BNL - 64502 CAP- 171-MISC-97C

WAVELET APPROACH TO ACCELERATOR PROBLEMS: 11. METAPLECTIC WAVELETS*

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A. Fedorova and M. Zeitlin

Russian Academy of Sciences S. Petersburg, Russia

Inst. of Problems of Mechanical Engineering JuL 2 !I l;z7

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2. Parsa Department of Physics

Brookhaven National Laboratory Upton, NY 11973

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* T h i s w o r k w a s p e r f o r m e d under the a u s p i c e s of the U . S . D e p a r t m e n t o f E n e r g y under Contract N o . DE-AC02-76CH00016.

May 1997

M CENTER FOR ACCELERATOR PHYSICS

B R O O K H A V E N N A T I O N A L L A B O R A T O R Y A S S O C I A T E D U N I V E R S I T I E S , I N C .

Under Contract No DE-AC02-76CH00016 with the

U N I T E D S T A T E S D E P A R T M E N T O F ENERGY Accelerator i3nferEnce, Acc- science, Techndogy and A p p w ' , V m o u v a , aC., -, M9' 12-16.1997.

DISCLAIMER

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WAVELET APPROACH TO ACCELERATOR PROBLEMS, 11. METAPLECTIC WAVELETS

A. Fedorova and M. Zeitlin, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Russia, 199 178, St. Petersburg,

V.O., Bolshoj pr., 6 1 , e-mail: zeitlin@math.ipme.ru 2. Parsa, Dept. of Physics, Bldg. 901A, Brookhaven National Laboratory,

Upton, N Y 1 1973-5000, USA, e-mail: parsa@bnl.gov

/

Abstmci

This is the second part of a series of talks in which we present applications of wavelet analysis to polynomial ap- proximations for a number of accelerator physics problems. According to the orbit method and by using construction from the geometric quantization theory we construct the symplectic and Poisson structures associated with gener- alized wavelets by using metaplectic structure and corre- sponding polarization. The key point is a consideration of semidirect product of Heisenberg group and metaplectic group as subgroup of automorphisms group of dual to sym- plectic space, which consists of elements acting by affine transformations.

1 INTRODUCTION

In this paper we continue the application of powerful meth- ods of wavelet analysis to polynomial approximations of nonlinear accelerator physics problems. In part 1 we con- sidered our main example and general approach for con- structing wavelet representation for orbital motion in stor- age rings. But now we need take into account the Hamilto- nian or symplectic structure related with system (1) from part 1. Therefore, we need to consider instead of com- pactly supported wavelet representation from part 1 the generalized wavelets, which allow us to consider the corre- sponding symplectic structures. By using the orbit method and constructions from the geometric quantization theory we consider the symplectic and Poisson structures associ- ated with Weyl- Heisenberg wavelets by using metaplec- tic structure and the corresponding polarization. In part 3 we consider applications to construction of Melnikov func- tions in the theory of homoclinic chaos in perturbed Hamil- tonian systems.

In wavelet analysis the following three concepts are used now: 1). a square integrable representation U of a group G, 2). coherent states over G 3). the wavelet transform associ- ated to U.

a) the affine (as + b) group, which yields the usual wavelet analysis

We have three important particular cases:

b). the Weyl-Heisenberg group which leads to the Gabor functions. i.e. coherent states associated with windowed

Fourier transform.

In both cases time-frequency plane corresponds to the phase space of group representation. c). also, we have the case of bigger group, containing both affine and Weyl-Heisenberg group, which interpolate be- tween affine wavelet analysis and windowed Fourier anal- ysis: affine Weyl-Heisenberg group [7]. But usual repre- sentation of it is not square-integrable and must be mod- ified: restriction of the representation to a suitable quo- tient space of the group (the associated phase space in that case) restores square - integrability: G=wH+ homoge- neous space. Also, we have more general approach which allows to consider wavelets corresponding to more general groups and representations [8], [9]. Ourgoal is applications of these results to problems of Hamiltonian dynamics and as consequence we need to take into account symplectic nature of our dynamical problem. Also, the syrnplectic and wavelet structures must be consistent (this must be resem- ble the symplectic or Lie-Poisson integrator theory). We use the point of view of geometric quantization theory (or- bit method) instead of harmonic analysis. Because of this we can consider (a) - (c) analogously.

2 METAPLECTIC GROUP AND REPRESENTATIONS

Let Sp( n) be symplectic group, M p ( n) be its unique two- fold covering - metaplectic group. Let V be a symplec- tic vector space with symplectic form ( , ), then R @ V is nilpotent Lie algebra - Heisenberg algebra:

[R,V]=O, [ u , w ] = ( u , w ) E R , [V,V]=R.

Sp( V ) is a group of automorphisms of Heisenberg algebra. Let N be a group with Lie algebra R e V , i.e. Heisenberg

group. By Stone- von Neumann theorem Heisenberg group has unique irreducible unitary representation in which 1 c) i. This representation is projective: U,, UgZ = c(gl,g2) . UgIg2, where c is a map: Sp( V ) x Sp( V ) -b S', i.e. c is S' -cocycle.

But this representation is unitary representation of uni- versal covering, Le. metaplectic group M p ( V ) . We give this remesentation without Stone-von Neumann theorem.

Consider a new group F = N' w M p ( V ) , w is semidi- rect product (we consider instead of N = R @ V the N' = S' x V, S' = (R /27rZ) ) . Let V' be dual to V, G( V') be automorphism group of V'.Then F is subgroup of G( V'), which consists of elements, which acts on V' by affine transformations. This is the key point!

Let q i , ..., q n ; p l , ..., pn be symplectic basis in V, Q = pdq = C p i d q i and d o be symplectic form on V'. Let M be fixed affme polarization, then for u E F the map u c) Q, gives unitary representation of G: 0, : X ( M ) +

Explicitly we have for representation of N on H(M): H ( W

( @ q f ) ' ( X ) = e - i q z f ( x ) , Q , f ( x ) = f ( x - P)

The representation of N on HW) is irreducible. Let A,, A, be infinitesimal operators of this representation

then

Now we give the representation of infinitesimal basic ele- ments. Lie algebra of the group F is the algebra of all (non- homogeneous) quadratic polynomials of (p,q) relatively Poisson bracket (PB). The basis of this algebra consists of elements 1 , q i , ..., qn. p i , . . . , p n ? qiqjyqipj, PiPj, i , j = 1 ,..., n, i < j , -

so, we have the representation of basic elements f e A f : 1 C , i , q k +hiXk,

This gives the structure of the Poisson manifolds to rep- resentation of any (nilpotent) algebra or in other words to continuous wavelet transform.

3 THE SECAL-BARCMAN REPRESENTATION

Let

p = ( p i , ..., P n ) , tions of n complex variables with (f, f ) < 00, where

F n is the space of holornorphic func-

Consider a map U : H + Fn , where H is with real polarization,F,, is with complex polarization, then we have

i.e. the Bargmann formula produce wavelets.We also have the representation of Heisenberg algebra on Fn :

j

andalso: w = d o = d p A d q , wherep = i z d z .

4 ORBITAL THEORY FOR WAVELETS

Let coadjoint action be

< g . f , Y >=< f , Ad(g)-'Y >, where<,>ispair inggEG, f E g ' , Y E G .

The orbit is O f = G . f 5 G/G(f). AlsoJet A=A(M) be algebra of functions, V(M) is A-

module of vector fields, AJ' is A-module of pforms. Vector fields on orbit is

where 4 E A(O) , symplectic manifolds with 2-form

f E 0. Then Of are homogeneous

and dSZ = 0. PB on 0 have the next form

where p is A'(O) + V(0) with definition Sl(p(a) , X) = i ( X ) a . Here 41, 42 E A ( 0 ) and A(O) is Lie algebra with bracket { ,}. Now let N be a Heisenberg group. Consider adjoint and coadjoint representations in some particular case. N = ( z , t ) E C x R, z = p + iq; compositions in N are (z,t)-(z', t') = (z+z',t+t'+B(z, 2)). where B ( z , z') = pq' - qp'. Inverse element is (4, -z) . Lie algebra n of N is (C, T ) E C x R with bracket [ (C, T ) , (<', T') ] = (0, B(<, C')) . Centre is i E n and generated by (0.1); 2 is a subgroup exp 2. Adjoint representation N on n is given by formula

1 1 z = - ( p - i q ) , Z = - ( p + i q ) , . . . . . , Jz fi

Coadjoin t: forf E n', g = ( z , t ) ,

( 9 . f)(C, C) = f (C1 - C ) f ( O , 1)

then orbits for which f l ; # 0 are plane in n' given by equation f(0, I ) = /..A . If X = (C,O), Y = (C', 0), X, Y E n then symplectic structure is

Q(a(O, XI, , 40, Y ) f ) =< f, [X, YI >= f (0 , B(C, C'))PB(C, C')

Also we have for orbit 0, = N / Z and 0, is Hamiltonian G-space.

5 KIRILLOV CHARACTER FORMULA OR ANALOGY OF GABOR WAVELETS

Let U denote irreducible unitary representation of N with condition U(0 , t ) = exp(it1) . 1, where k' # 0,then U is equivalent to representation which acts in L2(B) ac- cording to

Z ( Z , W.) = exp (W + P.)) 4(z - Q) If instead of N we consider E(2)/R we have SI case and we have Gabor functions on S' .

6 OSCILLATOR GROUP

Let 0 be an oscillator gr0upj.e. semidirect product of R and Heisenberg group N.

Let H,P,Q,I be standard basis in Lie algebra o of the group0 and H', P*, Q', I' be dual basis in 0.. Let func- tional f=(ab,c,d) be

al' + bP' + cQ' + d H ' .

Let us consider complex polarizations

h = ( H , I , P + i Q ) , 6 = ( I , H , P - io) Induced from h representation, corresponding to functional f (for a > 0), unitary equivalent to the representation

is an operator, which according to Stone-von Neumann theorem has the property

/ U t ( n ) = v( t )~(n)v( t ) - ' .

This is our last private case, but according to our ap- proach we can construct by using methods of geometric quantization theory many "symplectic wavelet construc- tions" with corresponding symplectic or Poisson structure on it. Very useful particular spline-wavelet basis with uni- form exponential.contro1 on stratified and nilpotent Lie groups was considered in [9].

Extended version and related results may be found in [ 11- [61. This research was supported in part under "New Ideas

for Particle Accelerators Program" NSF-Grant no. PHY94- 07194.

7 REFERENCES

A.N. Fedorova, M.G. Zeitlin: Proc. of 22 Summer School'Nonlinear Oscillations in Mechanical Systems' St. Petenburg (1995) 97. A.N. Fedorova, M.G. Zeitlin: Proc. of 23 Summer School 'Nonlinear Oscillations in Mechanical Systems' St. Peters- burg (1996) 322. AN. Fedorova, M.G. Zeitlin: Proc. 4 Int Congress on Sound and Vibration, Russia (1996) 1483. AN. Fedorova, M.G. Zeitlin: Proc. 7th IEEE DSP Workshop, Noway (1 996) 409. AN. Fedorova. M.G. Zeitlin: Proc. 2nd WACS Symp. on Math. Modelling, ARGESIM Report 11, Austria (1997) 1083. AN. Fedorova, M.G. Zeitlin: EUROMECH-2nd European Nonlinear Oscillations Conf. (1996) 79. C. Kalisa, B. Torresani: Ndimensional Affine Weyl- Heisenberg Wavelets preprint CPT-92 P.2811 Marseille (1 992). T. Kawazoe: F'roc. Japan Acad. 71 Ser. A (1995) 154. P.G. Lemarie: Proc. Int. Math. Congr., Satellite Symp. (1991) 154.

[IO] AN. Fedorova, M.G. Zeitlin. 2. Parsa, This Proceedings. w(t, n)f(y) = exp(it(h - 1/2)) . Ua(n)V( t ) ,

where

V(t) = exp[-it(P2 + Q 2 ) / 2 a ] , p = - d / d x , Q = i a x ,

and W , (n) is irreducible representation of N, which have the form U, (2) = exp( iaz) on the center of N. Here we have: U(n=(x,y,z)) is Schrijdinger representa- tion, Ut(n) = V ( t ( n ) ) is the representation,which ob- tained from previous by automorphism (time translation) R -+ t (n ) ; U t ( n ) = U ( t ( n ) ) is also unitary irreducible representation of N.

~ ( t ) = e x p ( i t ( P + Q~ + /t - 1/21)

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