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\begin{document}
\title{ WAVELET APPROACH TO HAMILTONIAN, CHAOTIC AND
QUANTUM CALCULATIONS IN ACCELERATOR PHYSICS}
\author{A.~Fedorova and  \underline{M.~Zeitlin},
Institute of Problems of Mechanical Engineering,\\
 Russian Academy of Sciences, Russia, 199178, St.~Petersburg,\\
  V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru\\
Z.~Parsa, Dept. of Physics, Bldg.~901A, Brookhaven National Laboratory,\\
 Upton, NY 11973-5000, USA, e-mail: parsa@bnl.gov}
\maketitle
\begin{abstract}
This is the second part of  two our papers in which we
present applications of wavelet
analysis to polynomial approximations for a number of
accelerator physics problems. We consider applications of very
useful fast wavelet transform technique to calculations in
symplectic scale of spaces and to quasiclassical evolution dynamics.
\end{abstract}

\section{INTRODUCTION}
This is the second part of two our presentations in which we consider
applications of methods from wavelet analysis to nonlinear accelerator
physics problems. This is a continuation of our results from [1], [2],
which is based on approach of two of us from [3], [4] to investigation
of nonlinear problems -- general, with additional structures (Hamiltonian,
symplectic or quasicomplex), chaotic, quasiclassical, quantum, which are
considered in the framework of local(nonlinear) Fourier analysis, or wavelet
analysis. Wavelet analysis is a relatively novel set of mathematical
methods, which gives us a possibility to work with well-localized bases in
functional spaces and with the general type of operators (differential,
integral, pseudodifferential) in such bases.
Now we consider applications of very useful and powerful method of fast wavelet
transform to the problems which appear in nonlinear orbital dynamics in
storage rings [5]. The first problem is the explicit calculation of
quasiclassical evolution which we consider in section 2 and the second problem,
which we consider in section 3, is calculations in (perturbed) Hamiltonian
systems in cases when we need to consider multiresolution expansion not in one
functional space but in infinite scale of spaces with underlying symplectic
structure. In section 4 we consider the key point  of this approach which gives
useful maximally sparse
representation of differential operator that allows us to take into account
contribution from each level of resolution.


\section{QUASICLASSICAL EVOLUTION}

Let we consider classical and quantum dynamics in phase space
$\Omega=R^{2m}$ with coordinates $(x,\xi)$ and generated by
Hamiltonian ${\cal H}(x,\xi)\in C^\infty(\Omega;R)$.
If $\Phi^{\cal H}_t:\Omega\longrightarrow\Omega$ is (classical) flow then
time evolution of any bounded classical observable or
symbol $b(x,\xi)\in C^\infty(\Omega,R)$ is given by $b_t(x,\xi)=
b(\Phi^{\cal H}_t(x,\xi))$. Let $H=Op^W({\cal H})$ and $B=Op^W(b)$ are
the self-adjoint operators or quantum observables in $L^2(R^n)$,
representing the Weyl quantization of the symbols ${\cal H}, b$ [5]
\begin{eqnarray*}
(Bu)(x)&&=\frac{1}{(2\pi\hbar)^n}\int_{R^{2n}}b\left(\frac{x+y}{2},\xi\right)
\cdot\\
&&e^{i<(x-y),\xi>/\hbar}u(y)dyd\xi,
\end{eqnarray*}
where $u\in S(R^n)$ and $B_t=e^{iHt/\hbar}Be^{-iHt/\hbar}$ be the
Heisenberg observable or quantum evolution of the observable $B$
under unitary group generated by $H$. $B_t$ solves the Heisenberg equation of
motion
$$\dot{B}_t=\frac{i}{\hbar}[H,B_t].$$
Let $b_t(x,\xi;\hbar)$ is a symbol of $B_t$ then we have
 the following equation for it
\begin{equation}
\dot{b}_t=\{ {\cal H}, b_t\}_M,
\end{equation}
with initial condition $b_0(x,\xi,\hbar)$ $=b(x,\xi)$.
Here $\{f,g\}_M(x,\xi)$ is the Moyal brackets of the observables
$f,g\in C^\infty(R^{2n})$, $\{f,g\}_M(x,\xi)=f\sharp g-g\sharp f$,
where $f\sharp g$ is the symbol of the operator product and is presented
by the composition of the symbols $f,g$
\begin{eqnarray*}
&&(f\sharp g)(x,\xi)=\frac{1}{(2\pi\hbar)^{n/2}}\int_{R^{4n}}
e^{-i<r,\rho>/\hbar+i<\omega,\tau>/\hbar}\\
&&\cdot f(x+\omega,\rho+\xi)g(x+r,\tau+\xi)d\rho d\tau drd\omega.
\end{eqnarray*}
For our problems it is useful that $\{f,g\}_M$ admits the formal
expansion in powers of $\hbar$:
$\{f,g\}_M(x,\xi)\sim \{f,g\}+2^{-j}
\sum_{|\alpha+\beta|=j\geq 1}(-1)^{|\beta|}$.
$(\partial^\alpha_\xi fD^\beta_x g)\cdot(\partial^\beta_\xi
 gD^\alpha_x f)$, where $\alpha=(\alpha_1,\dots,\alpha_n)$ is
a multi-index, $|\alpha|=\alpha_1+\dots+\alpha_n$,
$D_x=-i\hbar\partial_x$.
So, evolution (1) for symbol $b_t(x,\xi;\hbar)$ is
\begin{eqnarray}
\dot{b}_t=&&\{{\cal H},b_t\}+\frac{1}{2^j}
\sum_{|\alpha|+\beta|=j\geq 1}(-1)^{|\beta|}\\
&&\cdot\hbar^j
(\partial^\alpha_\xi{\cal H}D_x^\beta b_t)\cdot
(\partial^\beta_\xi b_t D_x^\alpha{\cal H}).\nonumber
\end{eqnarray}

At $\hbar=0$ this equation transforms to classical Liouville equation
\begin{equation}
\dot{b}_t=\{{\cal H}, b_t\}.
\end{equation}
Equation (2) plays a key role in many quantum (semiclassical) problem.
We note only the problem of relation between quantum and classical evolutions
or how long the evolution of the quantum observables is determined by the
corresponding classical one.
Our approach to solution of systems (2), (3) is based on our technique
from [1]-[4] and very useful linear parametrization for differential operators
which we present in section 4.


\section{Symplectic Hilbert Scales via Wavelets}

We can solve many important dynamical problems such that KAM
perturbations, spread of energy to higher modes, week turbulence, growths of
solutions of Hamiltonian equations only if we consider scales of spaces instead
of one functional space. For Hamiltonian system and their perturbations
for which we need take into account underlying symplectic structure we
need to consider symplectic scales of spaces. So, if
$\dot{u}(t)=J\nabla K(u(t))$
is Hamiltonian equation we need wavelet description of symplectic or
quasicomplex structure on the level of functional spaces. It is very
important that according to [8] Hilbert basis is in the same time a
Darboux basis to corresponding symplectic structure.
We need to provide Hilbert scale $\{Z_s\}$ with symplectic structure [7], [9].
All what we need is the following.
 $J$  is a linear operator, $J : Z_{\infty}\to Z_\infty$,
$J(Z_\infty)=Z_\infty$, where $Z_\infty =\cap Z_s$.
$J$ determines an isomorphism of  scale $\{Z_s\}$ of order $d_J\geq 0$.
 The operator $J$ with domain of definition $Z_\infty$ is
antisymmetric in $Z$:
$
<J z_1,z_2>_Z=-<z_1,J z_2>_Z, z_1,z_2 \in $ $ Z_\infty
$.
Then the triple
$$\{Z,\{Z_s|s\in R\},\quad
\alpha=<\bar J dz,dz>\}
$$
 is symplectic Hilbert scale. So, we may consider
any dynamical Hamiltonian problem on functional level.
As an example, for KdV equation we have
$ Z_s=\{u(x)\in H^s(T^1)|\int^{2\pi}_0 u(x)dx=0\},\
s\in R,\quad J=\partial/\partial x,$
J is isomorphism  of the scale of  order one, $\bar J=-(J)^{-1}$ is
isomorphism of order $-1$.
According to [10] general functional spaces and scales of spaces such as
Holder--Zygmund, Triebel--Lizorkin and Sobolev can be characterized
through wavelet coefficients or wavelet transforms. As a rule, the faster
the wavelet coefficients decay, the more the analyzed function is
regular [10]. Most important for us example is the scale of Sobolev spaces.
Let $H_k(R^n)$ is the Hilbert space of all distributions with finite norm
$
\Vert s\Vert^2_{H_k(R^n)}=\int d\xi(1+\vert\xi\vert^2)^{k/2}\vert
\hat s(\xi)\vert^2.
$
Let us consider wavelet transform
$$
W_g f(b,a)=\int_{R^n}dx\frac{1}{a^n}\bar g\left(\frac{x-b}{a}\right) f(x),
$$
$ b\in R^n, \quad a>0$,
w.r.t. analyzing wavelet $g$, which is strictly admissible, i.e.
$$
C_{g,g}=\int_0^\infty\frac{da}{a}\vert\bar{\hat g(ak)}\vert^2<\infty.
$$
Then there is a $c\geq 1$ such that
\begin{eqnarray*}
c^{-1}\Vert s \Vert^2_{H_k(R^n)}&\leq&\int_{H^n}
\frac{dbda}{a}(1+a^{-2\gamma})\vert W_gs(b,a)\vert^2\\
&\leq&
c\|s\|^2_{H_k(R^n)}
\end{eqnarray*}
This shows that localization of the wavelet coefficients at small
scale is linked to local regularity.


\section{FAST WAVELET TRANSFORM FOR DIFFERENTIAL OPERATORS}

Let us consider multiresolution representation
$\dots\subset V_2\subset V_1\subset V_0\subset V_{-1}
\subset V_{-2}\dots$ (see our other paper from this proceedings for
details of wavelet machinery). Let T be an operator $T:L^2(R)
\rightarrow L^2(R)$, with the kernel $K(x,y)$ and
$P_j: L^2(R)\rightarrow V_j$ $(j\in Z)$ is projection
operators on the subspace $V_j$ corresponding to j level of resolution:
$(P_jf)(x)=\sum_k<f,\varphi_{j,k}>\varphi_{j,k}(x)$. Let
$Q_j=P_{j-1}-P_j$ is the projection operator on the subspace $W_j$ then
we have the following "microscopic or telescopic"
representation of operator T which takes into account contributions from
each level of resolution from different scales starting with
coarsest and ending to finest scales:
$$
T=\sum_{j\in Z}(Q_jTQ_j+Q_jTP_j+P_jTQ_j)
$$
We remember that this is a result of presence of affine group inside this
construction.
The non-standard form of operator representation [11] is a representation of
an operator T as  a chain of triples
$T=\{A_j,B_j,\Gamma_j\}_{j\in Z}$, acting on the subspaces $V_j$ and
$W_j$:
$$
 A_j: W_j\rightarrow W_j, B_j: V_j\rightarrow W_j,
\Gamma_j: W_j\rightarrow V_j,
$$
where operators $\{A_j,B_j,\Gamma_j\}_{j\in Z}$ are defined
as $A_j=Q_jTQ_j, B_j=Q_jTP_j, \Gamma_j=P_jTQ_j$.
The operator $T$ admits a recursive definition via
$T_j=
\left(\begin{array}{cc}
A_{j+1} & B_{j+1}\\
\Gamma_{j+1} & T_{j+1}
\end{array}\right)$,
where $T_j=P_jTP_j$ and $T_j$ works on $ V_j: V_j\rightarrow V_j$.
It should be noted that operator $A_j$ describes interaction on the
scale $j$ independently from other scales, operators $B_j,\Gamma_j$
describe interaction between the scale j and all coarser scales,
the operator $T_j$ is an "averaged" version of $T_{j-1}$.


The operators $A_j,B_j,\Gamma_j,T_j$ are represented by matrices
$\alpha^j, \beta^j, \gamma^j, s^j$
\begin{eqnarray}
\alpha^j_{k,k'}&=&\int\int K(x,y)\psi_{j,k}(x)\psi_{j,k'}(y)dxdy\nonumber\\
\beta^j_{k,k'}&=&\int\int K(x,y)\psi_{j,k}(x)\varphi_{j,k'}(y)dxdy\\
\gamma^j_{k,k'}&=&\int\int K(x,y)\varphi_{j,k}(x)\psi_{j,k'}(y)dxdy\nonumber\\
s^j_{k,k'}&=&\int\int K(x,y)\varphi_{j,k}(x)\varphi_{j,k'}(y)dxdy\nonumber
\end{eqnarray}
We may compute the non-standard representations of operator $d/dx$ in the
wavelet bases by solving a small system of linear algebraical
equations. So, we have for objects (4)
\begin{eqnarray*}
\alpha^j_{i,\ell}&=&2^{-j}\int\psi(2^{-j}x-i)\psi'(2^{-j}-\ell)2^{-j}dx\\
&=&2^{-j}\alpha_{i-\ell}\\
\beta^j_{i,\ell}&=&2^{-j}\int\psi(2^{-j}x-i)\varphi'(2^{-j}x-\ell)2^{-j}dx\\
&=&2^{-j}\beta_{i-\ell}\\
\gamma^j_{i,\ell}&=&2^{-j}\int\varphi(2^{-j}x-i)\psi'(2^{-j}x-\ell)2^{-j}dx\\
&=&2^{-j}\gamma_{i-\ell},
\end{eqnarray*}
where
\begin{eqnarray*}
\alpha_\ell&=&\int\psi(x-\ell)\frac{d}{dx}\psi(x)dx\\
\beta_\ell&=&\int\psi(x-\ell)\frac{d}{dx}\varphi(x)dx\\
\gamma_\ell&=&\int\varphi(x-\ell)\frac{d}{dx}\psi(x)dx
\end{eqnarray*}
then by using refinement equations
\begin{eqnarray*}
\varphi(x)&=&\sqrt{2}
\sum^{L-1}_{k=0}h_k\varphi(2x-k),\\
 \psi(x)&=&\sqrt{2}\sum_{k=0}^{L-1}g_k\varphi(2x-k),
\end{eqnarray*}
$g_k=(-1)^kh_{L-k-1}, k=0,\dots,L-1$ we have in terms of filters
$(h_k,g_k)$:
\begin{eqnarray*}
\alpha_j&=&2\sum^{L-1}_{k=0}\sum^{L-1}_{k'=0}g_kg_{k'}r_{2i+k-k'},\\
\beta_j&=&2\sum^{L-1}_{k=0}\sum^{L-1}_{k'=0}g_kh_{k'}r_{2i+k-k'},\\
\gamma_i&=&2\sum^{L-1}_{k=0}\sum^{L-1}_{k'=0}h_kg_{k'}r_{2i+k-k'},
\end{eqnarray*}
where $r_\ell=\int\varphi(x-\ell)\frac{d}{dx}\varphi(x)dx, \ell\in Z$.
Therefore, the representation of $d/dx$ is completely determined by the
coefficients $r_\ell$ or by representation of $d/dx$ only on
the subspace $V_0$. The coefficients $r_\ell, \ell\in Z$ satisfy the
following system of linear algebraical equations
$$
r_\ell=2\left[ r_{2l}+\frac{1}{2} \sum^{L/2}_{k=1}a_{2k-1}
(r_{2\ell-2k+1}+r_{2\ell+2k-1}) \right]
$$
and $\sum_\ell\ell r_\ell=-1$, where $a_{2k-1}=$
$2\sum_{i=0}^{L-2k}h_i h_{i+2k-1}$, $k=1,\dots,L/2$
are the autocorrelation coefficients of the filter $H$.
If a number of vanishing moments $M\geq 2$ then this linear system of equations
has a unique solution with finite number of non-zero $r_\ell$,
$r_\ell\ne 0$ for $-L+2\leq\ell\leq L-2, r_\ell=-r_{-\ell}$.
For the representation of operator $d^n/dx^n$ we have the similar reduced
linear system of equations.
Then finally we have for action of operator $T_j(T_j:V_j\rightarrow V_j)$
on sufficiently smooth function $f$:
$$
(T_j f)(x)=\sum_{k\in Z}(2^{-j}\sum_{\ell}r_\ell f_{j,k-\ell})
\varphi_{j,k}(x),
$$
where $\varphi_{j,k}(x)=2^{-j/2}\varphi(2^{-j}x-k)$ is wavelet basis and
$$
f_{j,k-1}=2^{-j/2}\int f(x)\varphi(2^{-j}x-k+\ell)dx
$$
are wavelet coefficients. So, we have simple linear para\-met\-rization of
matrix representation of our differential operator in wavelet basis
and of the action of
this operator on arbitrary vector in our functional space.

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