Floquet operators with singular spectrum. III

A NNALES DE L ’I. H. P., SECTION A

J ^AMES S. H ^OWLAND

Floquet operators with singular spectrum. III

Annales de l’I. H. P., section A, tome 69, n

^o

2 (1998), p. 265-273

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265

Floquet operators ^with singular spectrum, ^III

James S.

HOWLAND

¹

Department ^of Mathematics, University ^of Virginia, Charlottesville, ^VA22903.

E-mail:

[email protected]

Ann. Inst. Henri Poincaré,

Vol. 69, ^n° 2, 1998, Physique théorique

ABSTRACT. - The

quasienergy

^{for the}

time-periodic

Hamiltonian

on

L2[0,27r]

^{has no}

absolutely

continuous

spectrum

^{if 0 a 1 and}

v(0, t)

^is

^{C° ,} although

the gap between successive

eigenvalues of

^is

decreasing.

^©

Elsevier,

^Paris

Key ^words: Singular spectrum, Floquet theory, quasienergy, quantum stability, gap theorem.

RESUME. -

L’ operateur

^de

quasi-energie correspondant

^au Hamiltonien

dependant

^du

temps

sur

L2 ~0, 2~r~

n’ a pas de

spectre

absolument continu si 0 a 1 et est

C°,

bien que I’ écart ^entre valeurs propres

de

^soit decroissant.

@Elsevier,

^Paris

1. INTRODUCTION

Let H be a

positive

^discrete

self-adjoint operator

^{on a}

separable

^Hilbert

space

~C,

^with

non-degenerate eigenvalues

1 Supported by NSF Contract MDS-9002357

Annales de l’Institut Henri Poincaré - Physique théorique - ^{0246-0211 1}

Vol. 69/98/02/@ Elsevier, ^Paris

266 J. S. HOWLAND

and define the _gap between

eigenvalues

If

V(t)

^{is a bounded}

strongly

continuous

perturbation

^of

^H, 27r-periodic

ⁱⁿ

time,

then the behavior of the

system

^{under the}

time-dependent

Hamiltonian

is

governed by

^the

quasienergy

on H (g)

£2[0, 27r],

where ~ _ - 2

-it

^with

^{periodic boundary}

^condition

~c(0) _

^{in t.}

In

[3],

the author

proved

^the

following

^result.

Gap

^Theorem.

If V (t) is strongly ^C°°,

^and

for

^{some a} &#x3E;

0,

^{then K has no}

absolutely

continuous component.

This result ^was extended to

degenerate eigenvalues by

the author

[4],

Nenciu

[6, 7]

^and

Joye ^[5].

The

question naturally

^{arises as to} how essential the

increasing

gap condition is to this result.

Hagedorn, Loss,

^and

Slawny [2]

^show

by explicit computation

that the forced harmonic oscillator

has a

quasienergy

^with

absolutely

continuous

spectrum

^{in the resonant}

case cv ⁼ 03C90.

Here,

^of course,

AAn

⁼ cvo is constant. On the other

hand,

numerical

experiments

^{with the}

operator

where _p ^{= on}

Z/2

^{of the.}

circle,

^{showed no} evidence of

absolutely

continuous

spectrum, although n -! _{[ 1 ].}

In

fact,

^we shall prove the

following

^theorem.

Annales de l’Institut Henri Poincaré - Physique théorique

267 FLOQUET ^OPERATORS

THEOREM B. - Let

t)

^{be C°° and}

203C0-periodic

ⁱⁿ t, ^and

satisfy

has no

absolutely

continuous component.

The

proof

^{is a} variant of the

operator

gauge transformation method of

[3JI].

Transformation of K

by leads,

up ^to first-order terms in G and

V,

to the

operator

In

[3,11], G(t)

^{was chosen so that the}

first

^{two terms in the braces}

cancel, effectively replacing by G(t).

^{In the}

present

paper, the last ^{two terms}

are made to

cancel, effectively replacing V ( t) by z[~,G(~)].

^Iteration

eventually

^{leads to the case that}

V (t)

^{is trace}

class,

and the result follows from

scattering theory.

The author thanks Alain

Joye

^{and Jean} Bellissard for useful comments.

2. MAIN THEOREM

Let H be a

positive

^discrete Hamiltonian with

eigenvalues

Assume that

We shall write operators ^as matrices in the

representation

in which H is

diagonal.

^For p &#x3E; 1 and a &#x3E;

0,

^define

X (p, a)

^to be the space of all infinite matrices

Vol. 69, n° 2-1998.

268 J. S. HOWLAND

satisfying

X(p, a)

^{is a} Banach space under the ^norm

For a ⁼

0,

^{A defines a bounded}

operator

^on

~2.

^since

~n~ -P

^{is summable.}

For a &#x3E;

0,

every A E

:B:’(0, a)

^can be written as

where A is the

diagonal

^{matrix with}

and

Ao

^{E The}

operators

^{A in}

X(p, a)

^{are therefore}

^compact

^for

a &#x3E;

0, and,

ⁱⁿ

fact,

for

2aq ~ ^1,

^where

Zv

is the Shatten class. In

particular,

^{A E}

X(p, a)

^is

trace class if ^a &#x3E;

~.

Define

X(a)

^to be the space of all A such that A E

X(p, a)

^{for all}

p &#x3E; 1.

Again,

^{A E}

X(a)

^{is trace class if a} &#x3E;

~.

LEMMA 1. -

If

^{A E}

X(p, a)

^{and B E}

X(p,

^{then the}

^product

^{AB is}

in

X (r,

^{a +}

(3) if

Proof. -

^{We note in}

preparation

^{the two}

elementary inequalities

and

which hold for _n, 1. These follow from the

triangle inequality

^and

the fact that a + b 2ab if _a, b &#x3E; 1.

Annales de l’lnstitut Henri Poincaré - Physique théorique

FLOQUET ^OPERATORS 269 We have

Since the

exponents

^{in the sum are}

negative,

^{it follows}

by

^Holder’s

inequality

^{that the sum is}

uniformly

^{bounded if}

that

is,

^if

COROLLARY I . -

If

^{A E}

X(a),

^{and B E}

x ~,C3~,

^{then the}

product

^{AB is}

in +

{3).

LEMMA 2. -

If

^{A E}

x(p, a)

^{and H}

satisfies (2.1),

^{then the commutator}

[H, A]

^{is in}

x (p - 1, cx

⁺

q).

Proof -

^{We have}

COROLLARY 2. -

If

^{A E}

X(a)

^{and H}

satisfies (2.1 ),

^{then the commutator}

[H, A]

^{is in}

X (a

⁺

q).

Let

V (t)

^{be a}

203C0-periodic operator-valued

^{function of t.} We say that

V (t)

is in a Banach space x

uniformly

^{is a} bounded function of t.

We say that is in

X(a) uniformly

^iff

V(t)

^{is in}

x(p, a) uniformly

for all _p &#x3E; 1.

LEMMA 3. - Let H

satisfy (2.1 ).

^{Let W E and} be

203C0-periodic, strongly continuous,

^{and in}

uniformly,

^where ~y &#x3E; 0. Then

Vol. 69, n° 2-1998.

270 J. S. HOWLAND

is

unitarily equivalent

^to

where

Wl

^E

~(03B3), Y1 (t)

^is

203C0-periodic, strongly

continuous and

uniformly

in +

1’),

^{crnd is}

uniformly

^{in trace class.}

Proof. -

^Let

where

Define

so that

G(t)

^is

203C0-periodic,

^and

Note that V is in

X(a)

^and

G(t)

^{is in}

^{X(a) uniformly.}

If

then

But

is in

X(a) uniformly by hypothesis,

^while

[G(t), H]

^{is in +}

’"Y) uniformly by Corollary ^2,

^and

Annales de l ’lnstitut Henri Poincaré - Physique théorique

271 FLOQUET ^OPERATORS

is in +

~y) uniformly by Corollary

^{1. It follows from} Corollaries 1 and 2 that every ^term in

(2.8)

^{is in} +

~), except

^for

Moreover,

^{the terms} of the series are all in trace class if na &#x3E;

2. ^Hence,

(2.8)

^is

equal

^to

with

W1

^{= W} + V E E +

q),

^{and in trace}

class

uniformly.

^{Trace norm} convergence of the series

presents

^no

problem

because of the factor n!. D

THEOREM B. - Let H

satisfy (2.1 ) for

some 03B3 &#x3E;

0,

and _suppose that

for

^{some 03B1} &#x3E;

0, W(t)

^is

203C0-periodic, strongly continuous,

and in

uniformly.

^Then

has no

absolutely

continuous component.

Proof -

^If

(2.1 )

^{holds for some}

positive

^{then it holds} for any smaller

positive

^number

q’.

^{Since also} c if a

/3,

^{it follows that}

we may ^assume for

simplicity

^{that a =} ~.

By

^Lemma

3,

^{K is therefore}

unitarily equivalent

^to

with

Wi

^E and E

X(2)’),

^and

T1(t)

^{in trace class}

_uniformly.

From

scattering theory,

^{and hence also}

K,

^{have the same}

absolutely

continuous component ^as

But

[(1

^{satisfies the}

hypotheses

of Theorem A with ae =

2~y. Continuing

this process, ^we find that K has the same

absolutely

continuous component

as an

operator

with

WN

^E

VN(t)

^E

1)’r).

^{But if}

(N +

&#x3E;

~,

^then

VN(t)

is trace

class,

^{so that} and hence also K have the same

absolutely

continuous component ^{as D}

+ H

+

WN

which is pure

point.

^D

VoL 69, no 2-1998.

272 J. S. HOWLAND

3. PROOF OF THEOREM B

Theorem B follows from Theorem A. The

operator H =

^has

eigenvalues

where

Matrices are taken in the basis

1,

^... in which H is

diagonal.

We shall show that H satisfies

(2.1 ), with ~

⁼

( 1 - a) /2.

^We

^have, for j

&#x3E;

k,

by

^{the mean} value theorem and

convexity

^of

~a .

^If

^{~n -} = j a - ^ka,

then n -

m &#x3E; (2~

⁺

1) -

^2k

&#x3E; j - ^k,

^{and so}

By (1.3),

^we may write

for some

y(A; t,)

ⁱⁿ C°°. Since is C°° in

~

^the

operators v(B, t)

and

g(B, t,)

^{are in}

X (0, p)

^{for all p. The}

^operator

^{K is therefore}

^unitarily equivalent

^to

where

The

operator Ko

^will

satisfy

^the conditions of Theorem A with ^a = ~

provided

^{we show that}

V(t)

^is

^uniformly

ⁱⁿ

Write

Annales de l’lnstitut Henri Poincaré - Physique théorique

FLOQUET ^OPERATORS 273

Now g

^{and are C°} and hence in

x(0~,

^so

[g, by Corollary

^2.

By Corollary ^1,

^the

right

^{side of}

(3.1 )

^{is in}

uniformly

in t and s.

Regarding ^(3.1)

^{as a} differential

equation

in the Banach space

x(p, ~y~,

^{we find that}

V(t)

⁼

W (1, t)

^{is in}

^uniformly

^{for all p. 0}

Remark. -

Actually,

^it is clear from the

proof

^that

differentiability

^{in t}

is not

actually required. Moreover, only

^{a finite}

degree

^of

differentiability

in 0 is

required, depending although

^{it did not seem} worthwhile to

quantify

^this.

REFERENCES [1] J. BELLISSARD, ^Private communication, 1987.

[2] ^G. HAGEDORN, ^M. Loss, and J. SAWNYl. Non-stochasticity ^of time-dependent Hamiltonians and the spectra ^{of canonical} transformations, ^J. Phys. A, Vol. 19, 1986, pp. 521-531.

[3] ^{J. S.} HOWLAND, Floquet operators ^with singular spectrum, Ann. Inst. H. Poincare, Vol. 50, 1989, I, pp. 309-323; II, pp. 325-334.

[4] ^J. S. HOWLAND, Quantum stability ⁱⁿ "Schrödinger Operators: ^The Quantum ^Mechanical Many Body Problem," Lecture Notes in Physics, ^Vol. 403, pp. 100-122. Springer Verlag, ^New York, ^1992.

[5] ^A. JOYE, Absence of absolutely continuous spectrum ^of Floquet operators, ^{J. Stat.} Phys., Vol. 75, 1994, pp. 929-952.

[6] ^G. NENCIU, Floquet operators ^without absolutely continuous spectrum, Ann. Inst.

H. Poincare, ^Vol. 59, 1993, _pp. 91-97.

[7] ^G. NENCIU, ^Adiabatic theory: Stability ^of systems ^with increasing gaps. Preprint ^1995.

(Manuscript ^received May 14th, 1996;

Revised version received July 22, 1997.)

Vol. 69, n° 2-1998.