From Resonances to Master Equations

Vojkan Jaksic and ClaudeAlain Pillet

Departmentof Mathematics and Statistics UniversityofOttawa OttawaON KN N Canada

DepartementdePhysique Theorique Universitede Geneve CH Geneve Switzerland

Abstract Westudy the longtime b ehavior of the dynamics of a level

atom coupled to a scalar radiation eld at p ositive temp erature Wediscuss

the deep relations b etween results proven recently in JP JP and the

Davies master equation technique D D

From Resonances to Master Equations

Intro duction

This pap er is a note on the results we recently proved in JPJP There

we studied the dynamics of an op en quantum system Acharacterized by

discrete set of energy levels fe g allowed to interact with a large reservoir B

i

The reservoir is an innite free Bose gas at inverse temp erature without

BoseEinstein condensate Using algebraic and sp ectral techniques wehave

shown that in the weakcouplinghightemp erature regime the interacting

system A B has strong ergo dic prop erties In particular it approaches

thermal equilibrium exp onentially fast The techniques weemployed shed a

new lighton the origin and derivation of master equations for this class of

mo dels and it is this asp ect that wewould liketodiscuss here

The basic idea of our approachistoreduce ergo dic prop erties of the sys

tem to sp ectral problems for a distinguished selfadjointoperator the Liou

villean This op erator is dened in abstract terms We use Tomita Takesaki

theory to compute the Liouvillean and complex deformation techniques to

study its sp ectrum It turns out that complex resonances of the Liouvillean

carry critical information concerning physical mechanism of thermal relax

ation For instance it is wellknown that the time evolution of an op en

system is Markovian if the time variable is suitably rescaled the Van Hove

weakcoupling limit Weshowthat the generator of this Markovapproxi

mation arises as Fermis Golden Rule for the resonances of the Liouvillean

Our technique is not restricted to the second order p erturbation theoryand

gives an exact transp ort equation with convergentexpansion in the p owers

of the coupling constant The rst nontrivial contribution to this expansion

is the generator of the Markovapproximation

Although our results can b e presented in an abstract form as in D

weprefer to b e concrete and develop the theory on a sp ecic physically

imp ortantmodelThus inthispap er wewillstudy the dissipativedynamics

of a level atom or spin interacting with a free Bose gas In the

V Jaksic and CA Pillet

physics literature this mo del is known as the spinb oson system The master

equation whichgoverns the spin relaxation is the wellknown Blo chequation

The literature on the sub ject is enormous and is partially listed in JP We

remark that the results of this pap er havea straightforward extension to

mo dels in whichthe spin is replaced byan N level atom An extension of

our results to a varietyof related mo dels including nonrelativistic QED

will b e presented in JP

Originally this pap er was meanttobeapart of JP The mathematical

analysis of the mo dels in nonrelativistic QED is however very technical It

has b een suggested that results discussed here mightbeofinterest to a wider

audience and we decided to present them separately

Acknow ledgments We are grateful to H Sp ohn for useful discussions The

rst author was supp oted in part bytheNSERC the second by the Fonds

National Suisse This pap er was completed while the rst author was a

visitor at Californa Institute of Technology VJ is grateful to B Simon and

C Peckfor their hospitality

The mo del

Werecall that a W dynamical system is a pair M where M is a von

Neumann algebra weakly closed algebra of b ounded op erators on some

t

separable Hilb ert space and R t aweakly continuous group of

automorphisms of MTheelements of M are asso ciated with observables of

t

the quantum mechanical system under consideration The group sp ecies

their time evolution The physical states of the system are represented by

normalized continuous p ositivelinear functionals ie statesover MAstate

S is normal if there is a densitymatrix a p ositivetrace class op erator of

unit trace such that S ATrA A state S is faithful if S A A

implies A A quantum dynamical system is a triple M S where S

From Resonances to Master Equations

is a faithful normal invariant state

Thermal equilibrium states of quantum systems are characterized bythe

KMS condition

Denition Let M beaW dynamical system and Astate

S on M is a KMS state if it satises

S is normal

For any A B M thereexist a function F z analytic in the strip

AB

Imz continuous and boundedonits closure and satisfying the

KMS boundary conditions

t

F t S A B

AB

t

F t i S B A

AB

for t R

Remark AKMSstate is invariant and a unique KMS state is automat

ically faithful

In the sequel werestrict ourselves to quantum dynamical systems of the

form M S where S is a unique KMS state Tocharacterize the

ergo dic prop erties of suchsystems Robinson RO RO intro duced the

following notion

Denition A quantum dynamical system M S has the property of

return to equilibrium if for any A M and any normal state S one has

t

lim S A S A

jtj

For a detailed discussion of this denition and its various reformulations we

refer the reader to RO

Wenow describ e the spinb oson system Weintro duce rst the isolated

spin and the reservoir The Hilb ert space of the isolated spin is C Its

algebra of observables is M the algebra of all matrices on C We

V Jaksic and CA Pillet

denote by the usual Pauli matrices The Hamiltonian of this system

x y z

is chosen to b e H This Hamiltonian induces a ow

s z

t itH itH

s s

A e Ae

s

The eigenenergies of the spin are e and wedenote the corresp onding

eigenstates by Atinverse temp erature the equilibrium state of the

spin is dened by the Gibbs Ansatz

H

s

Tr e A S A

s

Z

s

where Z is a normalization factor It is wellknown that S is a unique

s s

KMS state on M

s

The heat reservoir is an innite gas of free massless b osons at p ositive

temp erature without BoseEinstein condensate The detailed mathematical

description of this system is presented in JP Thus here wewill just

intro duce the necessary notation referring the reader to JP for details

and additional information

The reservoir is describ ed bya triple fH H g where H is a Hilb ert

B B B B

space aunitvector in H and H a selfadjointoperator on H We

B B B B

denote by ktheenergy of a b oson with momentum k R We are

interested in the physically realistic case

kjkj

Our metho d easily accommo dates other disp ersion laws as long as the b osons

remain massless The equilibrium momentum distribution of the b osons is

given bythe Plancklaw

k

k

e

The space H carries a regular cyclic representation of Weyls algebra CCR

B

over the space of test functions

D ff k f L R g

From Resonances to Master Equations

suchthat

Z

kf k

W f exp jf kj kd k

B B B

R

i f

B

for f DWedenote by f the eld op erators W f e The

B B

op erator H is uniquely sp ecied bytherequirements

B

itH itH it

B B

e W f e W e f

B B

H

B B

Wedenote by M the Von Neumann algebra generated by fW f f Dg

B B

The triple fH H g and M are up to unitary equivalence uniquely

B B B B

determined by We remark that these structures can b e explicitly

identied see eg AW or JP Let

S A A

B B

B

and

itH itH t

B B

A e Ae

B

Then S is a unique KMS state on M Thequantum dynamical

B B

B

system M S denes the heat reservoir

B B

B

The spinb oson system is dened as follows The Hilb ert space of the

combined system is C H and its algebra of observables is MM M

B B

The Hamiltonian of the system is

H H I I H Q

s B B

where is a real constant Q and D In the sequel wewill

x

refer to as the form factor In JP wehaveshown that H is essentially

selfadjointon C D H foreach R provided

B

k k L R

V Jaksic and CA Pillet

Thus under this assumption

t itH itH

A e Ae

isaweakly continuous group of automorphisms of M and the spinb oson

mo del denes a W dynamical system M

The following result was proven for the rst time in FNV see also

Theorem in JP

Prop osition For any R and there exists a unique

KMS state S on M

In JP weinvestigated the ergo dic prop erties of the quantum dynamical

system M S To state the result we need some additional notation

Let S be the unit sphere in R and d its surface measure Let H L S

be the Hardy class of L S valued functions in the strip fz jImz j g

For a given function f on R wedene a new function f on R S by the

formula

s f sk if s

f s k

f jsjk if s jsj

We set the Hyp othesis

H The formfactor in Equation satises

k k L R

H There exists suchthat

H L S

R

jk j d k H



S

The Hyp othesis H is a technical condition related to the use of the

complex deformation technique The Hyp othesis H ensures that the spin

eectively couples to the reservoir at Bohrs frequency e e

Wehaveproven the following twotheoremsinJP

From Resonances to Master Equations

Theorem Suppose that Hypotheses HH hold Then for

thereexists a constant l depending only on the formfactor such

that the spinboson system has property of return to equilibrium for any real

satisfying jj l

The return to equilibrium is exp onentially fast in the following sense

Theorem Suppose that Hypotheses HH hold and let l beas

in Theorem Thereexists a norm dense set of normal states N and a

strongly dense algebra M Mboth independent of and such that for

jj l SN and A M one has

jtj t

AjC S Ae A S jS

for some C S A independent of and The function is positive for

jj l and satises

Z

jk j d k O



tanh

S

as

Remark By the b est estimate wehave l O as Thus the

ab ove results do not yield anyinformation concerning the zerotemp erature

mo del

The traditional approachtothe dynamics of op en quantum systems is

based on the use of master equations In this pap er wewould liketodiscuss

the relation b etween the master equation technique and the techniques we

intro duced in JP JP to proveTheorems and Although the

quantum mechanical master equations are extensively studied in the physics

literature HA KTH the rigorous results on the sub ject are scarce We will

discuss here only the mathematically rigorous results of Davies D D

These results played an imp ortantrole in our understanding of the sub ject

V Jaksic and CA Pillet

Davies theory

Davies theory is most conveniently develop ed in the Schrodinger picture

Weintro duce rst the necessary notation Let M be the predual of M ie

the Banach space of all normal linear functionals over MEvery such linear

functional is represented byatrace class op erator over C H LetM

B

and M be the preduals of M and M Clearly

B B

M M M

B

Throughout this section wewillassume that Hyp othesis H holds Let L

be an operator on M dened by

L S AS H A

For each real L is a closed densely dened op erator It generates a

strongly continuous group of isometries

t tL

T e

of M suchthat

t t

T S A S A

Let

L S A S Q A

I B

Then L is a closed densely dened op erator and

I

L L L

I

Weintro duce the partial trace

P M M

by the formula

P S X S X I X M

From Resonances to Master Equations

Identifying M with a subspace of M by the injection S P

B

b ecomes a pro jection The reduced dynamics of the system is obtained by

pro jecting out the reservoir variables

P U t PT

t

Let Q P Dening

tL QL Q



I

V t e

we derive the integral form of NPRZ Naka jimaPrigogineResib oisZwanzig

equation as follows First since PL is a b ounded op erator wehaveawell

I

dened equation

Z

t

V t sPL QT Pds U t V t

I

t

see eg Theorem in D Wehaveused that PL P Since L P is

I I

also b ounded wehave

Z

t

V t sQL PU sds P QT

I

t

Combining the ab oveequations and using that V tP U t weobtain

the NPRZ equation

Z Z

s t

ds U t sPL QV s s QL PU s ds U t U t

I I

This equation is also known as the generalized master equation Intro ducing

the new variables t t and u s weget

Z

t

U t U t U t u K t uU u du

where

Z



t

K t U sPL QV sQL Pds

I I

V Jaksic and CA Pillet

This op erator acts on M One can immediately conjecture that the Van

Hove limit t t twill yield the generator

Z

K U sPL QU sQL Pds

I I

The matter is however more complicated Let L be dened on M ac

s

cording to

L S X S H A X M

s s

and let

Z

T

tL tL

s s

e K lim K e dt

T

T

T

Wenow state Davies results D D sp ecialized to the spinb oson system

Note that M can b e identied with the vector space of matrices

equipp ed with the trace norm kk

Theorem For any a

  

tL tK tL tK

s s

lim sup ke e k



ta

Theorem Suppose that for some the formfactor satises

Z Z

it k

e jkj d k dt t

R

Then for any a



tL tK

s

lim sup kU t e k



ta



tK

lim sup kU t U t e k

ta



tK

Furthermore for t the semigroup e is positivity and tracepreserving

on M Italsoleaves the Gibbs state S invariant

s



tK

e S S

s s

From Resonances to Master Equations

Remark Since in general K K SD Relations and

imply that the Markovapproximation is unambiguously dened only in the

interaction picture This ambiguityisclearly related to the fact that the

master equation technique do es not provide suciently sharp estimates which

can distinguish b etween K and K The matter is aggravated by the fact

that the Markovgenerator commonly used in the physics literature diers

from b oth K and K For a detailed discussion we refer the reader to SD

We will return to this p ointinthenextsection

Notation In the sequel all the matrices are written in the basis

The generator K can b e explicitly computed Let

Z

e

jk j d k



sinh

S

Z

s

j s k j e

PV dsd k



j sinhsj s

RS

and

If

then

i

B C

K

A

i

is the probabilityper unit Note that to second order the co ecient

time that the spin will makea transition byemitting or absorbing

one quantum resp ectively This probabilityisdierentfrom zero i the

is the Lambshift of the energy Hyp othesis H holds The co ecient

V Jaksic and CA Pillet

level These co ecients dier from the usual physics textb o ok values by

a factor whichhasbeen absorb ed in tNotealso that inEquation

satises O Werefer the reader to JP JP for a

detailed discussion of the co ecients

It is instructivetoreformulate Relation in the Heisenb erg picture

Assume that the initial state of the system is given by the densitymatrix

j ih j

B B

If holds then for any X M

 



t t

tK

X I Tr e lim Tr X

Wedenote the righthand side of by hX tiThespinb oson mo del is

completely analogous to the following nuclear magnetic resonance problem

Aspin particle in a constant magnetic eld p ointing in the z direction

interacts with scalar radiation eld in the xdirection only LCD For such

a problem the phenomenological Blo ch equations are commonly used in the

physics literature to describ e the evolution of the exp ectation values h i

x

h i h i BL BLW KTH PU The Blo chequations for the spinb oson

y z

system followimmediately from

dh ti h ti

x x

h ti

y

dt T

dh ti h ti

y y

h ti

x

dt T

h ti dh ti

z z

dt T

Note that these equations and uniquely sp ecify K Theconstants

T and T are the longitudinal and transversal relaxation times resp ectively

and

T T

From Resonances to Master Equations

If the initial state of the spin system is diagonal then

d t

K t

dt

reduces to Paulis master equation P P

p t p t p t

p t p t p t

Here p tistheprobability that at the rescaled time tthe spin is in the

state j ih j

The original derivations of Pauli and Blo chequation were based on the

statistical assumption of random phases at all times whichis similar in

spirit to Boltzmann Stosszahlansatz and sub ject to similar ob jections VH

The rst derivation not using this assumption was given byVan HoveVH

VH see also PR M Z Z Mathematically rigorous derivations

were given for the rst time byPule PU and Davies D

Wenish this section with a number of remarks concerning the results of



tK

Davies The matrix e can b e computed If M is a state then



tK

D t e S

s

where

t t i

e e

C B

C B

D t

A

t i t

e e

Weconclude that if Hyp othesis H holds then at the time scale t the

spin system approaches thermal equilibrium exp onentially fast The linear

transp ort equation of thermal relaxation is the Blo chequationasexpected

The technical assumption is very mild and the metho d works equally

well at zerotemp erature ClearlyTheorem is a p owerful result

V Jaksic and CA Pillet

On the other hand the nal conclusions of the theory yield only a crude

understanding of the problem of thermal relaxation It has b een realized

quite early VH that a natural next step is to abandon the simplifying

assumption t t nite and to study the exp ectation values

t

Tr A for b oth and t nite This problem cannot b e treated by the

old techniques and the following two questions were op en for some years

Do es the quantum dynamical system M S returns to equilibrium

for suciently small nonzero

What are the higher order corrections to the linear transp ort equation Is

it p ossible to derivean exact transp ort equation with convergentexpansion

in p owers of such that its rst nontrivial term is the Blo chequation

We remark that Theorems and answered the Question In the

sequel wewill discuss howourtechniques resolvethe Question

Sp ectral theory of thermal relaxation

The sp ectral approachtodynamics of innite quantum systems is based on

the noncommutativeanalog of Ko opmans lemma We briey recall some

basic facts concerning this approach For details werefer the reader to Section

inJP

Let M S beaquantum dynamical system We denote byH

the GNS representation of M asso ciated to S There is a unique selfadjoint

op erator L on H suchthat

t itL itL

A e Ae

L

see eg Corollary in BR We call L the Liouvillean of the system

It is a simple exercise to showthatifM is ab elian then L reduces to the

familiar Ko opman op erator In JP weproved Theorem

From Resonances to Master Equations

Theorem A quantum dynamical system M S returns to equilib

rium if and only if

itL

w lim e P

jtj

where P is the orthogonal projection of H along the cyclic vector In

particular if the Liouvil lean has absolutely continuous spectrum except for a

simple eigenvalue then the system returns to equilibrium

TomitaTakesakis theory relates the Liouvillean of the system to its mo dular

structure Since the vector is cyclic and separating for M the basic

construction of mo dular theory applies see eg BR Section Let

and J be the mo dular op erator and mo dular conjugation asso ciated to the

pair M The Liouvillean L is related to the mo dular op erator by

the formula

L

e

The mo dular conjugation plays a critical role in the p erturbation theory of

the Liouvillean A

Since the reservoir is given in the cyclic representation it follows from

and that its Liouvillean is

L H

B B

The explicit construction of the triple H H isgiven in AW see also

B B B

JP JP BR From this construction it follows that L has purely ab

B

solutely continuous sp ectrum lling the real axis except for a simple eigen

value Thus the isolated free reservoir is an ergo dic system Wedenote by

J the mo dular conjugation of the pair M

B B B

The cyclic representation H ofthe spin system asso ciated to the

s s s

Gibbs state S can b e explicitly constructed for the details see JP or

s

the Section V in H Let

H M

s

V Jaksic and CA Pillet

with the inner pro duct Tr

X X

s

H

s

q

e

s

Z

s

The mo dular conjugation of the pair M isJ andthe

s s s

Liouvillean of the spin system is

L H

s s

In the absence of the interaction the state S S S is a unique

s B

KMS state on MThe corresp onding cyclic representation is given

byH where

H H H

s B

X W f X W f

B s B

B

s

In the absence of the interaction the Liouvillean of the combined system is

L L I I L

s B

In JP wehave computed the Liouvillean of the interacting system by

invoking Arakis p erturbation theory of W dynamical systems A see also

BR Theorem and A Wequote the nal result Theorem in

JP

be the cyclic representation of the noninteracting Theorem Let H

spinboson system M S For any R thereisacyclic and separat

ing vector H such that

A A S

From Resonances to Master Equations

is the unique KMS state on M The Liouvil lean of the interacting

spinboson system is given by

L L L

I

where

L Q J QJ J J

I s B s s s B B B

Remark The Liouvillean can b e written in a completely explicit form see

Section in JP

The basic idea of our approachistodeduce thermo dynamic prop erties

of the combined system spin reservoir from the sp ectral prop erties of L

The resulting picture can b e roughly summarized as follows The sp ectrum

of L is

L R

ac

L

sc

L f g

pp

where are simple eigenvalues while is twofold degenerate eigenvalue

These eigenvalues are emb edded in the continuous sp ectrum After the p er

turbation term is switched on all these eigenvalues turn into complex

resonances except for whichremains a simple eigenvalue Thus for small

nonzero the sp ectrum of L is absolutely continuous except for a simple

eigenvalue zero Theorem then implies that the spinb oson system has

the prop ertyofreturn to equilibrium The complex resonances of L deter

mine the transp ort equation of thermal relaxation In particular the Fermis

Golden Rule for the complete set of resonances yields the Blo chequations

while the Fermis Golden Rule for the degenerate eigenvalue yields Paulis

equation

To provethese results wehavedevelop ed in JP a eldtheoretic version

of the sp ectral deformation technique The general strategy of this argument

V Jaksic and CA Pillet

is wellknown AC BC S One tries to construct a oneparameter group

of unitary op erators U H Hsuchthatthe op erators

L U L U

can b e extended to an analytic family for and complex The construction

should b e suchthat for complex the essential sp ectrum of L moves away

from the real axis unveiling the resonances whichcanthen b e computed by

the usual RayleighSchrodinger expansion The formula

L z U L z U

relates these resonances to the p oles of the analytic continuation of matrix el

ements of the resolventinto the unphysical Riemann sheet The construction

of U and other details of this analysis are technically involved They are

presented in JP JP Wequote the nal result Theorem in JP

Recall that are given by

Theorem Suppose that Hypotheses HH are satised Then there

exists a dense subspace EHand for each aconstant

such that for and Ethe functions

z L z

have a meromorphic continuation from the upper halfplane onto the region

Ofz Imz g

The poles of matrix elements in O areindependent of and They

are identical to the eigenvalues of a quasienergy operator on H This

s

operator is analytic for jj withapower expansion of the form

X

n n

L

s

n

From Resonances to Master Equations

The matrix can be explicitly computed Denoting by P the eigenprojec

E

tions of L we have P P and

s E E

P P i

for simple eigenvalues and

i i e

P P

i e

for the degenerate one

Remark The formula for the matrix in JP Relation has a

typ ographical error the factors e in the odiagonal elements are inter

changed

Hyp othesis H ensures that zero is the only real eigenvalue of An

immediate consequence is that if HH hold then for suciently

small the sp ectrum of L is purely absolutely continuous except for the simple

eigenvalue Thus invoking Theorem wederiveTheorem

We remark that in principle all terms in the expansion can b e

computed The formulas from whichthese terms are generated are given in

the pro of of Prop osition in JP These formulas are cumb ersome and

we will not repro duce them here

It should b e clear bynowthat and K are closely related The

road from resonances to master equations starts with the following set of

observations see also the pro of of Theorem in JP Since is a

separating vector for M M is dense in H One can further show

is dense and Z that there is a subalgebra Z M such that Z

E see Relation in JP The set N of vector states asso ciated to Z

is total in the norm top ology in the set of all normal states over M Let

S b eastate asso ciated to a normalized vector C C ZWe then

have

itL t itL

Ae S A e

V Jaksic and CA Pillet

itL

A e C C

Furthermore one can show that if A J Z J then A E In

particular this will b e true for A X I X M By construction

C C E as well Wenowinvokethe dynamical consequence of Theorem

Theorem in JP and Theorem in JP

Theorem Suppose that Hypothesis HH hold and let E beasin

Theorem Then for each thereisaconstant with the

fol lowing property For jj therearetwo maps W E H such

s

that for any Eonehas W W and

itL it t

e W e W O e

as t

Remark Wewill commentonthecase t shortly

Combining Relations and Theorem wederivethatfor the

dense set of states N and the dense set of observables M J Z J the

following fundamental relation holds

t t it

W C C O e S A W X e

Note that Theorem follows immediately from Wepro ceed to

simplify the quantity

it

W X e W C C

O inthe strong we know that W From the construction of W

top ologySimilarly from the construction of we know that

O Finallyanyvector H can b e written as C for some C

s s s

s

M Thus if the initial state S is of the form S wehave the

s

B

following estimate

t it

C C O S X I X e

s s

s s s

From Resonances to Master Equations

for some C whichdepends on Since is a separating vector for M

s s

s

M Y Y H

s s

s

is an isomorphism Thus the relation

it t

e Y Y Y M

s s

s s

t

denes a semigroup of automorphisms of M Dening

t t

Y Y

we can write

t it

C C X C X e C

s s s s

s s s s s s

t

C X C

s s s

s s

t

Tr X

t t

Let M M be the dual semigroup to andlet K be its

generator

t tK

e

Since is analytic for small itisa simple exercise to showthatK is

also analytic

X

n n

K K L

s

n

Any term in the expansion can b e computed from the corresp onding

term of the series The calculations are easier in the tensor pro duct

realizations

H C C

s

X X I

s

e e

s

Z s

V Jaksic and CA Pillet

it

one Following the steps with i in instead of e

computes K from As exp ected

K K

Wesummarize these results in

Theorem Suppose that Hypotheses HH hold and that the initial

state is of the form j ih jIfK is given by then for any

B B

X M

t tK

lim sup jTr X I Tr e X j

t

In particular we recover Davies result

 



t t

tK

lim Tr X I Tre X

The Relations and improveDavies result They provide sharp er

estimates and a generator K dened to all orders in This generator is

completely determined by the resonances of the Liouvillean The Markovian

generator K arises as the rst nontrivial contribution to the expansion

of K in p owers of These results also clarify the ambiguities concerning

the choice of the Markovian generator whichare inherentinthe traditional

theory of master equations SD Finally they givea complete justication

for regarding the equation

d t

K t

dt

as the exact transp ort equation of thermal relaxation

Wewould liketoadd however that our central technical condition H

is more stringentthenDavies condition The master equation technique

is more robust and applies to the situation as zerotemp erature or p ositive

mass mo dels for whichourtechnique fails

Wenish with a few remarks concerning a choice of the timedirection

Theorem asserts that the combined system relaxes to equilibrium as t

From Resonances to Master Equations

Wehavederived the transp ort equation when t butasimilar

argument applies when t Thenwe study the analytic continuation

of the matrix resolventelements from the lower halfplane onto the

region

Ofz Imz g

for The p oles of the matrix elements are again indep endentof

Eand are identical to the eigenvalues of a quasienergy op erator

with a p ower expansion

X

n

n

L

s

n

The matrix can b e explicitly computed in fact and one

derives equations analogous to and arguing as b efore

References

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