On the relation between quantum walks and zeta functions

Definition 8.1 .

Given two smooth $p$ -forms $\alpha$ , $\beta\in\Omega^{p}(M)$ on a Riemannian manifold $M$ , their pointwise inner product at point $x\in M$ is defined by

\langle\alpha(x),\beta(x)\rangle\,\mu=\alpha(x)\wedge\operatorname{\ast}\beta(% x)\,,

(8.1)

where $\mu=\operatorname{\ast}1$ is the volume form associated with the metric induced by the inner product on $M$ .

Definition 5.4 .

For $A\in\textup{LCN}(E)$ we call a crystalline deformation $\rho:G_{\Sigma}\rightarrow\operatorname{GL}_{n}(A)$ $\tau$ -self-dual or simply self-dual if $\tau$ is clear from the context if

\textup{tr}\hskip 2.0pt\rho=\textup{tr}\hskip 2.0pt\rho\circ\tau.

Definition 3.1

A right quasigroup is a set $X$ with a binary operation $\circ$ such that for each $a,b\in X$ there exists a unique $x\in X$ such that $a\,\circ\,x\,=\,b$ . We write the solution of this equation $x\,=\,a\,\bullet\,b$ .

An idempotent right quasigroup (irq) is a right quasigroup $(X,\circ)$ such that for any $x\in X$ $x\,\circ\,x\,=\,x$ . Equivalently, it can be seen as a set $X$ endowed with two operations $\circ$ and $\bullet$ , which satisfy the following axioms: for any $x,y\in X$

(R1)

$\displaystyle x\,\circ\,x\,=\,x\,\bullet\,x\,=\,x$
(R2)

$\displaystyle x\,\circ\,\left(x\,\bullet\,y\right)\,=\,x\,\bullet\,\left(x\,% \circ\,y\right)\,=\,y$

Definition 2.1 .

A Hom-Lie algebra $(V,[\cdot,\cdot],\alpha)$ is a vector space $V$ together with a skew-symmetric bilinear map $[\cdot,\cdot]:V\times V\rightarrow V$ and a linear map $\alpha:V\rightarrow V$ satisfying

\displaystyle[\alpha(x),[y,z]]=[[x,y],\alpha(z)]+[\alpha(y),[x,z]]

for all $x,y,z\in V$ .

Definition 15 .

Fix a map $.^{-1}:\mathcal{U}\ni\alpha\mapsto\alpha^{-1}\in\mathcal{U}$ . A input-output map $f$ is said to be reversible with respect to the map $.^{-1}$ , if for all $\alpha\in\mathcal{U}$ , $s,w\in\mathcal{U}^{*}$ , $|sw|>0$ ,

f(s\alpha\alpha^{-1}w)=f(sw).

Definition 2.1 ( [ E ] ) .

For $N\geqslant 1$ , the Lie algebra $\mathfrak{grtm}_{(\bar{1},1)}(N,\mathbb{k})$ is defined to be the set of pairs $(\varphi,\psi)\in\mathfrak{t}^{0}_{3}\times\mathfrak{t}^{0}_{3,N}$ satisfying $\varphi\in\mathfrak{grt}_{1}(\mathbb{k})$ ,
the mixed pentagon equation in $\mathfrak{t}^{0}_{4,N}$

(2.1)

\psi^{1,2,34}+\psi^{12,3,4}=\varphi^{2,3,4}+\psi^{1,23,4}+\psi^{1,2,3},

the octagon equation in $\mathfrak{t}^{0}_{3,N}$

(2.2)			$\displaystyle\psi\bigl{(}A,B(0),B(1),\dots,B(i),\dots,B(N-1)\bigr{)}$
	$\displaystyle-$	$\displaystyle\psi\bigl{(}A,B(1),B(2),\dots,B(i+1),\dots,B(0)\bigr{)}$
	$\displaystyle+$	$\displaystyle\psi\bigl{(}C,B(1),B(0),\dots,B(N+1-i),\dots,B(2)\bigr{)}$
	$\displaystyle-$	$\displaystyle\psi\bigl{(}C,B(0),B(N-1),\dots,B(N-i),\dots,B(1)\bigr{)}=0$

with $A+\sum_{a\in\mathbb{Z}/N\mathbb{Z}}B(a)+C=0$ ,
the special derivation condition in $\mathfrak{t}^{0}_{3,N}$

(2.3)		$\displaystyle\sum_{a\in\mathbb{Z}/N\mathbb{Z}}$	$\displaystyle\Bigl{[}\psi\bigl{(}A,B(a),B(a+1),\dots,B(a+i),\dots,B(a-1)\bigr{% )},B(a)\Bigr{]}$
	$\displaystyle+\Bigl{[}\psi\bigl{(}$	$\displaystyle A,B(0),B(1),\dots,B(i),\dots,B(N-1)\bigr{)}$
		$\displaystyle-\psi\bigl{(}C,B(0),B(N-1),\dots,B(N-i),\dots,B(1)\bigr{)},C\Bigr% {]}=0$

and $c_{B(0)}(\psi)=0$ . ² ² 2 For our convenience, we slightly change the original definition by adding the small condition $c_{B(0)}(\psi)=0$ . The relation to the original Lie algebra is the direct sum decomposition of Lie algebras ${\mathfrak{g}}^{\text{original}}={\mathfrak{g}}\oplus\mathbb{k}\cdot B(0)$ , where ${\mathfrak{g}}=\mathfrak{grtm}_{(\bar{1},1)}(N,\mathbb{k})$ .

Definition 2.3 ( [ E ] ) .

For $N\geqslant 1$ , the group $\mathrm{GRTM}_{(\bar{1},1)}(N,\mathbb{k})$ is defined to be the set of pairs $(g,h)\in\exp\mathfrak{t}^{0}_{3}\times\exp\mathfrak{t}^{0}_{3,N}$ satisfying $g\in\mathrm{GRT}_{1}(\mathbb{k})$ , $c_{B(0)}(h)=0$ ,
the mixed pentagon equation in $\exp\mathfrak{t}^{0}_{4,N}$

(2.5)

h^{1,2,34}h^{12,3,4}=g^{2,3,4}h^{1,23,4}h^{1,2,3},

the octagon equation in $\exp\mathfrak{t}^{0}_{3,N}$

(2.6)		$\displaystyle h\bigl{(}$	$\displaystyle A,B(1),B(2),\dots,B(0)\bigr{)}^{-1}h\bigl{(}C,B(1),B(0),\dots,B(% 2)\bigr{)}\cdot$
		$\displaystyle h\bigl{(}C,B(0),B(N-1),\dots,B(1)\bigr{)}^{-1}h\bigl{(}A,B(0),B(% 1),\dots,B(N-1)\bigr{)}=1$

with $A+\sum_{a\in\mathbb{Z}/N\mathbb{Z}}B(a)+C=0$ and
the special action condition in $\exp\mathfrak{t}^{0}_{3,N}$

(2.7)

A+\sum_{a\in\mathbb{Z}/N\mathbb{Z}}Ad(\tau_{a}h^{-1})(B(a))+Ad\Bigl{(}h^{-1}% \cdot h\bigl{(}C,B(0),B(N-1),\dots,B(1)\bigr{)}\Bigr{)}(C)=0

where $\tau_{a}$ ( $a\in\mathbb{Z}/N\mathbb{Z}$ ) is the automorphism defined by $A\mapsto A$ and $B(c)\mapsto B(c+a)$ for all $c\in\mathbb{Z}/N\mathbb{Z}$ .

Definition 4.1 .

Let

\theta(x,y)=\dfrac{-x}{1+xy}\,,

and define the region of large coefficients of Diophantine approximation to be

\mathcal{D}=\big{\{}\,P\in\Gamma\;|\;\theta(P)>\tau,\,\theta(\,\mathcal{T}^{-1% }(P)\,)>\tau\,\big{\}}\,.

Definition 3 .

Let the function $s$ be defined on $\mathbb{N}_{0}$ by

s(n)=\begin{cases}1\quad\text{if\ }n=m^{2},\\ 0\quad\text{otherwise.}\end{cases}

Definition 0 .

Let $(\Lambda_{0},\Lambda_{1})$ be a cleanly intersecting pair of Lagrangians in codimension $k$ in $T^{*}(X)\backslash 0$ . The space of paired Lagrangian distributions of order $p,l\in\mathbb{R}$ associated to $(\Lambda_{0},\Lambda_{1})$ , denoted by $I^{p,l}(\Lambda_{0},\Lambda_{1})$ , is the set of all locally finite sums of elements of $I^{p+1}(\Lambda_{0})+I^{p}(\Lambda_{1})$ and distributions of the form

u(x)=\int e^{i\phi(x;\theta;\sigma)}a(x;\theta;\sigma)d\theta d\sigma,

(2.3)

where $a\in S^{\tilde{p},\tilde{l}}(X\times(\mathbb{R}^{N}\backslash 0)\times\mathbb{% R}^{k})$ , with $p=\tilde{p}+\tilde{l}+\frac{N+k}{2}-\frac{\dim X}{4}$ , $l=-\tilde{l}-\frac{k}{2}$ , and $\phi(x;\theta;\sigma)$ is multiphase parametrizing $(\Lambda_{0},\Lambda_{1})$ on a conic neighborhood of a point $\lambda_{0}\in\Lambda_{0}\cap\Lambda_{1}$ .

Definition 2.1 .

An algebra $L$ over a field $F$ is called a Leibniz algebra if for any $x,y,z\in L$ the Leibniz identity

[[x,y],z]=[[x,z],y]+[x,[y,z]]

is satisfied, where $[-,-]$ is the multiplication in $L$ .

Definition 3.1 .

For a $q$ -difference equation

a(x)u(xq^{2})+b(x)u(xq)+c(x)u(x)=0,

a shearing transformation is the following transformation:

x=t^{2},\ p=\sqrt{q},\ v(t)=u(x).

The shearing transform of the $q$ -difference equation is given by

a(t^{2})v(tp^{2})+b(t^{2})v(tp)+c(t^{2})v(t)=0.

Definition 2.2 .

An MV-algebra is a BL-algebra satisfying

x=\sim\sim x.

(inv)

A well known example of MV-algebra is the standard MV-algebra $[0,1]_{\text{\L}}=\\ \left\langle[0,1],*,\Rightarrow,\min,\max,0,1\right\rangle$ , where $x*y=\max(0,x+y-1)$ and $x\Rightarrow y=\min(1,1-x+y)$ .

Definition 2.5 ( [ Fer92 , BF00 ] ) .

A hoop is a structure $\mathcal{A}=\langle A,*,\Rightarrow,1\rangle$ such that $\langle A,*,1\rangle$ is a commutative monoid, and $\Rightarrow$ is a binary operation such that

x\Rightarrow x=1,\hskip 14.226378ptx\Rightarrow(y\Rightarrow z)=(x*y)% \Rightarrow z\hskip 14.226378pt\mathrm{and}\hskip 14.226378ptx*(x\Rightarrow y% )=y*(y\Rightarrow x).

Definition 3 .

We say that the vector field $X$ has the Lipschitz periodic shadowing property ( $X\in\operatorname{LipPerSh}$ ) if there exist $d_{0}$ and ${\cal L}>0$ such that if $y:\mathds{R}\mapsto M$ is a periodic $d$ - pseudotrajectory for $d\leq d_{0}$ , then $y(t)$ is ${\cal L}d-$ shadowed by a periodic trajectory , that is, there exists a trajectory $x(t)$ of $X$ and an increasing homeomorphism $\alpha(t)$ of the real line satisfying inequalities ( 1 ) and ( 2 ) and such that

x(t+\omega)=x(t)

for some $\omega>0$ .

The last equality implies that $x(t)$ is either a closed trajectory or a rest point of the flow $\phi$ .

Definition 2.1 .

Let $R\in(0,\,\infty]$ be fixed. We denote by $\mathfrak{L}_{R}$ the class of smooth functions $\vartheta\colon\mathbb{R}\to\mathbb{R}$ for which a quotient, and hence all quotients (see comment at the beginning of the Proof of Theorem 1 ) of independent solutions of

(2.5)

y^{\prime\prime}(x)+\vartheta(x)\,y(x)\,=\,0

have meromorphic extensions to the $R$ -strip

\{z\,\in\mathbb{C}\,,|\Im z|\,<\,R\,\}

that are holomorphic and with non-vanishing imaginary part for $0\,<\,\Im z\,<\,R$ .

1.1 Definition .

Let $V$ be a graded vector space, $n\geq 2$ , and $\mu:{V}^{\otimes n}\to V$ a degree $d$ linear map. The couple $A=(V,\mu)$ is a degree $d$ totally associative $n$ -ary algebra if, for each $1\leq i,j\leq n$ ,

(1)

\mu\left(\hbox{$1\hskip-3.0pt1$}^{\otimes i-1}\otimes\mu\otimes\hbox{$1\hskip-% 3.0pt1$}^{\otimes n-i}\right)=\mu\left(\hbox{$1\hskip-3.0pt1$}^{\otimes j-1}% \otimes\mu\otimes\hbox{$1\hskip-3.0pt1$}^{\otimes n-j}\right),

where $\hbox{$1\hskip-3.0pt1$}:V\to V$ denotes the identity map.