarrow{W_{q_{2}\ldots q_{R}q_{1}}}.$ (26)

And the diagonal matrix
${\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)$ of (21) acts on any
basis vector $\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$ of (24), leaving it
unchanged and only multiplying by the coefficient

${\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)*\overleftarrow{W_{q_{1}q_{2}\ldots
q_{R}}}=\begin{cases}\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}&\text{if
$q_{R}=0$},\\\ \lambda*\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}&\text{if
$q_{R}=1$}.\end{cases}$ (27)

Hence, for any $R$, there are two singleton invariant subspaces of the
elementary action $\overleftrightarrow{Z}$ of (22):

  * •

Subspace called the vacuum with the vacuum eigenvector
$\overleftarrow{W_{00\ldots 0}}$ and eigenvalue $\Lambda\left(00\ldots
0\right)=1$

$\overleftrightarrow{Z}*\overleftarrow{W_{00\ldots 0}}=\Lambda\left(00\ldots
0\right)*\overleftarrow{W_{00\ldots 0}}=1*\overleftarrow{W_{00\ldots 0}}.$
(28)

It will be shown later in this subsection that this eigenvalue is greater in
absolute value than any other eigenvalue.

  * •

Subspace called the full with the full eigenvector $\overleftarrow{W_{11\ldots
1}}$ and eigenvalue $\Lambda\left(11\ldots 1\right)=\lambda,\quad|\lambda|<1$

$\overleftrightarrow{Z}*\overleftarrow{W_{11\ldots 1}}=\Lambda\left(11\ldots
1\right)*\overleftarrow{W_{11\ldots 1}}=\lambda*\overleftarrow{W_{11\ldots
1}}.$ (29)

It will be shown later in this subsection that this eigenvalue is smaller in
absolute value than any other eigenvalue.

Each of the other basis vectors $\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$
is an eigenvector of the elementary action $\overleftrightarrow{Z}$ powered
$R$

$\overleftrightarrow{Z}^{R}*\overleftarrow{W_{q_{1}q_{2}\ldots
q_{R}}}=\lambda^{Q}*\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}},$ (30)

where $Q$ is the amount of quanta in the basis vector
$\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$

$Q=q_{1}+q_{2}+\ldots+q_{R},\quad 0\leq Q\leq R.$ (31)

Now one can divide the state-space into subspaces according to the values of
$Q$. Each subspace has the amount of basis vectors equal to the binomial
coefficients $\displaystyle\binom{R}{Q}=\frac{R!}{Q!*(R-Q)!}$. And eigenvalues
$\Lambda\left(q_{1}q_{2}\ldots q_{R}\right)=\Lambda\left(\overline{q}\right)$
satisfying, based on (30), the following equation

$\Lambda^{R}\left(\overline{q}\right)=\lambda^{Q}.$ (32)

Thus the eigenvalue $\Lambda\left(\overline{q}\right)$ of (32) has $R$ roots

$\Lambda_{k}\left(\overline{q}\right)=\sqrt[R]{\lambda^{Q}}=\exp\left(\cfrac{Q*Ln\left(\lambda\right)+2*\pi*k*i}{R}\right),\
k\in[0,R-1],$ (33)

where $Ln\left(\lambda\right)$ is the principal value of the logarithm.

###  3.2 The simplest cases

Based on (33), the simplest cases of R = 1 … 6 are shown in Table 1, which has
the following columns:

  1. 1.

R - the amount of circle-space points (see Figure 2);

  2. 2.

WR - the amount of basis vectors $\overleftarrow{W}$ (see (24)) corresponding
to $R$, which is equal to $2^{R}$;

  3. 3.

Q - the amount of quanta in the basis vector (see (31));

  4. 4.

WQ - the amount of basis vectors $\overleftarrow{W}$ corresponding to $Q$,
which is equal to $\displaystyle\binom{R}{Q}=\frac{R!}{Q!*(R-Q)!}$;

  5. 5.

$\displaystyle\overline{q}$ \- the compound numbers of basis vectors
$\displaystyle\overleftarrow{W_{\overline{q}}}$, for example, vector
$\displaystyle\overleftarrow{W_{100}}$ has the compound number $100$;

  6. 6.

$\displaystyle\Lambda\left(\overline{q}\right)$ \- the eigenvalues
corresponding to $\displaystyle\overline{q}$ (see (33));

  7. 7.

$A\Lambda$ \- the amount of the eigenvalues corresponding to
$\displaystyle\Lambda\left(\overline{q}\right)$, which is equal to the degree
of the root.

R | WR | Q | WQ | $\displaystyle\overline{q}$ | $\displaystyle\Lambda\left(\overline{q}\right)$ | $A\Lambda$
---|---|---|---|---|---|---
1 | 2 | 0 | 1 | 0 | 1 | 1
1 | 1 | 1 | $\lambda$ | 1
2 | 4 | 0 | 1 | 00 | 1 | 1
1 | 2 | 01, 10 | $\sqrt{\lambda}$ | 2
2 | 1 | 11 | $\lambda$ | 1
3 | 8 | 0 | 1 | 000 | 1 | 1
1 | 3 | 001, 010, 100 | $\sqrt[3]{\lambda}$ | 3
2 | 3 | 011, 110, 101 | $\sqrt[3]{\lambda^{2}}$ | 3
3 | 1 | 111 | $\lambda$ | 1
4 | 16 | 0 | 1 | 0000 | 1 | 1
1 | 4 | 0001, 0010, 0100, 1000 | $\sqrt[4]{\lambda}$ | 4
2 | 6 | 0011, 0110, 1100, 1001, 0101, 1010 | $\sqrt[4]{\lambda^{2}}$ | 4
3 | 4 | 0111, 1110, 1101, 1011 | $\sqrt[4]{\lambda^{3}}$ | 4
4 | 1 | 1111 | $\lambda$ | 1
5 | 32 | 0 | 1 | 00000 | 1 | 1
1 | 5 | 00001, 00010, 00100, 01000, 10000 | $\sqrt[5]{\lambda}$ | 5
2 | 10 | 00011, 00110, 01100, 11000, 10001, 00101, 01010, 10100, 01001, 10010 | $\sqrt[5]{\lambda^{2}}$ | 5
3 | 10 | 00111, 01110, 11100, 11001, 10011, 01011, 10110, 01101, 11010, 10101 | $\sqrt[5]{\lambda^{3}}$ | 5
4 | 5 | 01111, 11110, 11101, 11011, 10111 | $\sqrt[5]{\lambda^{4}}$ | 5
5 | 1 | 11111 | $\lambda$ | 1
6 | 64 | 0 | 1 | 000000 | 1 | 1
1 | 6 | 000001, 000010, 000100, 001000, 010000, 100000 | $\sqrt[6]{\lambda}$ | 6
2 | 15 | 000011, 000110, 001100, 011000, 110000, 100001 | $\displaystyle\sqrt[6]{\lambda^{2}}$ | 6
000101, 001010, 010100, 101000, 010001, 100010, 001001, 010010, 100100
3 | 20 | 000111, 001110, 011100, 111000, 110001, 100011, 001101, 011010, 110100, 101001 | $\displaystyle\sqrt[6]{\lambda^{3}}$ | 6
010011, 100110, 001011, 010110, 101100, 011001, 110010, 100101, 010101, 101010
4 | 15 | 001111, 011110, 111100, 111001, 110011, 100111 | $\displaystyle\sqrt[6]{\lambda^{4}}$ | 6
010111, 101110, 011101, 111010, 110101, 101011, 011011, 110110, 101101
5 | 6 | 011111, 111110, 111101, 111011, 110111, 101111 | $\sqrt[6]{\lambda^{5}}$ | 6
6 | 1 | 111111 | $\lambda$ | 1
Table 1: The division of state-space into subspaces according to the values of
Q for the simplest cases of R = 1 … 6

##  4 The eigenvectors of the elementary action

###  4.1 Splitting the state-space subspaces into ordered sub-subspaces closed
under elementary action

In the previous section, the state-space was divided into subspaces according
to the values of Q of (31). The subspaces may be splitted further into sub-
subspaces and ordered with respect for the elementary action
$\overleftrightarrow{Z}$ of (22) for $R>1$ and $0<Q<R$. To do this, one may
use the following algorithm in each subspace:

  1. 1.

Any basis vector $\displaystyle W_{q_{1}q_{2}\ldots q_{R}}$ is selected and
ordered at number 1.

  2. 2.

The numbered 1 basis vector $\displaystyle W_{q_{1}q_{2}\ldots q_{R}}$ is
acted upon by the elementary action $\overleftrightarrow{Z}$ of (22). Which is
the product of the diagonal matrix
${\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)$ of (21) and the
cyclic shift matrix $\overleftrightarrow{P_{r}}$ of (5). The matrix
$\overleftrightarrow{P_{r}}$ transforms the basis vector $\displaystyle
W_{q_{1}q_{2}\ldots q_{R}}$ into the basis vector $\displaystyle
W_{q_{2}\ldots q_{R}q_{1}}$ according to (26). And the matrix
${\overleftrightarrow{\lambda}}_{[R]}\left(\lambda\right)$ just multiplies the
basis vector by the coefficient: 1 or $\lambda$, depending on the last
subnumber. The transformed basis vector $\displaystyle W_{q_{2}\ldots
q_{R}q_{1}}$ has the same $Q$ according to (31). So let it be ordered at
number 2.

  3. 3.

The numbered 2 basis vector $\displaystyle W_{q_{2}\ldots q_{R}q_{1}}$ is also
acted upon by the elementary action $\overleftrightarrow{Z}$ of (22), etc. The
action continues until vector numbered 1 is received, which does not exceed
$R$ according to (30). Then the splitting off of the sub-subspace is
completed.

  4. 4.

If unselected vectors remain, the splitting off of a new sub-subspace begins
from step 1 of the algorithm.

Based on the above algorithm, the simplest cases of R = 2 … 6 are shown in
Table 2. In which, in comparison with Table 1, two columns are added: №- table
row number, WS - the amount of vectors of the sub-subspace.

R | Q | WQ | № | $\displaystyle\overline{q}$ | WS | $\displaystyle\Lambda\left(\overline{q}\right)$ | $A\Lambda$
---|---|---|---|---|---|---|---
2 | 1 | 2 | 1 | 01, 10 | 2 | $\sqrt{\lambda}$ | 2
3 | 1 | 3 | 2 | 001, 010, 100 | 3 | $\sqrt[3]{\lambda}$ | 3
2 | 3 | 3 | 011, 110, 101 | 3 | $\sqrt[3]{\lambda^{2}}$ | 3
4 | 1 | 4 | 4 | 0001, 0010, 0100, 1000 | 4 | $\sqrt[4]{\lambda}$ | 4
2 | 6 | 5 | 0011, 0110, 1100, 1001 | 4 | $\sqrt[4]{\lambda^{2}}$ | 4
6 | 0101, 1010 | 2 | $\sqrt{\lambda}$ | 2
3 | 4 | 7 | 0111, 1110, 1101, 1011 | 4 | $\sqrt[4]{\lambda^{3}}$ | 4
5 | 1 | 5 | 8 | 00001, 00010, 00100, 01000, 10000 | 5 | $\sqrt[5]{\lambda}$ | 5
2 | 10 | 9 | 00011, 00110, 01100, 11000, 10001 | 5 | $\sqrt[5]{\lambda^{2}}$ | 5
10 | 00101, 01010, 10100, 01001, 10010 | 5
3 | 10 | 11 | 00111, 01110, 11100, 11001, 10011 | 5 | $\sqrt[5]{\lambda^{3}}$ | 5
12 | 01011, 10110, 01101, 11010, 10101 | 5
4 | 5 | 13 | 01111, 11110, 11101, 11011, 10111 | 5 | $\sqrt[5]{\lambda^{4}}$ | 5
6 | 1 | 6 | 14 | 000001, 000010, 000100, 001000, 010000, 100000 | 6 | $\sqrt[6]{\lambda}$ | 6
2 | 15 | 15 | 000011, 000110, 001100, 011000, 110000, 100001 | 6 | $\sqrt[6]{\lambda^{2}}$ | 6
16 | 000101, 001010, 010100, 101000, 010001, 100010 | 6
17 | 001001, 010010, 100100 | 3 | $\sqrt[3]{\lambda}$ | 3
3 | 20 | 18 | 000111, 001110, 011100, 111000, 110001, 100011 | 6 | $\sqrt[6]{\lambda^{3}}$ | 6
19 | 001101, 011010, 110100, 101001, 010011, 100110 | 6
20 | 001011, 010110, 101100, 011001, 110010, 100101 | 6
21 | 010101, 101010 | 2 | $\sqrt{\lambda}$ | 2
4 | 15 | 22 | 001111, 011110, 111100, 111001, 110011, 100111 | 6 | $\sqrt[6]{\lambda^{4}}$ | 6
23 | 010111, 101110, 011101, 111010, 110101, 101011 | 6
24 | 011011, 110110, 101101 | 3 | $\sqrt[3]{\lambda^{2}}$ | 3
5 | 6 | 25 | 011111, 111110, 111101, 111011, 110111, 101111 | 6 | $\sqrt[6]{\lambda^{5}}$ | 6
Table 2: The splitting of the state-space subspaces into sub-subspaces for the
simplest cases of R = 2 … 6

###  4.2 Determining eigenvectors using a simple example

Let us determine eigenvectors for sub-subspace №5 (see Table 2) having: the
amount of circle-space points $R=4$, the amount of quanta $Q=2$, four basis
vectors $\\{\overleftarrow{W_{0011}},\ \overleftarrow{W_{0110}},\
\overleftarrow{W_{1100}},\ \overleftarrow{W_{1001}}\\}$, four eigenvalues
$\sqrt[4]{\lambda^{2}}$.

For some eigenvalue $\Lambda_{k}$ of (33), the eigenvector
$\overleftarrow{e_{k}}$ is the linear combination of its state-space
coordinates with respect for the four basis vectors.

$\overleftarrow{e_{k}}=f_{k}\left(0\right)*\overleftarrow{W_{0011}}+f_{k}\left(\Delta
u\right)*\overleftarrow{W_{0110}}+f_{k}\left(2*\Delta
u\right)*\overleftarrow{W_{1100}}+f_{k}\left(3*\Delta
u\right)*\overleftarrow{W_{1001}},$ (34)

where the sought-for state-space coordinates $f_{k}$ may be interpreted as
follows: $f_{k}\left(0\right)$ describes the location of the quanta
configuration 0011 at the initial circle-space point $u_{1}$ having circle-
space coordinate 0 (see Figure 2), $f_{k}\left(\Delta u\right)$ describes the
location of the same quanta configuration at the next circle-space point
$u_{R}$ having circle-space coordinate $\Delta u$, etc.

The characteristic equation for the elementary action $\overleftrightarrow{Z}$
of (22) is

$\overleftrightarrow{Z}*\overleftarrow{e_{k}}={\overleftrightarrow{\lambda}}_{[4]}\left(\lambda\right)*\overleftrightarrow{P_{r}}*\overleftarrow{e_{k}}=\Lambda_{k}*\overleftarrow{e_{k}}.$
(35)

Let the subnumbers of the first basis vector (which are now 0011) be written
as a function $\phi$ of points in circle-space

$\phi\left(0\right)=0,\ \phi\left(\Delta u\right)=0,\ \phi\left(2*\Delta
u\right)=1,\ \phi\left(3*\Delta u\right)=1,$ (36)

where (31) gives

$\phi\left(0\right)+\phi\left(\Delta u\right)+\phi\left(2*\Delta
u\right)+\phi\left(3*\Delta u\right)=Q.$ (37)

Accounting (26) and (27) with (36) gives

$\displaystyle\overleftrightarrow{Z}*\overleftarrow{W_{0011}}={\overleftrightarrow{\lambda}}_{[4]}\left(\lambda\right)*\overleftarrow{W_{0110}}=\lambda^{\phi\left(0\right)}*\overleftarrow{W_{0110}},$
(38)
$\displaystyle\overleftrightarrow{Z}*\overleftarrow{W_{0110}}={\overleftrightarrow{\lambda}}_{[4]}\left(\lambda\right)*\overleftarrow{W_{1100}}=\lambda^{\phi\left(\Delta
u\right)}*\overleftarrow{W_{1100}},$
$\displaystyle\overleftrightarrow{Z}*\overleftarrow{W_{1100}}={\overleftrightarrow{\lambda}}_{[4]}\left(\lambda\right)*\overleftarrow{W_{1001}}=\lambda^{\phi\left(2*\Delta
u\right)}*\overleftarrow{W_{1001}},$
$\displaystyle\overleftrightarrow{Z}*\overleftarrow{W_{1001}}={\overleftrightarrow{\lambda}}_{[4]}\left(\lambda\right)*\overleftarrow{W_{0011}}=\lambda^{\phi\left(3*\Delta
u\right)}*\overleftarrow{W_{0011}}.$

Substitution of (34) into (35) taking into account (38) gives

$\displaystyle\left(\lambda^{\phi\left(0\right)}*f_{k}\left(0\right)-\Lambda_{k}*f_{k}\left(\Delta
u\right)\right)*\overleftarrow{W_{0110}}$ (39) $\displaystyle+$
$\displaystyle\left(\lambda^{\phi\left(\Delta u\right)}*f_{k}\left(\Delta
u\right)-\Lambda_{k}*f_{k}\left(2*\Delta
u\right)\right)*\overleftarrow{W_{1100}}$ $\displaystyle+$
$\displaystyle\left(\lambda^{\phi\left(2*\Delta u\right)}*f_{k}\left(2*\Delta
u\right)-\Lambda_{k}*f_{k}\left(3*\Delta
u\right)\right)*\overleftarrow{W_{1001}}$ $\displaystyle+$
$\displaystyle\left(\lambda^{\phi\left(3*\Delta u\right)}*f_{k}\left(3*\Delta
u\right)-\Lambda_{k}*f_{k}\left(0\right)\right)*\overleftarrow{W_{0011}}=0.$

The coefficients for the basis vectors must be equal to zero. Assuming the
initial coordinate $f_{k}\left(0\right)=1$, the first three coefficients
determine the other coordinates, and the fourth one should naturally equal
zero

$\displaystyle f_{k}\left(0\right)$ $\displaystyle=1,$ (40) $\displaystyle
f_{k}\left(\Delta u\right)$
$\displaystyle={\Lambda_{k}}^{-1}*\lambda^{\phi\left(0\right)}*f_{k}\left(0\right)={\Lambda_{k}}^{-1}*\lambda^{\phi\left(0\right)}={\Lambda_{k}}^{-1},$
$\displaystyle f_{k}\left(2*\Delta u\right)$
$\displaystyle={\Lambda_{k}}^{-1}*\lambda^{\phi\left(\Delta
u\right)}*f_{k}\left(\Delta
u\right)={\Lambda_{k}}^{-2}*\lambda^{\phi\left(0\right)+\phi\left(\Delta
u\right)}={\Lambda_{k}}^{-2},$ $\displaystyle f_{k}\left(3*\Delta u\right)$
$\displaystyle={\Lambda_{k}}^{-1}*\lambda^{\phi\left(2*\Delta
u\right)}*f_{k}\left(2*\Delta
u\right)={\Lambda_{k}}^{-3}*\lambda^{\phi\left(0\right)+\phi\left(\Delta
u\right)+\phi\left(2*\Delta u\right)}=\lambda*{\Lambda_{k}}^{-3},$
$\displaystyle 1$
$\displaystyle={\Lambda_{k}}^{-1}*\lambda^{\phi\left(3*\Delta
u\right)}*f_{k}\left(3*\Delta
u\right)={\Lambda_{k}}^{-4}*\lambda^{\phi\left(0\right)+\phi\left(\Delta
u\right)+\phi\left(2*\Delta u\right)+\phi\left(3*\Delta u\right)}.$

The last equation of (40) is true due to (37) and (32).

Let the numerical calculations be performed for $\lambda$ equal to
$\lambda_{21}$ of (17). Then $\lambda$ and eigenvalues $\Lambda_{k}$ of (33)
are:

$\lambda=\lambda_{21}=\cfrac{9-2\sqrt{73}}{9+2\sqrt{73}}\approx-0.3100,\
\Lambda_{0}=\sqrt[4]{\lambda^{2}}\approx 0.5568,\ \Lambda_{1}\approx
i*0.5568,\ \Lambda_{2}\approx-0.5568,\ \Lambda_{3}\approx-i*0.5568.$ (41)

For $R=4$, the eigenvectors have 16 elements each, numbered from 0 to
$1111_{2}=15_{10}$. Their nonzero elements are calculated with (40), taking
into account (36) and (41). The calculation results are shown in Table 3,
which has the following columns:

  * •

$n_{2}$ and $n_{10}$ \- nonzero element number: $n_{2}$ \- binary and $n_{10}$
\- decimal;

  * •

$\overleftarrow{e_{0}},\ \overleftarrow{e_{1}},\ \overleftarrow{e_{2}},\
\overleftarrow{e_{3}}$ \- eigenvectors.

The last table row $\Lambda$ shows the eigenvalues.

$n_{2}$ | $n_{10}$ | $\overleftarrow{e_{0}}$ | $\overleftarrow{e_{1}}$ | $\overleftarrow{e_{2}}$ | $\overleftarrow{e_{3}}$
---|---|---|---|---|---
0011 | 3 | $f_{0}\left(0\right)=1$ | $f_{1}\left(0\right)=1$ | $f_{2}\left(0\right)=1$ | $f_{3}\left(0\right)=1$
0110 | 6 | $f_{0}\left(\Delta u\right)={\Lambda_{0}}^{-1}$ | $f_{1}\left(\Delta u\right)={\Lambda_{1}}^{-1}$ | $f_{2}\left(\Delta u\right)={\Lambda_{2}}^{-1}$ | $f_{3}\left(\Delta u\right)={\Lambda_{3}}^{-1}$
$\approx 1.796$ | $\approx-i*1.796$ | $\approx-1.796$ | $\approx i*1.796$
1001 | 9 | $f_{0}\left(3*\Delta u\right)=\lambda*{\Lambda_{0}}^{-3}$ | $f_{1}\left(3*\Delta u\right)=\lambda*{\Lambda_{1}}^{-3}$ | $f_{2}\left(3*\Delta u\right)=\lambda*{\Lambda_{2}}^{-3}$ | $f_{3}\left(3*\Delta u\right)=\lambda*{\Lambda_{3}}^{-3}$
$\approx-1.796$ | $\approx-i*1.796$ | $\approx 1.796$ | $\approx i*1.796$
1100 | 12 | $f_{0}\left(2*\Delta u\right)={\Lambda_{0}}^{-2}$ | $f_{1}\left(2*\Delta u\right)={\Lambda_{1}}^{-2}$ | $f_{2}\left(2*\Delta u\right)={\Lambda_{2}}^{-2}$ | $f_{3}\left(2*\Delta u\right)={\Lambda_{3}}^{-2}$
$\approx 3.226$ | $\approx-3.226$ | $\approx 3.226$ | $\approx-3.226$
$\Lambda$ | $\Lambda_{0}=\sqrt[4]{\lambda^{2}}$ | $\Lambda_{1}=i*\sqrt[4]{\lambda^{2}}$ | $\Lambda_{2}=-\sqrt[4]{\lambda^{2}}$ | $\Lambda_{3}=-i*\sqrt[4]{\lambda^{2}}$
$\approx 0.5568$ | $\approx i*0.5568$ | $\approx-0.5568$ | $\approx-i*0.5568$
Table 3: The nonzero eigenvectors elements for $R=4$ sub-subspace №5 of Table
2

###  4.3 Generalization of determining eigenvectors

In Subsection 3.1, the state-space was divided into the subspaces according to
the amount of quanta $Q$ in the basis vector
$\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$ (see Table 1). In particular, for
any $R$ at $Q=0$ and $Q=R$ there were singleton subspaces having maximum
eigenvalue and minimum absolute value eigenvalue respectively.

In Subsection 4.1, at $R>1$ and $0<Q<R$, the subspaces were further splitted
into ordered sub-subspaces closed under elementary action (see Table 2) having
their dimentions at most $R$. In particular, for any $R>1$ at $Q=1$ and
$Q=R-1$ there was one R-dimensional subspace which was not splitted and just
ordered. And for $R>3$ at $1<Q<R-1$, the subspaces were always splitted as the
amount of their basis vectors $\displaystyle\binom{R}{Q}$ exceeded $R$.

When $R$ and $Q$ have a common divisor $d_{R,Q}$, then $R/d_{R,Q}$-dimensional
subspaces appear, similar to $R/d_{R,Q}$-dimensional subspaces at $Q/d_{R,Q}$.
See in Table 2: sub-subspaces №6 for $R=4$ at $Q=2$ and №21 for $R=6$ at $Q=3$
are similar to sub-subspace №1 for $R=2$ at $Q=1$, sub-subspace №17 for $R=6$
at $Q=2$ is similar to sub-subspace №2 for $R=3$ at $Q=1$, sub-subspace №24
for $R=6$ at $Q=4$ is similar to sub-subspace №3 for $R=3$ at $Q=2$.

Thus, when $R$ is a prime number, its sub-subspaces at any $1<Q<R-1$ are only
R-dimensional (see in Table 2: R = 2, 3, 5).

Now let the generalization of determining eigenvectors be performed on the
base of the previous subsection for some ordered R-dimensional sub-subspace
having any $R>1,\ 0<Q<R$, the initial basis vector
$\overleftarrow{W_{q_{1}q_{2}\ldots q_{R}}}$, $R$ eigenvalues $\Lambda_{k}$ of
(33) for $\ k\in[0,R-1]$.

The function $\phi\left(u\right)$ of (36) can be generalized to

$\phi\left(0\right)=q_{1},\ \phi\left(\Delta u\right)=q_{2},\ \ldots,\
\phi\left(\left(R-1\right)*\Delta u\right)=q_{R}.$ (42)

Let the function $\Phi\left(u\right)$ be introduced based on (42)

$\displaystyle\Phi\left(0\right)=0,\ \Phi\left(\Delta
u\right)=\phi\left(0\right),\ \Phi\left(2*\Delta
u\right)=\phi\left(0\right)+\phi\left(\Delta u\right),\ \ldots,$ (43)
$\displaystyle\Phi\left(\left(R-1\right)*\Delta
u\right)=\phi\left(0\right)+\phi\left(\Delta
u\right)+\ldots+\phi\left(\left(R-2\right)*\Delta u\right).$

Then the eigenvector $\overleftarrow{e_{k}}$ coordinates of (40) can be
generalized to

$\displaystyle f_{k}\left(0\right)$ $\displaystyle=1,$ (44) $\displaystyle
f_{k}\left(\Delta u\right)$
$\displaystyle={\Lambda_{k}}^{-1}*\lambda^{\Phi\left(\Delta u\right)},$
$\displaystyle f_{k}\left(2*\Delta u\right)$
$\displaystyle={\Lambda_{k}}^{-2}*\lambda^{\Phi\left(2*\Delta u\right)},$
$\displaystyle\ldots$ $\displaystyle f_{k}\left(\left(R-1\right)*\Delta
u\right)$
$\displaystyle={\Lambda_{k}}^{-\left(R-1\right)}*\lambda^{\Phi\left(\left(R-1\right)*\Delta
u\right)},$

or finally

$f_{k}\left(u\right)=\lambda^{\Phi\left(u\right)}*\exp{\left(-\cfrac{u}{\Delta
u}*\ln{\left(\Lambda_{k}\right)}\right)}.$ (45)

Let the eigenvector $\overleftarrow{e_{k}}$ with coordinates
$f_{k}\left(u\right)$ of (45) undergo several elementary actions
$\overleftrightarrow{Z}$ of (22). One elementary action on its eigenvector
leads to the multiplication of its coefficients by the eigenvalue
$\Lambda_{k}$, which was called the motion per time step $\Delta t$ in the
simplest quantum field model (see Subsection 2.2). Then several elementary
actions may be called the motion per time $t$, where the amount of the
elementary actions $t/\Delta t$ is a natural number. Thus, for the coordinates
of the eigenvector at time $t$, the wave function $F_{k}\left(u,t\right)$ may
be introduced as follows

$F_{k}\left(u,t\right)=f_{k}\left(u\right)*\exp{\left(\cfrac{t}{\Delta
t}*\ln{\left(\Lambda_{k}\right)}\right)}.$ (46)

Let the following universal values be introduced: the uncertainty $\delta$, as
well as the frequency step $\Delta\omega$ and the wavenumber step
$\Delta\kappa$

$\delta=\cfrac{2*\pi}{R},\quad\Delta\omega*\Delta
t=\delta,\quad\Delta\kappa*\Delta u=\delta.$ (47)

Substituting $f_{k}\left(u\right)$ of (45) and $\Lambda_{k}$ of (33) to (46),
taking into account $(\ref{eq:uncert})$, gives the wave function
$F_{k}\left(u,t\right)$ in the final form

$F_{k}\left(u,t\right)=\lambda^{\Phi\left(u\right)}*\exp{\left(\Omega_{k}*t-K_{k}*u\right)},$
(48)

where the full frequency $\Omega_{k}$ and the full wavenumber $K_{k}$ are

$\Omega_{k}=\cfrac{Q*Ln\left(\lambda\right)}{R*\Delta
t}+i*k*\Delta\omega,\quad K_{k}=\cfrac{Q*Ln\left(\lambda\right)}{R*\Delta
u}+i*k*\Delta\kappa.$ (49)

Note that $\Omega_{k}$ is directly proportional to $K_{k}$ through the
universal value c

$\Omega_{k}=c*K_{k},\quad c=\cfrac{\Delta u}{\Delta t}.$ (50)

###  4.4 A little fun

For real positive $0<\lambda<1$, there is real negative
$Ln\left(\lambda\right)<0$. Then (48) and (49) describe damped waves. This
entails various wave effects, some of which may be interpreted funnily:

  1. 1.

The group velocity of any wave function $F_{k}\left(u,t\right)$ of (48) is
equal to its phase velocity, which is equal to universal value $c$ of (50).
This may be interpreted as the simplest model of the special relativity.

  2. 2.

The wave functions propagate independently in homogeneous space-time. But when
there is heterogeneity, interaction arises between them. This may be
interpreted as the simplest model of the general relativity.

  3. 3.

(47) may be interpreted as the simplest model of the uncertainty principle.

  4. 4.

The basis vectors $\overleftarrow{W_{\overline{q}}}$ of (24) are not the
eigenvectors of the elementary action $\overleftrightarrow{Z}$ of (22)
(excluding the vacuum eigenvector $\overleftarrow{W_{00\ldots 0}}$ of (28) and
the full eigenvector $\overleftarrow{W_{11\ldots 1}}$ of (29)), so they behave
like waves. However, these same basis vectors
$\overleftarrow{W_{\overline{q}}}$ are the eigenvectors of the elementary
action $\overleftrightarrow{Z}$ powered $R$ (see (30)), so they behave like
particles. This may be interpreted as the simplest model of the wave-particle
duality.

##  5 The Jordan normal form of the elementary action

###  5.1 Determining the transformation matrix using a simple example

Let us consider a simple example: the amount of circle-space points is $R=4$.
Then the elementary action $\overleftrightarrow{Z}\left(\lambda\right)$ of
(22) has $2^{4}\times 2^{4}=16\times 16$ matrix, let its rows and columns be
numbered from $0000_{2}=0_{10}$ to $1111_{2}=15_{10}$. Let 16 basis vectors be
$\overleftarrow{W_{0000}},\ldots,\overleftarrow{W_{1111}}$ of (24). Let the
numerical calculations be performed for $\lambda$ equal to $\lambda_{21}$ of
(17).

The Jordan normal form is obtained by some similarity transformation
${\overleftrightarrow{S_{6}}}^{-1}*\overleftrightarrow{Z}\left(\lambda\right)*\overleftrightarrow{S_{6}}$.
Let the transformation matrix $\overleftrightarrow{S_{6}}$ be built of 16
column eigenvectors of the elementary action, the coordinates of which
$f_{k}\left(u\right)$ are calculated by (45). Further, only nonzero
transformation matrix elements will be shown.

  1. 0.

The vacuum basis vector $\overleftarrow{W_{0000}}$ is the eigenvector with the
eigenvalue equal to 1 (see (28)). Let it be the transformation matrix column
numbered $0000_{2}=0_{10}$, so this column has a single non-zero element at
the top equal to 1.

Similarly, the full basis vector $\overleftarrow{W_{1111}}$ is the eigenvector
with the eigenvalue equal to $\lambda$ (see (29)). Let it be the
transformation matrix column numbered $1111_{2}=15_{10}$, so this column has a
single non-zero element at the bottom equal to 1.

This is shown in Table 4, where $n_{2}$ and $n_{10}$ are binary and decimal
row/column number, row $\Lambda$ is the eigenvalue.

$n_{2}$ |  | 0
---|---|---
| $n_{10}$ | 0
0 | 0 | 1
$\Lambda$ | 1

$n_{2}$ |  | 1111
---|---|---
| $n_{10}$ | 15
1111 | 15 | 1
$\Lambda$ | $\lambda\approx-0.3100$

Table 4: The nonzero transformation matrix elements of columns 0 and 15

  2. 1.

The sub-subspace №4 of Table 2 has dimension 4, let its eigenvectors form the
transformation matrix columns numbered $0001_{2}=1_{10},\ 0010_{2}=2_{10},\
0011_{2}=3_{10},\ 0100_{2}=4_{10}$. The first basis vector is
$\overleftarrow{W_{0001}}$, so the functions $\phi\left(u\right)$ of (42) and
$\Phi\left(u\right)$ of (43) are

$\displaystyle\phi\left(0\right)=0,\ \phi\left(\Delta u\right)=0,\
\phi\left(2*\Delta u\right)=0,\ \phi\left(3*\Delta u\right)=1;$
$\displaystyle\Phi\left(0\right)=0,\ \Phi\left(\Delta u\right)=0,\
\Phi\left(2*\Delta u\right)=0,\ \Phi\left(3*\Delta u\right)=0.$

The calculation results for the eigenvector coordinates $f_{k}\left(u\right)$
of (45) are shown in Table 5.

$n_{2}$ |  | 0001 | 0010 | 0011 | 0100
---|---|---|---|---|---
| $n_{10}$ | 1 | 2 | 3 | 4
0001 | 1 | $f_{0}\left(0\right)=1$ | $f_{1}\left(0\right)=1$ | $f_{2}\left(0\right)=1$ | $f_{3}\left(0\right)=1$
0010 | 2 | $f_{0}\left(\Delta u\right)={\Lambda_{0}}^{-1}$ | $f_{1}\left(\Delta u\right)={\Lambda_{1}}^{-1}$ | $f_{2}\left(\Delta u\right)={\Lambda_{2}}^{-1}$ | $f_{3}\left(\Delta u\right)={\Lambda_{3}}^{-1}$
$\approx 0.9476-i*0.9476$ | $\approx-0.9476-i*0.9476$ | $\approx-0.9476+i*0.9476$ | $\approx 0.9476+i*0.9476$
0100 | 4 | $f_{0}\left(2*\Delta u\right)={\Lambda_{0}}^{-2}$ | $f_{1}\left(2*\Delta u\right)={\Lambda_{1}}^{-2}$ | $f_{2}\left(2*\Delta u\right)={\Lambda_{2}}^{-2}$ | $f_{3}\left(2*\Delta u\right)={\Lambda_{3}}^{-2}$
$\approx-i*1.796$ | $\approx i*1.796$ | $\approx-i*1.796$ | $\approx i*1.796$
1000 | 8 | $f_{0}\left(3*\Delta u\right)={\Lambda_{0}}^{-3}$ | $f_{1}\left(3*\Delta u\right)={\Lambda_{1}}^{-3}$ | $f_{2}\left(3*\Delta u\right)={\Lambda_{2}}^{-3}$ | $f_{3}\left(3*\Delta u\right)={\Lambda_{3}}^{-3}$
$\approx-1.702-i*1.702$ | $\approx 1.702-i*1.702$ | $\approx 1.702+i*1.702$ | $\approx-1.702+i*1.702$
$\Lambda$ | $\Lambda_{0}=\sqrt[4]{\lambda}$ | $\Lambda_{1}=i*\sqrt[4]{\lambda}$ | $\Lambda_{2}=-\sqrt[4]{\lambda}$ | $\Lambda_{3}=-i*\sqrt[4]{\lambda}$
$\approx 0.5276+i*0.5276$ | $\approx-0.5276+i*0.5276$ | $\approx-0.5276-i*0.5276$ | $\approx 0.5276-i*0.5276$
Table 5: The nonzero transformation matrix elements of columns from 1 to 4

  3. 2.

The sub-subspace №5 of Table 2 is calculated in Table 3. Let its 4 columns be
numbered $0101_{2}=5_{10},\ 0110_{2}=6_{10},\ 0111_{2}=7_{10},\
1000_{2}=8_{10}$.

  4. 3.

The sub-subspace №6 of Table 2 has dimension 2, so the calculation is similar
to the two-dimensional sub-subspace №1 of Table 2. Let sub-subspace №6
eigenvectors form the transformation matrix columns numbered
$1001_{2}=9_{10},\ 1010_{2}=10_{10}$. The first basis vector is
$\overleftarrow{W_{0101}}$, so the functions $\phi\left(u\right)$ of (42) and
$\Phi\left(u\right)$ of (43) are

$\phi\left(0\right)=0,\ \phi\left(\Delta
u\right)=1;\quad\Phi\left(0\right)=0,\ \Phi\left(\Delta u\right)=0.$

The calculation results for the eigenvector coordinates $f_{k}\left(u\right)$
of (45) are shown in Table 6.

$n_{2}$ |  | 1001 | 1010
---|---|---|---
| $n_{10}$ | 9 | 10
0101 | 5 | $f_{0}\left(0\right)=1$ | $f_{1}\left(0\right)=1$
1010 | 10 | $f_{0}\left(\Delta u\right)={\Lambda_{0}}^{-1}\approx-i*1.796$ | $f_{1}\left(\Delta u\right)={\Lambda_{1}}^{-1}\approx i*1.796$
$\Lambda$ | $\Lambda_{0}=\sqrt{\lambda}\approx i*0.5568$ | $\Lambda_{1}=-\sqrt{\lambda}\approx-i*0.5568$
Table 6: The nonzero transformation matrix elements of columns 9 and 10

  5. 4.

The sub-subspace №7 of Table 2 has dimension 4, let its eigenvectors form the
transformation matrix columns numbered $1011_{2}=11_{10},\ 1100_{2}=12_{10},\
1101_{2}=13_{10},\ 1110_{2}=14_{10}$. The first basis vector is
$\overleftarrow{W_{0111}}$, so the functions $\phi\left(u\right)$ of (42) and
$\Phi\left(u\right)$ of (43) are

$\displaystyle\phi\left(0\right)=0,\ \phi\left(\Delta u\right)=1,\
\phi\left(2*\Delta u\right)=1,\ \phi\left(3*\Delta u\right)=1;$
$\displaystyle\Phi\left(0\right)=0,\ \Phi\left(\Delta u\right)=0,\
\Phi\left(2*\Delta u\right)=1,\ \Phi\left(3*\Delta u\right)=2.$

The calculation results for the eigenvector coordinates $f_{k}\left(u\right)$
of (45) are shown in Table 7.

$n_{2}$ |  | 1011 | 1100 | 1101 | 1110
---|---|---|---|---|---
| $n_{10}$ | 11 | 12 | 13 | 14
0111 | 7 | $f_{0}\left(0\right)=1$ | $f_{1}\left(0\right)=1$ | $f_{2}\left(0\right)=1$ | $f_{3}\left(0\right)=1$
1011 | 11 | $f_{0}\left(3*\Delta u\right)={\lambda}^{2}/{\Lambda_{0}}^{3}$ | $f_{1}\left(3*\Delta u\right)={\lambda}^{2}/{\Lambda_{1}}^{3}$ | $f_{2}\left(3*\Delta u\right)={\lambda}^{2}/{\Lambda_{2}}^{3}$ | $f_{3}\left(3*\Delta u\right)={\lambda}^{2}/{\Lambda_{3}}^{3}$
$\approx-0.9476-i*0.9476$ | $\approx 0.9476-i*0.9476$ | $\approx 0.9476+i*0.9476$ | $\approx-0.9476+i*0.9476$
1101 | 13 | $f_{0}\left(2*\Delta u\right)=\lambda/{\Lambda_{0}}^{2}$ | $f_{1}\left(2*\Delta u\right)=\lambda/{\Lambda_{1}}^{2}$ | $f_{2}\left(2*\Delta u\right)=\lambda/{\Lambda_{2}}^{2}$ | $f_{3}\left(2*\Delta u\right)=\lambda/{\Lambda_{3}}^{2}$
$\approx i*1.796$ | $\approx-i*1.796$ | $\approx i*1.796$ | $\approx-i*1.796$
1110 | 14 | $f_{0}\left(\Delta u\right)={\Lambda_{0}}^{-1}$ | $f_{1}\left(\Delta u\right)={\Lambda_{1}}^{-1}$ | $f_{2}\left(\Delta u\right)={\Lambda_{2}}^{-1}$ | $f_{3}\left(\Delta u\right)={\Lambda_{3}}^{-1}$
$\approx 1.702-i*1.702$ | $\approx-1.702-i*1.702$ | $\approx-1.702+i*1.702$ | $\approx 1.702+i*1.702$
$\Lambda$ | $\Lambda_{0}=\sqrt[4]{{\lambda}^{3}}$ | $\Lambda_{1}=i*\sqrt[4]{{\lambda}^{3}}$ | $\Lambda_{2}=-\sqrt[4]{{\lambda}^{3}}$ | $\Lambda_{3}=-i*\sqrt[4]{{\lambda}^{3}}$
$\approx 0.2938+i*0.2938$ | $\approx-0.2938+i*0.2938$ | $\approx-0.2938-i*0.2938$ | $\approx 0.2938-i*0.2938$
Table 7: The nonzero transformation matrix elements of columns from 11 to 14

Using Wolfram Mathematica, the authors performed a similarity transformation
with the matrix constructed in this subsection. A strictly diagonal matrix was
obtained, consisting of these eigenvalues.

###  5.2 Generalization of determining the Jordan normal form

Of the various properties of the Jordan normal form of the elementary action
$\overleftrightarrow{Z}\left(\lambda\right)$ of (22), two are distinguished
for further analysis:

  1. 1.

There is a single dominant eigenvalue equal to 1, such that any other
eigenvalue in absolute value is strictly smaller than 1.
The associated subspace is one-dimensional of vacuum eigenvector
$\overleftarrow{W_{00\ldots 0}}$ (see (28)).
Let the dominant eigenvalue be the top left element of the Jordan normal form.
So the transformation matrix $\overleftrightarrow{S_{6}}\left(\lambda\right)$
has the top row and the left column equal to 0, excepting the top-left element
equal to 1.

  2. 2.

The normal Jordan form of the elementary action
$\overleftrightarrow{Z}\left(\lambda\right)$ of (22) is strictly diagonal of
the eigenvalues. Let it be denoted as
$\overleftrightarrow{\Lambda}\left(\lambda\right)$, thus

$\overleftrightarrow{\Lambda}\left(\lambda\right)={\overleftrightarrow{S_{6}}}^{-1}\left(\lambda\right)*\overleftrightarrow{Z}\left(\lambda\right)*\overleftrightarrow{S_{6}}\left(\lambda\right).$
(51)

Some of the eigenvalues may coincide (see $\Lambda_{0}$ of Table 6 and
$\Lambda_{1}$ of Table 3, also $\Lambda_{1}$ of Table 6 and $\Lambda_{3}$ of
Table 3). But they are in different sub-subspaces, and each sub-subspace
consists of different eigenvalues and is diagonalized independently.

###  5.3 The partition function and its analysis

Let the similarity transformation with the matrix $\overleftrightarrow{S_{0}}$
of (12) be performed over the partition function $Z$ of (8)

$Z=\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}\right)*{\left({\overleftrightarrow{S_{0}}}^{-1}*\overleftrightarrow{Z_{1}}*\overleftrightarrow{S_{0}}\right)}^{N}*\left({\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right).$
(52)

Accounting (13) gives the block-diagonal form

$Z=\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}\right)*{\begin{pmatrix}\overleftrightarrow{B_{0}}_{[R]}*\overleftrightarrow{P_{r}}&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\overleftrightarrow{B_{1}}_{[R]}*\overleftrightarrow{P_{r}}\end{pmatrix}}^{N}*\left({\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right).$
(53)

Let the similarity transformation with the matrix $\overleftrightarrow{S_{5}}$
of (19) be performed over the partition function $Z$ of (53). Accounting (20)
diagonalizes blocks $\overleftrightarrow{B_{0}}_{[R]}$ and
$\overleftrightarrow{B_{1}}_{[R]}$, then accounting (22) gives

$Z=\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}*\overleftrightarrow{S_{5}}\right)*{\begin{pmatrix}\lambda_{1}*\overleftrightarrow{Z}\left(\lambda_{21}\right)&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\lambda_{3}*\overleftrightarrow{Z}\left(\lambda_{43}\right)\end{pmatrix}}^{N}*\left({\overleftrightarrow{S_{5}}}^{-1}*{\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right).$
(54)

Let $2^{R+1}\times 2^{R+1}$ matrix $\overleftrightarrow{S_{7}}$ be constructed
of two $2^{R}\times 2^{R}$ matrices $\overleftrightarrow{S_{6}}$ of (51) as
follows

$\overleftrightarrow{S_{7}}=\begin{pmatrix}\overleftrightarrow{S_{6}}\left(\lambda_{21}\right)&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&\overleftrightarrow{S_{6}}\left(\lambda_{43}\right)\end{pmatrix}.$
(55)

Let the similarity transformation with the matrix $\overleftrightarrow{S_{7}}$
of (55) be performed over the partition function $Z$ of (54). Accounting (51)
gives strictly diagonal matrices
$\overleftrightarrow{\Lambda}\left(\lambda_{21}\right)$ and
$\overleftrightarrow{\Lambda}\left(\lambda_{43}\right)$, then finally

$Z=\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}*\overleftrightarrow{S_{5}}*\overleftrightarrow{S_{7}}\right)*\begin{pmatrix}{\lambda_{1}}^{N}*{\overleftrightarrow{\Lambda}}^{N}\left(\lambda_{21}\right)&\overleftrightarrow{0}\\\
\overleftrightarrow{0}&{\lambda_{3}}^{N}*{\overleftrightarrow{\Lambda}}^{N}\left(\lambda_{43}\right)\end{pmatrix}*\left({\overleftrightarrow{S_{7}}}^{-1}*{\overleftrightarrow{S_{5}}}^{-1}*{\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right).$
(56)

Let the partition function $Z$ of (56) be analysed for a large amount of cells
$N$. First, $\lambda_{1}$ is greater than $\lambda_{3}$ (see (17)). Second,
the strictly diagonal normal Jordan form $\overleftrightarrow{\Lambda}$ has a
dominant eigenvalue 1 as the top-left element. Then, in a strictly diagonal
matrix of (56), the top-left element ${\lambda_{1}}^{N}$ is much larger than
the rest. Third, the transformation matrix
$\overleftrightarrow{S_{6}}\left(\lambda\right)$ has the top row and the left
column equal to 0, excepting the top-left element equal to 1, and this is also
true for $\overleftrightarrow{S_{7}}$ due to (55). Then the partition function
$Z$ is close to

$Z_{\infty}={\lambda_{1}}^{N}*\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}*\overleftrightarrow{S_{5}}\right)_{0}*\left({\overleftrightarrow{S_{5}}}^{-1}*{\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right)_{0},$
(57)

where
$\displaystyle\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}*\overleftrightarrow{S_{5}}\right)_{0}$
is the left element numbered 0 of the row vector
$\displaystyle\left(\overrightarrow{Z_{N+1}}*\overleftrightarrow{S_{0}}*\overleftrightarrow{S_{5}}\right)$,
and
$\displaystyle\left({\overleftrightarrow{S_{5}}}^{-1}*{\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right)_{0}$
is the top element numbered 0 of the column vector
$\displaystyle\left({\overleftrightarrow{S_{5}}}^{-1}*{\overleftrightarrow{S_{0}}}^{-1}*\overleftarrow{Z_{0}}\right)$.

Note that (57) is the same as (38) of [2].

From the partition function one gets the free energy $A$ and the specific free
energy per spin $a$ (see (39) of [2])

$A=-k_{B}*T*ln(Z),\quad a=\frac{A}{2*N+R+1},$ (58)

where $2*N+R+1$ is the amount of spins, since each of the $N$ internal cells
has 2 spins and the finish cell has $R+1$ spins with spin first sub-number
equal to the cell number.

Taking into account (57), with a large amount of cells $N$, the free energy
$A$ and the specific free energy per spin $a$ of (58) are close to

$A_{\infty}=-k_{B}*T*N*\ln(\lambda_{1}),\quad
a_{\infty}=-k_{B}*T*\cfrac{\ln(\lambda_{1})}{2}.$ (59)

It is important that the properties of (59) do not depend on the number of
rows $R$ and the boundary conditions.

##  6 Conclusion

  1. 1.

The simplest quantum field model is built in Section 2 for the block-
diagonalizable two-dimensional generalized Ising systems, introduced in [1]
and [2]. In this model, the elementary action operator of (22) is introduced,
which is analyzed further in the paper.

  2. 2.

The eigenvalues of the elementary action were analyzed in Section 3. They may
be calculated by (33).

  3. 3.

The eigenvectors of the elementary action were analyzed in Section 4. Their
coordinates may be calculated by (45) and (46).

  4. 4.

It’s funny that the simplest quantum field model yields the simplest models
of: the special relativity, the general relativity, the uncertainty principle,
the wave-particle duality (see Subsection 4.4).

  5. 5.

The Jordan normal form of the elementary action was analyzed in Section 5. It
has a single dominant eigenvalue and is strictly diagonal of the eigenvalues,
although some of the eigenvalues may coincide.

  6. 6.

The partition function was analyzed in Subsection 5.3. With a large amount of
cells the specific free energy per spin does not depend on the number of rows
and the boundary conditions (see (59)).

## References

  * Sakhno and Sakhno [2020] Vadym Sakhno and Mykola Sakhno.  Exact matrix model for generalized ising model.  2020\.  URL https://arxiv.org/abs/2012.10364.
  * Sakhno and Sakhno [2021] Vadym Sakhno and Mykola Sakhno.  Block-diagonalizable two-dimensional generalized ising systems (bd2dgis): the free energy.  2021\.  URL https://arxiv.org/abs/2103.06634.