A wavepacket approach to periodically driven scattering

Definition 2.4 .

If $f(x)$ is a real-valued analytic function of $x$ in a neighborhood of $a$ , we define the sign of $f$ at $a$ $\operatorname{sgn}(f,a)$ to be the sign of the first nonvanishing Taylor series coefficient (around $a$ ), if there is such, and zero otherwise. In other words, if $f\neq 0$ , we have:

\operatorname{sgn}(f,a)=\operatorname{sgn}(f^{(n)}(a))\,\in\{-1,1\},

where $f^{(k)}(a)=0$ for $k<n$ and $f^{(n)}(a)\neq 0$ .

Definition 2.1

A Courant algebroid is a vector bundle $E\to M$ with a Loday bracket on $\Gamma E$ , i.e., an $\mathbb{R}$ -bilinear map satisfying the Jacobi identity,

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

for all $x,y,z\in\Gamma E$ , an anchor, $\rho:E\to TM$ , which is a morphism of vector bundles, and a field of non-degenerate symmetric bilinear forms $(~{}|~{})$ on the fibers of $E$ , satisfying

	$\displaystyle{\rm(i)}\quad\quad\quad\rho(x)(u\|v)$	$\displaystyle=(x~{}\|~{}[u,v]+[v,u])\ ,$
	$\displaystyle{\rm(ii)}\quad\quad\quad\rho(x)(u\|v)$	$\displaystyle=([x,u]~{}\|~{}v)+(u~{}\|~{}[x,v])\ ,$

for all $x$ , $u$ and $v\in\Gamma E$ .

Definition 1.5

A Poisson algebra $A$ over $k$ is called exact if there exists a derivation $\xi:A\to A$ such that

(1.3)

\xi(\{a,b\})=\{a,b\}+\{\xi(a),b\}+\{a,\xi(b)\}

for any $a,b\in A$ .

Definition 2

A Lie algebroid $(E,\rho,[\cdot,\cdot])$ is a vector bundle $\pi\colon E\to M$ together with a bundle map (“anchor”) $\rho\colon E\to TM$ and a Lie algebra structure $[\cdot,\cdot]\colon\Gamma(E)\times\Gamma(E)\to\Gamma(E)$ satisfying the Leibniz identity

[\psi,f\psi^{\prime}]=f[\psi,\psi^{\prime}]+\left(\rho(\psi)f\right)\psi^{\prime}

(9)

$\forall\psi,\psi^{\prime}\in\Gamma(E),\forall f\in C^{\infty}(M)$ .

Definition 2.1

A Poisson manifold $(M,\{\cdot,\cdot\})$ is a manifold endowed with a Poisson bracket, that is a bilinear antisymmetric composition laws defined on the space $C^{\infty}(M)$ satisfying:

1.

The Leibnitz rule: $\{fg,h\}=f\{g,h\}+g\{f,h\}$ ;
2.

The Jacobi identity $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$ .

Definition 2.1 .

An orthomodular lattice ( OML ) is an algebraic structure $\langle L,\cup,^{\perp}\rangle$ in which the following conditions are satisfied for any $a,b,c\in L$ :

L1. $a\leq a^{\perp\perp}\quad\&\quad a^{\perp\perp}\leq a$

L2. $a\leq a\cup b\quad\&\quad b\leq a\cup b$

L3. $a\leq b\quad\&\quad b\leq a\quad\Rightarrow\quad a=b$

L4. $a\leq 1$

L5. $a\leq b\quad\Rightarrow\quad b^{\perp}\leq a^{\perp}$

L6. $a\leq b\quad\&\quad b\leq c\quad\Rightarrow\quad a\leq c$

L7. $a\leq c\quad\&\quad b\leq c\quad\Rightarrow\quad a\cup b\leq c$

L8. $a\rightarrow_{i}b\>=\>1$ $\qquad\Rightarrow\qquad$ $a\leq b\qquad\qquad(i=1,\!...,5)$

where $a\leq b{\buildrel\rm def\over{\Leftrightarrow}}a\cup b=b$ , $1\ {\buildrel\rm def\over{=}}\ a\cup a^{\perp}$ . Also

a\cap b\ {\buildrel\rm def\over{=}}\ (a^{\perp}\cup b^{\perp})^{\perp},\qquad 0% \ {\buildrel\rm def\over{=}}\ a\cap a^{\perp}.

and the implications $a\rightarrow_{i}b\ (i=1,\dots,5)$ are defined as follows

$a\rightarrow_{1}b\ \ {\buildrel\rm def\over{=}}\ \ a^{\perp}\cup(a\cap b)$ (Sasaki)

$a\rightarrow_{2}b\ \ {\buildrel\rm def\over{=}}\ \ b\cup(a^{\perp}\cap b^{% \perp})$ (Dishkant)

$a\rightarrow_{3}b\ \ {\buildrel\rm def\over{=}}\ \ ((a^{\perp}\cap b)\cup(a^{% \perp}\cap b^{\perp}))\cup\bigl{(}a\cap(a^{\perp}\cup b)\bigr{)}$ (Kalmbach)

$a\rightarrow_{4}b\ \ {\buildrel\rm def\over{=}}\ \ ((a\cap b)\cup(a^{\perp}% \cap b))\cup\bigl{(}(a^{\perp}\cup b)\cap b^{\perp}\bigr{)}$ (non-tollens)

$a\rightarrow_{5}b\ \ {\buildrel\rm def\over{=}}\ \ ((a\cap b)\cup(a^{\perp}% \cap b))\cup(a^{\perp}\cap b^{\perp})$ (relevance)

Definition 1.2 .

A character $\varphi$ of a graded Hopf algebra ${\mathcal{H}}$ is said to be even if

\bar{\varphi}=\varphi

and it is said to be odd if

\bar{\varphi}=\varphi^{-1}\,.

Definition 2 (Quaternion Algebras) .

Let $F$ be a field of characteristic $\neq~{}2$ . For $a,b\in F^{*}$ , let $H(a,b)$ be the $F$ -algebra with basis $1,i,j,k$ (as an $F$ -vector space) and with multiplication rules

i^{2}=a,j^{2}=b,ij=k=-ji.

Then $H(a,b)$ is an $F$ -algebra which is called a quaternion algebra over $F$ .

Definition 1.1

A Poisson algebra over the field $k$ is a commutative algebra $A$ over $k$ equipped with an additional skew-linear operation $\{-,-\}:A\otimes A\to A$ such that

(1.1)

\{a,bc\}=\{a,b\}c+\{a,c\}b\quad,\quad 0=\{a,\{b,c\}\}+\{b,\{c,a\}\}+\{c,\{a,b% \}\},

for all $a,b,c\in A$ . An ideal $I\subset A$ is called a Poisson ideal if $\{i,a\}\in I$ for any $i\in I$ , $a\in A$ .

	$\displaystyle{\rm(i)}\quad\quad\quad\rho(x)(u\|v)$	$\displaystyle=(x~{}\|~{}[u,v]+[v,u])\ ,$
	$\displaystyle{\rm(ii)}\quad\quad\quad\rho(x)(u\|v)$	$\displaystyle=([x,u]~{}\|~{}v)+(u~{}\|~{}[x,v])\ ,$