Quantum dynamics of time periodic systems

Physica 124A(1984) 613-620 North-Holland,Amsterdam

613

QUANTUMDYNAMICSOF TIME PERIODIC SYSTEMS Kenji YAJIMA Department of Pure and Applied Sciences, University of Tokyo, 3-8-I Komaba, Maguroku, Tokyo, 153 Japan 1. THE FLOQUETHOWLANDTHEORY We study here the large time behavior of the quantum systems governed by the Schrodinger equations with time periodic Hamiltonians H(t) with period T>O; i(~u/~t) = H(t)u,

H(t+T) = H(t)

(I.1)

The purpose of my talk is to propose a new method for studying such systems which I call the Floquet-Howland theory as i t is a synthesis of Howland's method of stationary scattering theory for time dependent Hamiltonians and the Floquet theory for the periodic d i f f e r e n t i a l operators and to i l l u s t r a t e i t by some examples. Section l and a part of section 2 contain an abstract theory and in other parts H(t) = -(I/2)~ + V(t) in the Hilbert space L2(~n). Suppose that {H(t): - ~< t < ~} is a family of operators on the separable Hilbert space H which satisfies the following properties: (A.l) For each t , H(t) is a selfadjoint operator on H and the domain D(H(t)) = D is independent of t ; (A.2) H(t)(H(O)-i) -1 is strongly continuously differentiable in to

I t is

well known that under the conditions (A.l) and (A.2) the equation ( l . l ) generates a unique unitary propagator

U(t,s): -= < t,x<~}

which satisfies the following properties: (U.l) For each t and s U(t,s) is a unitary operator on H and is continuous in ( t , s ) ; (U.2) U(t,s)U(s,r) = U ( t , r ) , U(s,s) = I ; (U.3) For f c D, U(t,s)f ~ D and is strongly differentiable in t (for fixed s) and s (for fixed t ) .

Moreover

i ( ~ / ~ t ) U ( t , s ) f = H(t)U(t,s)f, i(~/~s)U(t,s)f = - U(t,s)H(s)f; (U.4) U(t+T,s+T) = U(t,s)

for all ( t , s ) .

Using the propagator {U(t,s)} we define a unitary group {U(o):- ~ < o < ~} on a big Hilbert space K = L2(R/TZ, H) by the equation Presented at the VIIth INTERNATIONALCONGRESSON MATHEMATICALPHYSICS, Boulder, Colorado, 1983. 0378-4371/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

614

K. YAJIMA

(U(u)f)(t)

(l .2)

: U(t,t-~)f(t-~).

By the properties

(U.I) - (U.4),

U(o) is w e l l - d e f i n e d on the space K and

is a s t r o n g l y continuous u n i t a r y grQup on i t .

Hence by Stone's theorem there

e x i s t s a unique s e l f a d j o i n t operator K on K such that U(o) : exp (-ioK). Definition I.I

(I.3)

We c a l l

K,U (o) and K the grand H i l b e r t space, the grand

group and the grand generator f o r the equation ( I . I ) . I t is c l e a r from the property (U.3) t h a t C = HI (IR/TZ, H)n L2(IRITZ, D) is a core of the operator K and f o r f c C

,

(l .4)

Kf : ( - i ~ / ~ t + H ( t ) ) f ( t ) .

The f o l l o w i n g lemma r e l a t e s the spectral property of the grand generator K and the one period propagator U(t+T,t) and plays the central r o l e in the method. Lemma 1.2

Let Us be the u n i t a r y operator on K defined by the equation

(Usf)(t) = U(t,s)f(t) exp(-iTK) : Us

for s ~t

< s+T.

(I ~£) U(T+s,s))

•

Then f o r any s,

• Us

This lemma reduces the study of the spectral property of the complicated opera t o r U(T+s,s) to the one of the concrete operator K expressed by (1.4) on i t s core. 2. THE CHARACTERIZATIONOF THE BOUND AND THE SCATTERING STATES In what f o l l o w s the H i l b e r t space is H = L2(~n) . For R > O, F(Ix I ~ is the m u l t i p l i c a t i o n

operator by the c h a r a c t e r i s t i c f u n c t i o n of the ball

Ixl ~ R of radius R.

EK(-) is the spectral measure f o r K.

R)

For the u n i t a r y

operator U, Hp(U), Hc(U) and Hac(U) are the p o i n t , the continuous and the a b s o l u t e l y continuous spectral subspace f o r the operator U, r e s p e c t i v e l y . D e f i n i t i o n 2.1

The equation ( I . I )

is said to have the local compactness

property i f f o r any R>O and a compact set I ~ pact operator on K.

,

F(Ix I ~ R)EK(1) is a com-

QUANTUM DYNAMICS OF TIME PERIODIC SYSTEMS

Theorem 2.2 property.

Suppose t h a t the equation ( I . I )

615

has the local compactness

Then f o r any s,

( I ) u ~ Hp(U(T+s,s)) i f and only i f f o r any ~ > O, there e x i s t s R > 0 such thatIIF(Ixl

~

R)U(t,s)ull

~ ( l - c ) i l u II f o r any ( t , s ) .

(2) u ~Hc(U(T+s,s)) i f and only i f f o r any R > 0

lira (I/T

Irl],,F(Ixl~R)U(t,s)u,,2 dt : 0.

T'-><°

S

Moreover i f u c Hac(U(T+s,s)), then for any R > 0 and any ~ > 0 (or equivalently for some c > O) lira

(I/2~)f r]F(Ixl~ R)U(t,s)ull 2

dt = O.

J

For the concrete case of H(t) = -(I/2)a + V(t), i t is not a t r i v i a l question when the equation ( l . l ) has the local compactness property. Proposition 2.3 Let H(t) = -(I/2)~ + V(t) satisfy the conditions (A.l) and (A.2). Then the equation ( l . l ) has the local compactness property i f V(t) is a linear combination of (1) the multiplication operator by an L~(~ n)-value bounded function of t , (2) the multiplication operator by an LP((Rn)N Lq(iRn) -valued bounded function of t with l ~p< n / 2 < q ~ , and (3) the f i r s t order differential operator in x with coefficients which decay at i n f i n i t y in x-space faster than ( l + I x I ) -l We note that i f the potential V(t,x) grows to i n f i n i t y as x grows to i n f i n i t y , then the equation does not in general have the local compactness property and the theorem 2.2 does not apply to such systems. This is a consequence of Weyl's theorem on the s t a b i l i t y of the essential spectrum by compact perturbations. To i l l u s t r a t e how the Floquet-Howland method may be applied, we shall give a proof of the i f part of (2) of the theorem 2.2. We write F(Ix I ~R) = J for simplicity. By the local compactness assumption on the equation ( l . l ) and Wiener's ergodic theorem on the continuous measure, we have that for any f ~ Kc(K) T lira ( I / T ) ; li J exp(-i~K)fll 2 d~ = 0

(2.1)

.Rewriting the relation (2.1), we have that lm i T_~ ~

dt { ( I / T ) / ;

"JU(t+~,t)f(t)" 2 d~ }

: 0

(2.2)

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K. YAJIMA

where S = R/TZ is the c i r c l e . equation f ( t )

Now by Lemma 1.2 the function f ( t )

defined by

= U(t,s)u for s <_I t < s+T belongs to Kc(K) when u ~ ffc(U(s+T,s))

Plugging t h i s i n t o t h e r e l a t i o n (2.2), we see that for any sequence ~n there e x i s t s a subsequence T' such that n ~lim ( I / z n )~ I I I J U ( t + ~ , s ) u l I 2 n-~

for almost a l l t ~ S.

d~ =

0

(2.3)

However (2.3) holds for a l l t ~ S provided i t holds f o r

a single point to~ S since the l i m i t is in fact independent of t .

This proves

the statement. 3. SCATTERINGTHEORY Next we show that the method may also be applied to the scattering theory for the equation ( I . I ) .

We take H(t) = - ( I / 2 ) A + V ( t , x ) and assume for

s i m p l i c i t y that the potential V ( t , x ) is short range:

V(t,x)

~

C(l+Ixl) -I-~

Then i t is very easy to show that the wave operators W±(s) = s-lim U ( t , s ) - l e x p ( - i ( t - s ) H O)

(3.1)

t+±~

e x i s t for any s, H0 = - ( I / 2 ) A

being the free Hamiltonian.

The problem here

is to i d e n t i f y the images R(W±(s)) of the wave operators W±(s) with the continuous subspace Hc(U(T+s,s)) for the one period propagator U(T+s,s), and here comes in the Floquet-Howland method. Once the l i m i t s of (3.1) are known to e x i s t , i t is a matter of t r i v i a l i t y to show that the following l i m i t s e x i s t and the equation holds: (W+f)(s) = s-lim exp(iaK)'exp(-i~Ko)f(s) = W+(s)f(s), -

0÷±~

where K0 = -i@/~t + HO.

(3.2)

Note that K and K0 are time independent in the sense

that they do not contain the time parameter o in t h e i r expressions.

We apply

the s t a t i o n a r y theory of scattering for the pair (K,Ko). I t can be shown that i f one writes M = ( l + I x l ) " ( I + ~ ) / 2 , the operator valued function M(Ko-z)-IM o r i g i n a l l y defined on the upper or lower plane {± has continuous extensions to the closed h a l f planes { ± U ~ { ( 2 ~ / T ) n : n ~ Z}.

Likewise by the perturba-

t i o n technique one can show that M(K_z)-IM : M(Ko-Z)-IM + M(Ko-Z)-IA-(I + B ( K o - Z ) - I A ) - I . B ( K o - Z ) - I M

QUANTUM DYNAMICS OF TIME PERIODIC SYSTEMS

617

also has a norm continuous extension to C, uR\{Op(K)U{(2~/T)n:z n=O, ± I , . . . } } . Then i t follows from Kato-Kuroda theory of scattering that the wave operators (3.2) exist and their images R(~) = Kc(K).

~(K)

By Lemma 1.2 we know that

2

(3.3)

= UsL (S,H (U(T+s,s)))

c

and by (3.2) R(W±) = UsL2(S,R(W±(s))).

(3.4)

Comparing the equations (3.3) and (3.4) we have the desired relation

R(W±(s))

: Hc(U(T+s,s)). 4. THE STABILITYAND THE INSTABILITYOF THE BOUNDSTATES Let H = -(I/2)A + V(x) be a stationary Hamiltonian.

Then i t is a well known

result of Kato-Rellich-Schrodinger's perturbation theory that the isolated eigenvaules are stable under a wide class of perturbations.

We ask whether

this remains true or not when the perturbations are periodic in time.

This

turns out to be a very d i f f i c u l t question and we report here only a few results on some specific examples. The reason for the d i f f i c u l t y is as follows.

By

virtue of Lemma 1.2 i t is clear that the problem is the perturbation of the eigenvalues of the operator K1 = -i@/@t + H by the time periodic potentials. Suppose f i r s t that the potential V(x) + 0 as continuous spectrum 0,~).

Ixl +~ and the operator H has a

Then every eigenvalue of H appears as an embedded

eigenvalue of the grand generator Kl and one has to deal with this embedded eigenvalue.

I f on the other hand H has only purely discrete eigenvalues, the

spectrum of Kl in general f i l l s up the whole real line which is a closure of the eigenvalues of KI .

Hence i f one tries the perturbation series, the small

denominators appear and there is l i t t l e hope that the series converges. This of course comes from the resonance between the states and the external force. 4.1 The AC-Stark effect I f a particle is subject to an alternating homogeneous electric f i e l d , i t is governed by the Schr6dinger equation i@u/@t : -(I/2)AU + V(X)U + {~E.xcos~t} Here E = (l,O,O),~,m > O.

u .

By the transformation

u ( t , x ) : exp(-i~E'xsin ~t/m+i~2sin2~t/8~3-iu2t/4~2) v(t,x-pEcos~t/m2),

(4.1)

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K. YAJIMA

the equation (4.1) can be transformed into the equation i ~ v / 3 t = - ( I / 2 ) A v + V(x+~Ecos ~t/~2)v = H(t)v.

(4.2)

Obviously H(t) can be regarded to be a small perturbation of H = - ( I / 2 ) A + V(x) uniformly in t with the perturbation V(x+~Ecos~t/~ 2) - V(x).

One can show by

applying the complex scaling technique to the grand generator K that the bound states are very unstable for the equation (4.2) and a l l bound states w i l l disappear as soon as the perturbation is switched on forming resonance states. One can also observe that there is a resonance between two states which have the energy difference exactly equal to integer multiples of ~ and there is a state which o s c i l l a t e s between these two states for a long time.

This r e s u l t

suggests that the bound states are in general instable i f the unperturbed system has a sea of continuous spectrum via the resonance with the states in t h i s sea which is always possible. 4.2 Perturbations of the harmonic o s c i l l a t o r We consider now the case when H has only discrete spectrum.

The simplest

such example is the one dimensional harmonic o s c i l l a t o r H = - ( I / 2 ) d 2 / d x 2 + (I/2)x 2 .

Enss and Veselic have shown by using the fact the equation can be

solved e x p l i c i t l y that the equation i 3 u / ~ t = -(I/2)~2u/~x 2 + (I/2)x2u + uxcos~t,u

(4.3)

has only bound states i f the frequency ~ is non-resonant whereas i f i t is resonant K has only the absolutely continuous spectrum.

They also have

shown that i f the perturbation has compact support in place of the l i n e a r force, K has only discrete spectrum i f the frequency is resonant. ever s t i l l

I t is how-

open i f i t has only bound states when the frequency is non-resonant.

REFERENCES I ) V Enss and K Veselic, Bound states and propagation states for time dependent Hamiltonians, Annales L'IHP (in p r i n t ) . 2) S Graffi and K Yajima, Exterior complex scaling and the AC-Stark e f f e c t in a Coulomb f i e l d , CMP (in p r i n t ) . 3) J S Howland, Stationary theory for time dependent Hamiltonians, Math. Ann. 207 (1974), 315-335. 4) J S Howland, Scattering theory f o r Hamiltonians periodic in time, Indiana Univ. Math. J. 28 (1979) 471-494. 5) T Kato and S T Kuroda, The abstract theory of scattering, Rocky Mt. J. Math I(1971) 121-171.

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6) H Kitada and K Yajima, A scattering theory for time dependent long range potentials, Duke J. Math. 49 (1982) 341-376. 7) H Kitada and K Yajima, Remarks on our paper 'A scattering theory for timedependent long range potentials', Duke J. Math (in print). 8) K Yajima, Scattering theory for Schr~dinger equations with potentials periodic in time, J. Math. Soc. Japan 29 (1977) 729-743. 9) K Yajima, Resonances for the AC-Stark effect, CMP 87 (1982) 331-352. IO) K Yajima and H Kitada, Bound states and scattering states for time periodic Hamiltonians, L'IHP (in print).