Math. Ann. 299, 117-125 (1994)

Irmm 9 Springer-Verlag1994

Results in L~ (IRd) for the Schr6dinger equation with a time-dependent potential Arne Jensen Department of Mathematics and Computer Science, Institute for Electronic Systems, Aalborg University, Fredrik Bajers Vej 7, DK-9220 Aalborg ~, Denmark* Received May 24, 1993; in revised form September 27, 1993

Mathematics Subject Classification (1991)." 35J10, 35P25, 47A40 1 Introduction. Main result Let Ho = - 89 on L2(~xd) denote the free Schr6dinger operator. It is selfadjoint on the domain ~(Ho) = H2(~-f), the usual Sobolev space. Let Uo(t) = exp( - itHo) be the unitary group generated by Ho. The evolution equation

dq, i ~ - = Ho~,, ~O(0)= 0o,

(1.1)

is called the free Schr6dinger equation. For ~ko~(Ho) the strong solution is given by r = Uo(t)~Oo, and conversely, by Stone's theorem. For general ~ko~LZ(IR a) it is a weak solution to (1.1). Let V(t, x) be a realvalued function on R x A n. We denote by V(t) multiplication by V(t, x). Under mild assumptions the operator H(t) = Ho + V(t) is selfadjoint in L2(IRa). Consider the full Schr6dinger equation with a timedependent potential 9dgo z--~-f = H(t)go, go(s)= goo. (1.2) For a large class of V there exists a unique unitary propagator U(t, s) associated with (1.2) such that go(t)= U(t, s)goo is a weak solution to the problem. Under additional assumptions on goo and V it is a strong solution, see [13, 16, 18] and references therein for such results. Here we are only interested in the propagator

U(t, s). For each selR the wave operators are given on L2($, d) by

W+(s) = s-lim U(s, t)Uo(t - s). t--* • ao

* Current address: Mittag-Leflter Institute, Auravfigen 17, S-182 62 Djursholm, Sweden

(1.3)

118

A. Jensen

There is a large literature on the existence and completeness of wave operators for Schr6dinger operators with time-dependent potentials, see for example [4, 6, 9, 10, 15], and references therein. We introduce the notation W(s; t) = U(s, t)Uo(t - s)

(1.4)

for all s, t e N . The Fourier transform of V(t, x) with respect to the x-variable is denoted l~(t, 4). The bounded operators on LP(lRd) are denoted N(LP). The finite complex regular measures on IRd are denoted YJI(IRd). The total variation norm on ~IR(IRd) is denoted [[. [l~(~d). Assumption 1.1. V(t, x) is a realvalued function such that I~eL1 (~; ~ ( N d ) ) . Under this assumption Ve LI (IR; L ~ (IRd)) and we can apply the results in [ 16, 18] to conclude that there exists a unique unitary propagator associated with the problem (1.2). Our main results are stated in the following theorem. Theorem 1.2. Let V satisfy Assumption 1.1. (i) Let selR. The limits We(s) = lim W(s; t) t ~

_+00

exist in operator norm in ~ ( L 2) and are unitary. (ii) The operators W• extend to bounded operators on LP(IRd), 1 <=p <= ~ . One has s u p ~ , [IW• < ~ for each p. The operators W• are invertible in g$(LP). (iii) For each s, telR the operator W(s; t) extends to a bounded operator on LP(p,a), 1 <=p <=~ . We have W•

= lim W(s; t) t ---) •

in operator norm in ~(LP). The proof of Theorem 1.2 is given in Sect. 2. It turns out to be quite simple. However, it seems to be the first result on LP-boundedness of the wave operators for Schr6dinger operators with genuinely time-dependent potentials. Previous results on scattering theory in a Banach space for a pair of operators (H, Ho) have assumed that H and Ho generate (semi-) groups. See for example [3, 11, 12]. This excludes the free Schr6dinger equation, since Uo(t) is known to be unbounded on LP(~g), p r 2, or equivalently, the problem (1.1) is not well-posed in LP(IRd), p r 2. Here we use the observation that W(s; t) given by (1.4) extends to a bounded operator on LP(g(d) and satisfies an integral equation (see (2.9)). This equation has previously been used in [2, Theorem 2.16]. Even though Uo(t) is unbounded on the LP-spaces (p ~ 2), it has nice mapping properties from a Besov space which is a subspace of L v, to L p. In Sect. 3 we use the result on the wave operators to give an analogous mapping property of the propagator U(t, s) in LP-spaces. Such results are based on the intertwining property U(t, s) W+(s) = W•

Uo(t - s) .

Thus Theorem 1.2 can be used to transfer results on the free propagator in LP-spaces to properties of U(t, s) in the same or analogous spaces. As another

The Schr6dinger equation in Lp (IRd)

119

example of this technique we obtain the result (see Theorem 3.2) that the propagator is bounded from LI(IR d) to L~ a) (t ~ s) and satisfies the optimal global estimate IIU (t, S) ll~,(C~(~d),L~(~)) <=clt - sl -a/2 , for all t, s~lR, t 4= s. For small It - s] this result was obtained by Yajima [17] for a class of both time-dependent and time-independent potentials, which has some overlap with our class. In the last section we extend some of our results to include a small timeindependent potential. In a recent paper Yajima [19] obtained LP-boundedness of the wave operators for time-independent potentials under several different kinds of assumptions. There are recent results on the Cauchy problem for a class of first order pseudodifferential operators, which are related to our results on the propagator, see 18].

2 Proof of Theorem 1.2 In this section we prove the main theorem. We first establish some results under slightly more general conditions on the potential than in Theorem 12. Assumption 2.1. V e L 1 (IR; L|

is a realvalued function.

Under this assumption the results in [16, 18] imply the existence of a unique unitary propagator U(t, s) for the problem (1.2). We recall some of its properties from these papers. We have U(t, r)U(r, s) = U(t, s) for all t, r, s~R, and U(t, 0 = 1 (the identity operator) for all t e R . For each uo~LE(1R d) the function u(t) = U (t, S)Uo solves the integral equation t

u(t) = Uo(t - S)Uo - i S Uo(t - z ) V ( z ) u ( z ) d z .

(2.1)

$

The function u(t) is a weak solution to the problem (1.2). Here we base our proof on the integral equation. It is not necessary to consider strong solutions to (1.2). We introduce the notation Z(s; t ) = U o ( s - t)U(t, s). In @(L 2) we have the relation W(s; t)* = Z(s; t). Here and in the sequel the adjoint is the L2-adjoint. The following simple result is probably well-known. We prove it for the sake of completeness, and because we need some of the computations in the proof of the next proposition.

Proposition 2.2. Let V satisfy Assumption W•

2.1. Let s~IR. Then the limits

= lim W(s; t)

(2.2)

1 - r :t:oo

exist in operator norm in

~(L2). The W•

operators W+(s) are unitary. We have =

lim Z(s; t)

(2.3)

t-'* •

in operator norm in ~(L2). Proof The proof is based on the usual Cook argument, except that we use the integral Eq. (2.1). Let Uo, voeL2(lRa). Let v(t) = U(t, s)vo be the solution to (2.1).

120

A. Jensen

Then we have

= (Uo(t - S)Vo, Uo(t - S)Uo) t

+ iS(Uo(t - z)V(r)v(z), Uo(t - S)Uo)dr s t

= (vo, Uo> + i S (Vo, U(s, r)V(~)Uo(r -s)uo)d~:. s

Since vosLZ(~, a) is arbitrary, we conclude t

W(s; t)Uo = Uo + i S U(s, z) V(z)Uo(z - S)Uod r .

(2.4)

s

It follows that for tl < tz t2

II W(s; tgUo - W(s; h)uo 112 < 11Uo112 S II V(t)I1~ dt. tl

Existence of the limits (2.2) follows from this estimate and Assumption 2.1. Analogous arguments prove (2.3). Unitarity of W+_(s) follows from W ( s ; t ) Z ( s ; t ) = Z(s; t) W(s; t) = 1 by taking limits. [] Let us explain our terminology concerning L'-boundedness of an operator T~ 9~(L2). Initially the operator T is densely defined on L 2 c~ L p, 1 < p < ~ . Thus there is at most one extension to a bounded operator on L p. For p = oo we consider the LZ-adjoint T * on L ~ c~ L 2 and then use duality. Here the duality between L p and L v' (p and p' are conjugate exponents) is obtained via the inner product ( - , . ) on L 2. Using this approach we get a uniquely determined extension also to L~176 Consider the following bounded operator on L 2 (IRa). 17(s; t) = Uo(s - t) V(t) Uo(t - s).

(2.5)

The idea of using this operator is related to the interaction representation in quantum mechanics. Assumption 2.3. Let Vsatisfy Assumption 2.1. Assume l?(s; t), defined on L 1 c~ L z, extends to a bounded operator on L~(R a) such that for each s e n i II V(s; t)ll~L~dt < ~ 9

(2.6)

--CO

Proposition 2.4. Let V satisfy Assumption 2.3. Then for each s, t e ~

and p, 1 < p < ~ , we have the following results: (i) The operators W(s; t) and Z(s; t) extend to bounded operators on LP(]Rd). For each s e n the operator norms are bounded uniformly in t~lR. (ii) The operators W+_(s)and W• extend to bounded operators on LV(IRd), and W•

= lim W(s; t) ,

(2.7)

t -'-~ •

W•

= lim Z(s; t), t-'* •

(2.8)

The Schr6dinger equation in Lp (F,a)

121

where the converoence is in the operator norm in ~(LV). These operators are invertible in ~(LV). We have W_+(s)- 1 = W+_(s)*. Proof. We fix s~IR throughout the proof. Let uo~L2(lRe). Then we have from (2.4) and (2.5) t

W(s; t)Uo = Uo + i ~ U (s, z)V(z)Uo(z - S)Uodz $ t

= Uo + i~ W(s; z) 17(s; Z)Uo dz.

(2.9)

s

This integral equation is the key to our proofs. Standard iteration technique and Assumption 2.3 imply that W(s; t) extends to a bounded operator on L 1(Ra) with a bound

II W(s; t) lt~L1) < exp

]1V(s;z) l[~Ll~dz

.

Then (2.6) implies II W(s; OII~L*) ~ C(s),

t ~ ]R .

(2.10)

We can now extend (2.9) to all u o ~ L l ( R e) to conclude for all tl < t2 t2

IIW(s, t2)Uo - W(s, tl)uoll 1 < Iluolll j" IIlY(s; z)II~L1)dz. II

It follows that (2.7) holds for p = 1. Analogous arguments prove the results for Z(s; t) and W• for p = 1. The general case then follows by interpolation and duality from the above results and the results in Proposition 2.2. []

Remark 2.5. The integral Eq. (2.9) was used in I-2, Theorem 2.16-] to prove L~~

of W(0; t) for a time-independent potential.

Remark 2.6. In the above results no special properties of Ho = - 89 have been used. The same arguments apply to Ho replaced by H1 = Ho + Vl(x), provided there is a unitary propagator U(t, s) with the above properties associated with H(t) = H1 + V(t, x). However, the verification of this property and of (2.6) may be difficult. An example is given in Sect. 4.

Proof of Theorem 1.2. Let V satisfy Assumption 1.1. Then VeLI(IR;

L~176

SO

Assumption 2.1 is satisfied. Using the notation p = - i V we have on the Schwartz space

Thus as operators on L2(R d) we have (t + s) V(s; t) = Uo(s - t)V(t)Uo(t - s)

= exp ( ~

x2 ) V(t, (t -- s ) p ) e x p ( ~

x2 ) .

The two exponential factors are isometric as multiplication operators on Lt(Rd). Using well-known properties of Fourier multipliers we see that V(s; t) extends to

122

A. Jensen

a bounded operator on LI(P.n), and we have I[17is; t)[tattLe) = ItVit,( t -- s)p)II~,(L,> = IIVit, p)II~r

< c IIlg(t, .)II~<~> 9

Hence

I1~7(s; t)II~,r

at <=c tt PltL*r162

--o0

and we conclude that Assumption 2.3 is satisfied. Since the bound above is independent of s e R , it follows that the constant C(s) in (2.10) can be chosen independent of s. The remaining results now follow from Proposition 2.4. This concludes the proof of Theorem 1.2.

3 Mapping properties of U(t, s) It is well-known that Uo(t) is unbounded on all spaces LP(IRd),p # 2. However, one can find a Banach space ~r, which is densely and continuously embedded in LP(IRd), such that Go(t): Ar ~ LP(~, a) is bounded with operator norm bounded by
~-

. Here
using the wave operators. Denote by B~ "~, a > 0, 1 <=p <= ee, 1 <=q <= o0, the usual Besov spaces (see for example [1]). Let V satisfy Assumption 1.I and let ~ ( s ) be the wave operators from Theorem 1.2, extended to the L~'-spaces. We define for stiR, a > 0, l__
~r;,qis) = {u ~LPi~ ~) I W+ (s)- ' u e~;q } with the norm Itult~.~) = II m+(s)- lu IIBg" "

In this manner we get a Banach space which is a subspace of LV(1Rd), and which is a "copy" of B~"q. Theorem 3.1. Let V satisfy Assumption 1.1. Let 1 <=1) < 0% 1 < q < ~ , and let fl = d 2 - 1 [ . Let t, s ~ .

Then U (t, s) is bounded from yf2"'(s) to LPORd) with norm

bounded by c (t - s) ~. Proof We have on L2(]Rd) the relation U(t, s)W+(s)Uo(s - t) = W+(0, hence U(t, s) = W+(t)Uo(t - s) W+(s) -~ .

(3.1)

By the results in [1] (see also [5] ) the operator Uo(t - s) is bounded from B 2t~'q to LP(R d) with norm bounded by c(t - s) ~. By Theorem 1.2 the operator W+(t) is bounded on LP(R d) with operator norm bounded uniformly in telR. The result now follows from (3.1) and the definition of the space 8q2~'q(s). [] Various other mapping properties of the propagator U(t, s) can be obtained using the relation (3.1) and well-known properties of the free propagator Uo(t). We give another result in this direction.

The Schr6dinger equation in L p (R a)

123

Theorem 3.2. Let V satisfy Assumption 1.1. Then U(t, s) is bounded from L I ( R ~) to

L~176

t ~ s, and

tl u(t, S)II~(L'(~),L~(~)) < clt - sl -a/z, t + s ,

(3.2)

with a constant c independent of s, telR. Proof Using the well-known estimate for the Laplacian

II Uo(t - S) II~(L't~d),Lo{~)) < clt - sl -a/2 the result follows from (3.1) and Theorem 1.2.

[]

The estimate (3.2) was extended from Ho to Ho + V(x), V small, d = 3, by Schonbek [14] and for a general class of time-independent potentials by Journ6 et al. [7]. Such estimates also follow from the results in 1-19]. For small It - sl the estimate (3.2) was obtained by Yajima [17, Lemma 3.1] for a class of both time-dependent and time-independent potentials, which has some overlap with our class. Theorem 3.2 seems to be the first global result for time-dependent potentials.

4 An extension

In this section we extend Theorem 1.2 to include a small time-independent potential. The results are obtained for a fairly small class of potentials. Assumption 4.1. Assume d > 3. Let Vbe decomposed as V(t, x) = Vx(x) + Vz(t, x) into realvalued functions. (i) ~ e L I ( I R a) and for some a > 2 / d the norm II("V~)^llL(d-',/(d-2)t~d)is sufficie~ly small such that [19, Theorem 1.1] holds. (ii) V2(t, .)~gJ~(R d) and IIl?2(t, ") II~(Rd) exp(21t141 ~ I[1)dt < ~ .

(4.1)

--oo

Let 1/1 satisfy Assumption 4.1(i). Let Ha = Ho + II1 and let Ul(t) = e x p ( - i t H O . Then W+_(H~, Ho) = s-lim UI ( - t) Uo(t) t~

~ot3

exist and are unitary on L2(Rd). Furthermore, by [19, Theorem 1.1], these operators extend to bounded operators on LP(IRd), 1 < p < ~ , with bounded inverse given by W+_(HI, Ho)*. Lemma 4.2. Let V satisfy Assumption 4.1. Then there exists a unique unitary propagator U(t, s) associated with the problem (1.2), with the properties U(t, t) = 1, U(t, s)U(s, r) = U(t, r), t, s, r~lR, and such that for each uo~L2(R d) the integral Eq. (2.1) is solved by u(t) = U(t, S)Uo. Proof We only sketch the proof. The inclusion of the time-independent potential V~ is not covered by the results in [16, 18]. One can obtain the estimate I1UI(t)II~(L,(~,L~(~)) < cltl -a/2 from [19] (see also the remarks after Theorem 3.2), and then one can repeat the proofs in [16,18] to construct a propagator t.71(t,s) for the pair (H~ + Vz(t,x), HO. Then the propagator in the lemma is given by U(t, s) = rYe(t, s)U~(t - s). []

124

A. Jensen

Related results have been obtained in [13], under different conditions on the potential. Theorem 4.3. L e t V satisfy A s s u m p t i o n 4.1. L e t U (t, s) be the associated propagator. Then f o r each s e ~ we have the f o l l o w i n g results: (i) The wave operators W e ( s ) = s-lim U (s, t) Uo(t - s) t -+ -I- c ~

exist and are unitary o n LE(IRa). (ii) We(S) e x t e n d to bounded operators on LP(IRd), 1 < p < ~ , which are invertible in ~ ( L P ) . Proof. Write W ( s ; t) = U(s, t)Uo(t - s) = U(s, t)U~(t - s ) U l ( s - t)Uo(t - s) .

The arguments in the proof of Proposition 2.2 can be repeated (see also Remark 2.6) to yield existence and unitarity of f2 +(s) =

lim U(s, t) Ul(t - s ) , t*~

-I-o0

where the limit is in operator norm in/~(L2). This proves part (i). Note that we have W•

= f2 •

W • (H1, Ho) .

(4.2)

To prove part (ii) we verify Assumption 2.3 with Ho replaced by H i and V by V2. We compute on LE(R d) : UI(S -- t ) V z ( t ) U l ( t - s) = UI(s -- t)Uo(t - s)Uo(s - t) VE(t)Uo(t - s)Uo(s - t ) U l ( t - s).

Assumption 4.1(i) and the proof of Proposition 2.4 yield the estimate IIUl(S - t)Uo(s

-

t)HatL, ) ~ exp(It - sl II ~ I1~).

Using this estimate and Assumption 4.1 we can repeat the proofs of Proposition 2.4 and Theorem 1.2 to conclude that f2• (s) extend to bounded operators on LP(R ~) for all p, 1 < p < oo. The proof is concluded using the properties of W • Ho) and (4.2). [] Acknowledgement. The author wishes to thank Professor Tosio Kato for helpful remarks on the

manuscript. References

[1] Brenner, P., Thom~e, V., Wahlbin, L.B.: Besov spaces and applications to difference methods for initial value problems. (Lect. Notes Math., vol. 434) Berlin Heidelberg New York: Springer 1974 [2] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schr6dinger operators. Berlin Heidelberg New York: Springer 1987 I-3] Evans, D.E.: Time dependent perturbations and scattering of strongly continuous groups on Banach spaces. Math. Ann. 221, 275-290 (1976) 1-4] Howland, J.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315-335 (1974)

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[5] Jensen, A., Nakamura, S.: Mapping properties of functions of Schr6dinger operators between LP-spaces and Besov spaces. Report. Djursholm: Institut Mittag-Leftter 1992 [6] Jensen, A., Ozawa, T.: Existence and non-existence results for wave operators for perturbations of the Laplacian. Rev. Math. Phys. 5, 601-629 (1993) [7] Journ6, J.L., Softer, A., Sogge, C.D.: Decay estimates for Schr6dinger operators. Commun. Pure Appl. Math. 44, 573-604 (1991) [8] Kapitanskii.: Some generalizations of the Strichartz-Brenner inequality. Leningr. Math. J. 1, 693-726 (1990) [9] Kitada, H., Yajima, K.: A scattering theory for time-dependent long-range potentials. Duke Math. J. 49, 341-376 (1982) [10] Kitada, H., Yajima, K.: Remarks on our paper "A scattering theory for time-dependent long-range potentials". Duke Math. J. 50, 1005-1015 (1983) [11] Lin, S.-C.: Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Am. Math. Soc. 139, 469-494 (1969) [12] Neidhardt, H.: On abstract linear evolution equations. I. Math. Nachr. 103, 183-298 (1981) [13] Ruiz, A., Vega, L.: On local regularity of Schr6dinger equations. Duke Math. J. (Int. Math. Res. Notices) 69 (no. 1), 13-27 (1993) [14] Schonbek, T.: Decay of solutions of Schroedinger equations. Duke Math. J. 46, 203-213 (1979) [15] Yafaev, D.R.: Scattering subspaces and asymptotic completeness for the time-dependent Schr6dinger equation. Math. USSR, Sb. 46, 267-283 (1983) [16] Yajima, K.: Existence of solutions for Schr6dinger evolution equations. Commun. Math. Phys. 110, 415-426 (1987) [17] Yajima, K.: On smoothing property of Schr6dinger propagators. In: Fujita, H., Ikebe, T., Kuroda, S.T. (eds.) Functional-analytic methods for partial differential equations. (Lect. Notes Math., vol. 1450, pp. 20-35) Berlin Heidelberg New York: Springer 1990 [18] Yajima, K.: Schr6dinger evolution equations with magnetic fields. J. Anal. Math. 56, 29-76 (1991) [19] Yajima, K.: The LP-continuity of wave operators for Schr6dinger operators. (Department of Mathematical Sciences, University of Tokyo, Preprint 1993)