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Propagation Theorems for Some Classes of Pseudo-Differential Operators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
211, 481]497 Ž1997.
AY975481
Propagation Theorems for Some Classes of Pseudo-Differential Operators Jaouad Sahbani Laboratoire de Physique mathematique et Geometrie, ´ ´ ´ Uni¨ ersite´ Paris 7, UMR 9994, Case 7012, 2, Place Jussieu, Paris Cedex 05, F-75251, France Submitted by James S. Howland Received August 26, 1996
We study continuity properties of the boundary values of the resolvent of perturbations of certain pseudo-differential operators by using recent versions of the conjugate operator method. Our results are optimal on the Holder]Zygmund ¨ scale. In particular, three physical situations are included, namely relativistic and non-relativistic Schrodinger operators and the Stark effect hamiltonian. We allow a ¨ large class of perturbations by giving an ‘‘optimal’’ compromise between regularity and decay at infinity. Q 1997 Academic Press
1. INTRODUCTION Recently considerable effort has been expended on the conjugate operator method in w5, 8, 9, 20x Žsee also w21x.. Our purpose here is to give an idea on the kind of results that can be obtained by using the theorems established in these works. More precisely, we shall study spectral and propagation properties of self-adjoint operators of the form H s H0 q V where H0 is a pseudo-differential operator with constant coefficients and V is an operator that tends to zero at infinity in some sense. Let Ž E, 5 ? 5. be a Banach space and f : R ª E a bounded continuous function. For each integer l G 1 we denote by w l the modulus of continuity of order l of f defined by w l Ž e . s sup x g R 5Ýlks0 Žy1. k Ž kl . f Ž x q k e .5. 481 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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JAOUAD SAHBANI
We say that f is of class La , p for some a ) 0, 1 F p F ` if there is an integer l ) a such that the function e ¬ ey a w l Ž e . belongs to L p ŽŽ0, 1., ey1 d e .; and we set La s La , ` . It is not difficult to show that this condition is independent of l. We denote C kq 0 , for an integer k G 0, the space of k times continuously differentiable functions whose kth order derivative is Dini continuous. Let A be a self-adjoint operator in a Hilbert space H . For each real s we denote by Hs A the Sobolev space associated to A and defined by the norm 5 f 5 s s 5² A:s f 5, with ² x :2 s 1 q x 2 . By interpolation we obtain the Besov space Hs,Ap associated to A, namely Hs,Ap s Ž Hs1A , Hs A2 .u , p if s1 - s2 , 0 - u - 1, 1 F p F `, and s s u s1 q Ž1 y u . s2 . If s ) 0 one can define Hs,Ap as the set of vectors f g H such that the function t ¬ e i At f g H is of class Ls, p. A self-adjoint operator H in H is of class C k Ž A. Žresp. C kq 0 Ž A. or s, p Ž .. C A if there is a number z g C _ s Ž H . such that the function yi t A Ž . yi t A t¬e R z e g B Ž H . is strongly of class C k Žresp. C kq 0 or Ls, p .. We have denoted RŽ z . s Ž H y z .y1 the resolvent of H. A detailed presentation of these notions can be found in w3x. If H is of class C 1 Ž A. then one may introduce the real open set m A Ž H . of points l such that EŽ J .w H, iA x EŽ J . G aEŽ J . for some number a ) 0 and neighborhood J of l; here E is the spectral measure of H. Denote by Pq the spectral projector of A associated to w0, `. and set Pys 1 y Pq. The regularity properties of the maps l ¬ RŽ l " i0., l ¬ P . RŽ l " i m ., considered as B Ž Hs,Ap , Ht,Aq .-valued functions on m A Ž H ., were studied in w9x Žsee also w5, 8, 21x.. The results of these works are optimal on the scales Hs,Ap 4 , C s, p Ž A.4 , and La , p 4 . We use these results in Section 2 for the treatment of the case when H0 s hŽ P . and h is a hypoelliptic function on X s R n, P is the momentum operator in H s L2 Ž X ., and V is in a class of symmetric operators which is described more precisely in our theorems. The simplest version of this situation is when h is an elliptic symbol of strictly positive degree. In particular, non-relativistic and relativistic Schrodinger operators with non¨ local perturbations are covered and we obtain for them sharp local decay and propagation properties. In the relativistic case one may compare our results with those of Umeda w22x and more recently Ben-Artzi and Nemirovsky w13x Žnote that in these references the order of Holder ¨ continuity of the boundary values of the hamiltonians is not specified.. In Section 3 we make a similar study of the perturbations of the Dirac hamiltonian. In both cases we give an ‘‘optimal’’ compromise between regularity and decay at infinity by allowing several components of the perturbations. More explicitly, we consider a perturbation V s Ý Vk such that: the first
PROPAGATION THEOREMS
483
component can be very singular but it must have a faster decay at infinity, this is the so-called ‘‘short range part’’; the kth component may have a weaker decay at infinity but is supposed to be more regular and its kth derivative has to decay at infinity as a short range component. In particular, we recover the usual decomposition of the perturbation in two components: the short and long range parts. We stress the fact that our results are best possible on the Holder]Zygmund scale La 4 . ¨ The main results of w9x are obtained under the condition that H has a spectral gap Ži.e., the spectrum of H is not equal to R.. This is rather annoying in some applications, e.g., it excludes the Stark effect hamiltonians Žsee w14x or w17x. or the simply characteristic operators Žsee w2x.. We have eliminated this condition in w20x for ‘‘small’’ orders of A-regularity classes. We prove the power of these results by studying hamiltonians of the form H s hŽ P . q V where h is a simply characteristic function and V is a pseudo-differential operator with admissible Weyl symbol Žin a sense to be explained.. We allow short and long range bounded perturbations with ‘‘optimal’’ decay at infinity conditions. This is the subject of Subsection 2.2. Similarly, in Section 4 we treat the Stark effect hamiltonian with ‘‘singular’’ perturbations. Indeed, our limiting absorption principle is better than those obtained, for example, in w17x or w4x. In fact, under their regularity condition on the potential V we obtain not only the existence of the boundary values of the resolvent of H but we also give the optimal Žon the Holder]Zygmund scale. order of regularity of these boundary values. ¨
2. PSEUDO-DIFFERENTIAL OPERATORS Let X s R n with n G 2 and H s L2 Ž X . the Hilbert space of all square integrable functions on X. Let Q s Ž Q1 , . . . , Q n . and P s Ž P1 , . . . , Pn . be the position and momentum operators on H given by Ž Q j f .Ž x . s x j f Ž x . and Ž Pf .Ž x . s yiŽ j f .Ž x .. For each multi-index a s Ž a 1 , . . . , a n . we set a s xa1 1 ??? xan n , < a < s a 1 q ??? qa n , Q a [ Q1a 1 ??? Q na n , and ad Qa [ ad Qa 11 ??? ad Qa nn, where ad 0AŽ B . s B, ad 1AŽ B . s w A, B x, and ad kAŽ B . s 1 Ž .. ad 1AŽad ky B ; and similarly we define ad aP . The Schwartz’s test functions A space will be denoted S Ž X .. This section is devoted to the study of the continuity properties of the boundary values of the resolvent of the self-adjoint operators of the form H s hŽ P . q V, when h: X ª R is a simply characteristic function and V belongs to some perturbation classes which are described in our statements.
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JAOUAD SAHBANI
2.1. Hypoelliptic Operators Let h: X ª R be a function of class C m , with m G 2, such that lim h Ž x . s q`,
< x <ªq`
Ý
a hŽ x . F C ;
Ž 2.1.
< a
Ý
a h Ž x . F C Ž1 q h Ž x . . .
Ž 2.2.
< a
These conditions are satisfied by all hypoelliptic polynomials, and for this reason we say that h is a function of hypoelliptic type. The self-adjoint operator H0 s hŽ P . in H is bounded from below Žin particular it has a spectral gap.. Let G s DŽ< H0 < 1r2 . be the form domain of H0 and G * be its adjoint space. We have S Ž X . ; G ; H ' H * ; G * ; S *Ž X . continuously and densely. From Ž2.1. and Ž2.2. it follows that the group e iQ x leaves invariant the spaces G and G * Žsee Lemma 7.6.7 of w3x.. Hence one may consider the Sobolev scale Gs 4 associated to this group in G : if s G 0 then Gs is the Hilbert space defined by the norm 5 u 5 Gs s 5² Q :s u 5 G ; if s - 0 then Gs is the completion of G for the norm 5 ? 5 Gs defined by the same formula. We have G0 s G and Gs ; Gt for t - s. By interpolation we obtain the weighted Besov space Gr, p s Ž Gs , Gt .u , p for 0 - u - 1, 1 F p F `, and r s Ž1 y u . s q u t. Note that Gs, 2 s Gs . Similarly we define Ž G *.s ' GsU , U Ž G U .s, p ' Gs,U p related to the preceding spaces by the formula Ž Gs .* s Gys U and Ž Gs, p .* s Gys, p9 for each s g R, 1 F p - `, and p9 s prŽ p y 1.. Let V: G ª G * be a symmetric operator such that H0 q V q i: G ª G * is an isomorphism and denote by H the associated self-adjoint operator in H . Suppose that Ž H q i .y1 y Ž H0 q i .y1 is a compact operator in H . Assumption 2.1. Let u g C0`Ž X . with u Ž x . ) 0 in an annulus 0 - a < x < - b - ` and u Ž x . s 0 otherwise. Let s ) 1 and suppose that V [ Ý0 F k - s Vk where Vk g X ' B Ž G , G *. is a symmetric operator such that
Ý
sup r sykq< g < u Ž Qrr . ad gP ad Qb Ž Vk .
< g
X
- `.
Ž 2.3.
In the particular case where V s Ý k - s Vk Ž Q . is a multiplication operator in H our assumption is quite simple and natural, as the next examples show: Ž1. If 1 - s - 2 then V s V0 Ž Q . q V1Ž Q . such that V0 Ž x . s O Ž< x
PROPAGATION THEOREMS
485
Ž2. If 2 - s - 3 then V s V0 Ž Q . q V1Ž Q . q V2 Ž Q . such that V0 , V1 are as above and V2 Ž x . s O Ž< x
a a Ž x . F Ca² x :y< a < .
Ž 2.4.
Let c " be two functions of class C0`ŽR n . such that < h9Ž j .< G a ) 0 on the supports of c " and let w " be two symbols of order zero such that "x ? h9 Ž j . G b < x < ? h9 Ž j .
on supp w "= supp c " Ž with b ) 0 . .
Ž 2.5. The next theorem shows that the pseudo-differential operators a " Ž Q, P . s w " Ž Q . c " Ž P . localize in non-propagation regions of the phase space with respect to the flow e i H t generated by H. THEOREM 2.2. In addition to the assumptions of Theorem 2.1 let the function h be of class C`Ž X .. Then a " Ž Q, P . RŽ l . i0. GsU ; Gsy1y a if 0 - a - s y 1r2 and l g m Ž H .. Moreo¨ er, the maps l ¬ a " Ž Q, P . RŽ l . i0. g B Ž GsU , Gsy1y a . are locally of class La on m Ž H .. Here again the orders of Holder continuity are optimal in the same ¨ sense as above. The case a s 0 is treated in the next theorem.
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JAOUAD SAHBANI
Assumption 2.2. With the notations of Assumption 2.1, let `
Ý
< g
H1 r
sykq< g <
dr
u Ž Qrr . ad gP ad Qb Ž Vk .
X
r
- `.
Ž 2.6.
THEOREM 2.3. Suppose that the function h is of class C` and that Assumption 2.2 is ¨ alid for s s s q 1r2. Then a " Ž Q, P . RŽ l . i0. GsU ; Gsy1 if l g m Ž H .. Moreo¨ er, the maps l ¬ a " Ž Q, P . RŽ l . i0. g B Ž GsU , Gsy1 . are weakly continuous on m Ž H .. The weak continuity cannot be replaced by norm-continuity in general Žsee again w8, 9x or w21x.. We emphasize the fact that from the theorems stated above one can deduce, via a Fourier transformation, important information concerning the dispersion and propagation properties of the flow e i H t generated by H as in w8, 9x Žsee also w21x.. For example, we have the following theorem: THEOREM 2.4. Under the assumptions of Theorem 2.2, for each w g C0`Ž m Ž H .. there exists a finite constant C such that t ² Q :ys eyi H w Ž H . ² Q:ys
yŽ sy1r2.
G *ª G F C ² t :
t ² Q :sy 1y a a . Ž Q, P . eyi H w Ž H . ² Q:ys
ya
G *ª G F C ² t :
;
Ž 2.7. if "t G 0.
Ž 2.8. In fact, by using a complex interpolation argument one can obtain more precise estimates, but we shall not state them here. Proof of Theorem 2.1. Ži. Let J be a bounded real open set such that J l Ž k Ž h. j sp Ž H .. s B. Then from Ž2.1., V [ hy1 Ž J . is a bounded open subset of X on which < h9Ž x .< ) c ) 0. So, one can construct a vector field F of class C0`Ž X . such that ŽŽ F ? = . h.Ž x . G c ) 0 in V. Let us set f s div F. It is clear that H0 is of class C m Ž A., where A is the self-adjoint operator As
1 2
n
n
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Q j Fj Ž P . y 2 f Ž P . . Ž 2.9.
js1
js1
In particular, A is strictly conjugate to H0 on J. Žii. We introduce now some notations which will be needed below. It is clear that e i A t leaves invariant the spaces G and G *. Then e i A t induces two strongly continuous groups of bounded operators on these spaces, and so the automorphism group W t on X defined by W t w T x s W t T s eyi A t Te i A t for all T g X . For s s 0 and p s 1, we say that T is of class
487
PROPAGATION THEOREMS
C 0, 1 Ž A; G , G *. ' Cq0 Ž A; G , G *. if the function t ¬ W t T g X is Dinicontinuous on R. For s ) 0 we say that T is of class C s , p Ž A; G , G *. Žresp. C k Ž A; G , G *., k g N. if the preceding function is of class Ls , p on R Žresp. strongly of class C k .. Similarly we define the regularity classes C s , p Ž A; G *, G ., C k Ž A; G *, G ., C kq 0 Ž A; G *, G .. Remark that if H g X is of class C s , p Ž A; G , G *. then Ž H q i .y1 is of class C s , p Ž A; G *, G . Žbecause H q i: G ª G * is an isomorphism., and this is better than H being of class C s , p Ž A.. So, in order to prove that H is of class C sq1r2 Ž A; G , G *., it suffices to prove that V is of class C sq 1r2 Ž A; G , G *. Ž H0 being of class C m Ž A; G , G *.., and this assertion will be shown with the help of the next criterion. THEOREM 2.5. Let L be a self-adjoint operator in H bounded from below by a strictly positi¨ e constant and such that: Ž1. e iLt G ; G and 5 e iLt 5 BŽ G . F C²t :N with N - `, and Ž2. there is an integer l G 1 such that AlLyl extends to a continuous operator in G *. Let 0 F s - l. Then a symmetric operator T g X is of class C s , p Ž A; G , G *. if there is a function u g C0`ŽR. with u Ž x . ) 0 if 0 - a - < x < - b - ` such that Ž with the usual con¨ ention if p s `. `
H1
r su Ž Lrr . T
p
dr
X
1rp
- `.
r
Ž 2.10.
In particular, if p s 1 or ` and if the operator T is of class C k Ž A; G , G *. for an integer 0 F k F s and `
H1
r
u Ž Lrr . A w T x
s yk
k
p
X
dr r
1rp
- `,
Ž 2.11.
then T is of class C s , p Ž A; G , G *.. The proof of this result is very similar to the proof of Theorem 7.5.8 of w3x up to minor modifications Žthe details can be found in w21x.. Žiii. By applying the preceding theorem to the operators L s ² Q : and A defined by Ž2.8., it is not difficult to establish the following proposition. PROPOSITION 2.1. If the operator V satisfies Assumption 2.1 or 2.2 then V is of class C s Ž A; G , G *. or C s , 1 Ž A; G , G *., respecti¨ ely. Živ. We then conclude that under the conditions of Theorem 2.1 the operator H is of class C sq 1r2 Ž A. and in particular, A is strictly conjugate to H on J. But H has a spectral gap Žbecause H0 is bounded from below and sessŽ H . s s Ž H0 ... Then the main theorem of w6x implies that the A . maps l ¬ RŽ l " i0. g B Ž Hs A , Hys are locally of class Lsy 1r2 .
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JAOUAD SAHBANI
Žv. Let Gs Ž A. and GsU Ž A. be the spaces of vectors f in G , resp. G *, such that the functions t ¬ e i At f with values in G , resp. G *, are of class Ls, 2 . Let l0 g R _ s Ž H .. Then by iterating the first resolvent equation we get 2
2
R Ž z . s R Ž l0 . q Ž l0 y z . R Ž l0 . q Ž l0 y z . R Ž l0 . R Ž z . R Ž l0 . . On the other hand, we have RŽ l 0 . GsU Ž A. ; Gs Ž A. Žbecause RŽ l 0 . is of class C s Ž A; G *, G .., so RŽ l0 . GsU Ž A. ; Hs A. Finally, from Theorem 3.7.8 in w3x, Gs Ž A. coincides Žas a topological vector space. with the space u g G N ² A:s u g G 4 endowed with the norm 5 u 5 Gs Ž A. s 5² A:s u 5 G ; similarly for GsU . So by taking into account the fact that ² A:s² Q :ys is a bounded operator in G and G * we deduce that GsU ; GsU Ž A. and so the maps l ¬ RŽ l " i0. g B Ž GsU , Gys . are locally of class Lsy 1r2 on J, hence on m Ž H .. This finishes the proof of Theorem 2.1. Žvi. By combining the arguments used above and an adapted version of Mourre’s proof of Theorem 3.1 from w18x to our case, and then by applying Theorem 2 Žresp. Theorem 1. of w8x we obtain easily Theorem 2.2 Žresp. Theorem 2.3.. Detailed proofs of these theorems can be found in w21x. Elliptic Case. In order to illustrate the preceding considerations in two physically important situations, namely the non-relativistic and relativistic Schrodinger operators, we shall first discuss the following particular case. ¨ Let h be an elliptic symbol of degree 2 r ) 0, i.e., h g C`ŽR n ., < hŽ a . Ž x .< F Ca² x :2 ry< a < for each multi-index a and < hŽ x .< G C² x :2 r, for some C ) 0, outside a compact set. In this case G s H r ŽR n ., G * s H yr ŽR n . the usual Sobolev spaces, and Gt, p s Ht,r p ŽR n ., G * s Ht,yrp ŽR n . are the standard weighted Besov spaces. Let V: H r ª H yr be a symmetric operator. We say that V is small at infinity if there is t ) r and a function j g C`ŽR n . with j Ž x . s 0 if < x < F 1 and j Ž x . s 1 if < x < G 2 such that lim R ª` 5 j Ž QrR.V 5 H t ª H y r s 0. In this case it is easy to see that Ž H y i .y1 y Ž H0 y i .y1 is a compact operator in H , where we denote by H the self-adjoint operator in H associated to the sum H0 q V Žsee w3, p. 348x.. So, this implicit condition is expressed now in terms of the decay of V at infinity. It is clear that all the theorems stated above are valid in this situation. But the advantage now is that all spaces are standard. Ži. Non-Relati¨ istic Schrodinger Operator. A trivial example of the ¨ elliptic symbol is the quadratic function hŽ j . s < j < 2 which is of degree 2 Ži.e., r s 1.. In this case we obtain sharp results concerning local decay and propagation properties for the non-relativistic Schrodinger operator H s ¨ yD q V with a non-local perturbation V satisfying Assumption 2.1 or 2.2.
PROPAGATION THEOREMS
489
Žii. Relati¨ istic Schrodinger Operator. Another physically important ¨ situation is obtained when hŽ x . s < j < 2 q 1 which is an elliptic symbol of degree 1 Ži.e., r s 1r2.. In this case the hamiltonian is the relativistic Schrodinger operator H s 'yD q 1 q V. Again we get sharp results ¨ concerning spectral and propagation properties of H. For instance, we get
'
THEOREM 2.6. Let s ) 1r2 and V: H 1r2 ª H y1 r2 be a symmetric and small at infinity operator which satisfies Assumption 2.1 for s s s q 1r2. Denote by H the self-adjoint operator in H associated to the sum 'yD q 1 q V. Then Ži. sp Ž H . is a countable set and has no accumulation points in R _ 14 . Žii. The boundary ¨ alues Ž H y l . i0.y1 s w* ylim m ª " 0 Ž H y r2 Ž n . 1r2 Ž n .. locally uniformly in l g l . i m .y1 exist in B Ž H y1 1r2, 1 R , Hy1r2, ` R R _ w 14 j sp Ž H .x. In particular, H has no singularly continuous spectrum. 1r2 Ž n .. Žiii. The maps l ¬ Ž H y l . i0.y1 g B Ž Hsy1 r2 ŽR n ., Hys R are sy1r2 locally of class L on R _ w 14 j sp Ž H .x. Theorems 2.2]2.4 also give interesting results in this particular context, but we shall not state them explicitly. The limiting absorption principle follows from the main theorem in w7x and is better than the similar result obtained in Umeda w22x or more recently ŽMay 1996. by Ben-Artzi and Nemirovsky w13x. Moreover, the order of regularity of the boundary values Ž H y l . i0.y1 as a function of l is optimal on the Holder]Zygmund scale even if V s 0 Žin the paper ¨ cited above the order of Holder continuity is not specified.. On the other ¨ hand our perturbations are non-local, and even in the local case our class is larger. Note, however, that in w13x the limiting absorption principle is also established near the threshold point 1 Žsee also w10, 11x.. 2.2. Simply Characteristic Operators Let us now consider h: X ª R of class C m , m G 4, such that lim
< x <ªq`
Ý
Ž hŽ x .
q h9 Ž x .
. s q`,
hŽ a . Ž x . F C Ž h Ž x . q h9 Ž x . . .
Ž 2.12. Ž 2.13.
< a
These conditions are satisfied by simply characteristic polynomials Žsee w16x for the definition.. The spectrum of H0 s hŽ P . is equal, in general, to R, for example, if hŽ x 1 , x 2 . s x 1 x 2 . In such a situation the results of w8, 9x are not useful, but rather the results of w20x.
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JAOUAD SAHBANI
Let V be a bounded symmetric operator in H such that the self-adjoint operator H s H0 q V satisfies that Ž H q i .y1 y Ž H0 q i .y1 is compact in H . We assume that V s VS q VL , with VS , VL bounded symmetric operators in H . The short range component VS of V will be submitted to the following type of condition sup r sx Ž Qrr . VS - `,
Ž 2.14.
r)1
where x is the characteristic function of the annulus 1 - < x < - 2. In order to introduce the long range assumptions we use the following DEFINITION 2.1. Let s be a real number such that 1 - s - 2. A function b: R 2 n ª C is a s-admissible symbol if its Fourier transform ˆ b is a measure satisfying
HH
ˆb Ž x, y . ? ² y :sy1
H< z <-1 ˆb Ž x q z, y . y ˆb Ž x, y .
q
? < z
If s s 1 then it is convenient to define the 1-admissibility by Žsee w6x.
HH
ˆb Ž x, y . log Ž 2 q < y < .
H< z <-1 ˆb Ž x q z, y . y ˆb Ž x, y .
q
? < z
We say that a pseudo-differential operator VL is s-admissible if its Weyl symbol is a distribution a: R 2 n ª R such that a j Ž x, y . s x j aŽ x, y . and a j, k Ž x, y . s Ž y k y ix k . a j Ž x, y . are s-admissible symbols for each j, k s 1, . . . , n. We consider the condition sup r sy1 x Ž Qrr . VL q x Ž Qrr . Q j , VL
4 - `.
Ž 2.17.
r)1
We know Žsee w6x. that if Ž2.14. and Ž2.17. are true for s s 1 with H1` dr in place of sup r ) 1 and if VL is a 1-admissible pseudo-differential operator then the boundary values RŽ l . i0. of H exist in optimal spaces. Let Hs 4 be the weighted Sobolev space defined by the norm 5 f 5 s 5² Q :s f 5. THEOREM 2.7. Let s be a real number such that 1r2 - s - 1. Assume that Ž2.14. and Ž2.17. are true for s s s q 1r2 and that VL is Ž s q 1r2.-
491
PROPAGATION THEOREMS
admissible. Then the maps l ¬ RŽ l . i0. g B Ž Hs , Hys . are locally of class Lsy 1r2 on m Ž H . [ R _ Ž sp Ž H . j k Ž h... Sketch of the Proof. Ži. It is clear that the function F: R n ª R n defined by FŽ j . s
h9 Ž j . 1 q hŽ j .
2
q h9 Ž j .
Ž 2.18.
2
is of class BC 1 ŽR n .. Let us set f s div F. Then As
1 2
n
n
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Fj Ž P . Q j q 2 f Ž P . Ž 2.19.
js1
js1
is a self-adjoint operator in H . It is easy to see that H0 is of class C 2 Ž A. and that A is locally strictly conjugate to H0 on R _ k Ž h. Že.g., see w1x or w6x.. Moreover, ² A:s² Q :ys is a bounded operator in H Žsee w6x., for each 0 - s F 2. Žii. As in the last section, from Theorem B in w20x, it suffices to show that H is of class C sq1r2 Ž A.. This assertion follows by mimicking the proof of w6x and by taking into account Theorem 2.5 of the last subsection. Let c " be two functions of class C0`ŽR n . such that supp c " are disjoint from the set of critical points of h; and let w " be two symbols of order zero such that for some b ) 0, "x ? h9 Ž j . G b < x < ? h9 Ž j .
if Ž x, j . g supp w "= supp c " . Ž 2.20.
If we set p "s w " Ž Q . c " Ž P . we then get THEOREM 2.8. Under the assumptions of Theorem 2.6 let a be a real number such that 0 - a - s y 1r2. Then p. Ž Q, P . RŽ l " i0. Hs ; Hsy1y a if l g m Ž H ., and the maps l ¬ p. Ž Q, P . RŽ l " i0. g B Ž Hs , Hsy1y a . are locally of class La on m Ž H ..
3. DIRAC OPERATOR Let E be a four dimensional Hilbert space endowed with 4 symmetric operators a 1 , a 2 , a 3 , b such that for j, k s 1, 2, 3 one has
¡a a
~
¢
j
k
q ak a j
a j b q ba j b
2
s 2 d jk s0 s1
Ž 3.1. .
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JAOUAD SAHBANI
We set H s L2 ŽR 3 . m E ( L2 ŽR 3, E . and we denote the tensor products of the preceding matrices with the identity of L2 ŽR 3 . by the same symbols. Let Q, P be the position and momentum operators, respectively, in L2 ŽR 3 .. Let us consider the free Dirac operator in H defined by 3
H0 [
Ý a j Pj q m b ' a ? P q m b ,
m ) 0.
Ž 3.2.
js1
H0 is a self-adjoint operator in H and s Ž H0 . s Žy`, ym x j w m, q`.. Denote by m Ž P ., p " the pseudo-differential operators associated to the symbols
¡m Ž k . ~ ¢p Ž k . "
1r2
s Ž k 2 q m2 . , 1 1 s " Ž a ? k q mb . . 2 mŽ k .
Ž 3.3.
It is clear that pq pys py pqs 0 and pqq pys 1, in particular, the symmetric operator pqy py is unitary in H . One has H0 s m Ž P .Žpqy py .. Let us set H s ŽR 3 . s H s ŽR 3 . m E, for s g R, with H s ŽR 3 . the usual Sobolev space. Remark that H 0 s H and that the form domain of H0 is H 1r2 . Let Hs " 1r2 be the weighted Sobolev space defined by the norm 5 u 5 s," 1r2 [ 5² P :" 1r2² Q :s u 5. We say that a symmetric bounded operator T : H 1r2 ª H y1 r2 is small at infinity if there is a function u g C0`ŽR 3 . with u Ž x . s 0 near zero and u Ž x . s 1 near infinity such that lim r ª` 5 u Ž Qrr .T 5 X s 0, where X s B Ž H 1r2 , H y1r2 .. For a positive number s , an integer 0 F k - s , and an operator Vk g X , let us consider the condition 3
Ý
Ý
sup r sykq< n < j Ž Qrr . ad Qm ad nP ad ab nb Ž Vk .
ns1 < m
X
- `, Ž 3.4.
where j is a function of C0`ŽR 3 . with j Ž x . ) 0 if a - < x < - b and j Ž x . s 0 otherwise. We know that Žsee w12x. if V s V0 q V1 with Vk g X satisfying condition Ž3.4. with s s 1 and H1` dr in place of sup r ) 1 then the boundary values of the resolvent of H0 q V exist in a certain topology. THEOREM 3.1. Let a number s ) 1r2 and let V: H 1r2 ª H y1 r2 be a symmetric bounded operator such that H0 q V q i: H 1r2 ª H y1 r2 is an
493
PROPAGATION THEOREMS
isomorphism, and denote by H the associated self-adjoint operator in H . Assume that V is small at infinity and V [ Ý0 F k - s Vk with Vk g X satisfying condition Ž3.4. with s s s q 1r2. Then the maps l ¬ Ž H y l . i0.y1 1r2 . g B Ž Hsy1 r2 , Hys are locally of class Lsy1r2 on the open set m Ž H . [ R_w sp Ž H . j "m4x. This result is the best possible on the Lipschitz]Zygmund scale La 4 , even if V s 0. Sketch of the Proof. Let J be a bounded open interval with closure included in Ž m, `., and let u be a function of class C0`ŽŽ m, `.. equal to 1 on J. Then the function F defined on R 3 by F Ž j . [ m2 Ž j . Ž u ( m .Ž j .Ž jr< j < 2 . is of class C0`ŽR 3 _ 04. Žbecause m Ž0. s m f supp u .. We set f s div F and we consider the self-adjoint operator in L2 ŽR 3 . defined by
A˜ [
1 2
3
3
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Fj Ž P . Q j q 2 f Ž P . . Ž 3.5.
js1
js1
It is easily shown that A s pq A˜pqq py A˜py is a self-adjoint operator in H , H0 is of class C`Ž A., and A is strictly conjugate to H0 on J. Moreover we have the following lemma: LEMMA 3.1. For each real positi¨ e number s, the operator Js s ² A:s² Q :ys is bounded in H . Indeed, it suffices Žinterpolation. to show it for s s 2 m an even integer. But this assertion is a simple consequence of the boundedness of A n ² Q :yn for each integer n G 1, which follows easily by writing A s Ý3js1 Fj Ž P . Q j q Ž ir2. f Ž P . q GŽ P . where the function G: R 3 ª E is of class C0` . Since V is small at infinity, the operator Ž H q i .y1 y Ž H0 q i .y1 is compact in H Žsee, for example, w12x.. As in Subsection 2.1, it suffices to show that H is of class C sq1r2 Ž A., and this is a simple consequence of the next proposition. PROPOSITION 3.1. Let Vk : H 1r2 ª H y1 r2 be a symmetric bounded operator. If Vk satisfies Ž3.4., then Vk is of class C s Ž A; H 1r2 , H y1r2 .. The proof is very similar to the proof of Proposition 2.1. It should be clear by now that one can easily obtain results similar to Theorems 2.2 and 2.3 of the last section, so we shall not state them explicitly here.
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JAOUAD SAHBANI
¨ 4. SCHRODINGER OPERATORS WITH CONSTANT ELECTRIC FIELD We consider the free Stark effect hamiltonian H0 defined in H s L2 ŽR. by H0 [ P 2 q Q,
Ž 4.1.
where P s yiŽ drdx . and Q is the operator of multiplication by x. 3 Let us set U [ eyi P r3 . It is clear that w Q, U x s P 2 . It follows that y1 H0 s U QU. Then H0 is essentially self-adjoint on S ŽR. in H and s Ž H0 . s R. Moreover
Ž H0 y z .
y1
s Uy1 Ž Q y z .
y1
U,
Ž 4.2.
and U commutes with P Žhence with all functions of P .. On the other hand, yP is conjugate to Q and Q is of class C`Ž P .. Denote P " the spectral projectors of P associated to the semi-axis .x G 0, and H s, p the usual Besov space, and C "s z g C N "I z ) 04 ; we set C " s C "j R for the closure of C ". PROPOSITION 4.1. Ž1. The holomorphic maps C "2 z ¬ Ž H0 y z .y1 g Ž B H 1r2,1 , H y1r2, ` . extend to weakly* continuous functions on C " . Let us set Ž H0 y l " i0.y1 [ lim m ªq0 Ž H0 y l . i m .y1 . Ž2. For each s ) 1r2, the maps l ¬ Ž H0 y l " i0.y1 g B Ž H s, H ys . are locally of class Lsy 1r2 on R. Ž3. For each s ) 1r2, we ha¨ e P " Ž H0 y l " i0.y1 H s ; H sy1 and the functions l ¬ P " Ž H0 y l " i0.y1 g B Ž H s, H sy1 . are weakly continuous on R. Ž4. For each a ) 0 and s ) a q 1r2, the maps l ¬ P " Ž H0 y l " i0.y1 g B Ž H s, H sy1y a . are locally of class La on R. Note that we have also P " Ž H0 y l " i0.y1 P .s 0. Obviously, these results are simple consequences of the known properties on the operators Q and P Žsee w21x for a detailed treatment .. We shall describe a class of potentials V such that the last proposition remains valid for the hamiltonians of the form H s H0 q V. For a non-negative integer k we denote by BC k ŽR. the Banach space of the functions of class C k such that the derivatives f Ž j. are bounded for each j s 0, . . . , k; and BC kq 0 ŽR. the space of the functions f of class BC k such that f Ž k . is Dini continuous. THEOREM 4.1. Let V Ž Q . be the operator of multiplication by a function V and assume that V is of class BC 1q 0 ŽR.. Denote H the self-adjoint operator in
PROPAGATION THEOREMS
495
H associated to the sum H0 q V. Then sp Ž H . has no accumulation points in R. Moreo¨ er, the boundary ¨ alues Ž H y l " i0.y1 [ lim m ªq0 Ž H y l " i m .y1 of the resol¨ ent of H exist locally uniformly in l g R _ sp Ž H . in the weak*-topology of B Ž H 1r2, 1 , H y1r2, ` .. Proof. Let us set A s yP. Then we have
w H , iA x s w H0 q V , iA x s w H0 , iA x q w V , iA x s 1 q V 9 Ž Q . , Ž 4.3. which is obviously bounded in H . So, we can easily Žcf. w14x. see that A is locally conjugate to H on R. From Theorem A of w20x, the limiting absorption principle is valid if H is of class C 1q 0 Ž A.. The last property is true if the function V is of class BC 1q 0 . This finishes the proof. Remark that the eigenvalues of H Žif they exist. are simple. In w4x the potential V is supposed of class BC 2 , which is more restrictive than our condition, but their condition implies absence of eigenvalues. If V is more regular with respect to A, the boundary values Ž H y l " . i0 y1 are Holder continuous in the appropriate topologies. We give the ¨ optimal order Žon the Besov scale. of continuity, with an optimal Žon the Holder]Zygmund scale. regularity condition on V. ¨ THEOREM 4.2. With the notations of the last theorem, let s be a real number such that 1r2 - s - 1. Suppose that the function V is of class Lsq 1r2 ŽR.. Then: Ž1. The maps l ¬ Ž H y l " i0.y1 g B Ž H s, H ys . are locally of class L on R _ sp Ž H .. Ž2. We ha¨ e P " Ž H y l " i0.y1 H s ; H sy1, ` if l f sp Ž H . and the maps l ¬ P " Ž H y l " i0.y1 g B Ž H s, H sy1, ` . are weak* continuous on R _ sp Ž H .. Ž3. For each a q 1r2 - s the maps l ¬ P " Ž H y l " i0.y1 g B Ž H s, H sy1y a . are locally of class La on R _ sp Ž H .. sy 1r2
In particular, if V is of class BC 2 ŽR. then all the assertions of that last theorem are true for each s and each a such that 0 - a - s y 1r2 - 1r2, and sp Ž H . s B. We ask that V be of class BC 2 in order to apply w15, Corollary 22, p. 1414x Žsee w4x., which implies that sp Ž H . s B. Note that in w4x this condition is imposed by the regularity requirement of the version of the conjugate operator method which they use. It is reasonable to conjecture that if V is a Holder continuous function ¨ of order a - 1 then the singularly continuous spectrum of H can cover all the real axis; and so the Lipschitz class could be the optimal regularity class for H to have purely absolutely continuous spectrum Žsee also w19x..
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ACKNOWLEDGMENTS I am extremely grateful to Professors A. Boutet de Monvel and V. Georgescu for helpful discussions and stimulating suggestions.
REFERENCES 1. G. Arsu, Spectral analysis for simply characteristic operators by Mourre’s method, 3, J. Operator Theory 34 Ž1995., 177]187. 2. S. Agmon and L. Hormander, Asymptotic properties of solutions of differential equations ¨ with simple characteristics, J. Analyse Math. 30 Ž1976., 1]38. 3. W. Amrein, A. Boutet de Monvel, and V. Georgescu, ‘‘C0-groups, commutator methods and spectral theory of N-body Hamiltonians,’’ in ‘‘Progress in Math.,’’ Vol. 135, Birkhauser, Basel, 1996. ¨ 4. F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, and R. Weder, Schrodinger ¨ operator with an electric field and random or deterministic potentials, Comm. Math. Phys. 88 Ž1983., 387]397. 5. A. Boutet de Monvel and V. Georgescu, Boundary values of the resolvent of a selfadjoint operator: Higher order estimates, in ‘‘Algebraic and Geometric Methods in Mathematical Physics, Kaciveli Workshop, Crimee, ´ 1993’’ ŽA. Boutet de Monvel and V. Marchenko, Eds.., Kluwer Academic, DordrechtrNorwell, MA, 1996. 6. A. Boutet de Monvel and V. Georgescu, Limiting absorption principle for long range perturbations of pseudo-differential operators, in ‘‘Advances in Dynamical Systems and Quantum Phys., Conference in Honour of G. Dell’Antonio, Capri, 1993,’’ World Scientific, Singapore, 1994. 7. A. Boutet de Monvel and V. Georgescu, Spectral theory and scattering theory by the conjugate operator method, Algebra i Analiz 4, No. 3 Ž1992., 73]116; Leningrad Math. J. 4, No. 3 Ž1993., 469]501. 8. A. Boutet de Monvel, V. Georgescu, and J. Sahbani, Boundary values of resolvent families and propagation properties, C.R. Acad. Sci. Paris Ser. ´ I Math. 322 Ž1996., 289]294. 9. A. Boutet de Monvel, V. Georgescu, and J. Sahbani, Higher order estimates in the conjugate operator theory, preprint, No. 59, Institut de Mathematiques de Jussieu, 1996. ´ 10. A. Boutet de Monvel, G. Kazantseva, and M. Mantoiu, Some anisotropic Schrodinger ¨ operators without singular spectrum, Hel¨ . Phys. Acta 69 Ž1996., 13]25. 11. A. Boutet de Monvel and M. Mantoiu, The method of the weakly conjugate operator, preprint. 12. A. Boutet de Monvel, D. Manda, and R. Purice, Limiting absorption principle for the Dirac operator, Ann. Inst. H. Poincare´ 58 Ž1993., 413]431. 13. M. Ben-Artzi and J. Nemirovsky, Remarks on relativistic Schrodinger operators and their ¨ ´ extensions, preprint, No. 342, Ecole Polytechnique, May, 1996. 14. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, ‘‘Schrodinger Operators with ¨ Applications to Quantum Mechanics and Global Geometry,’’ Springer-Verlag, Berlin, 1987. 15. N. Dunford and J. T. Schwartz, ‘‘Linear Operators. Part 2. Spectral Theory, Self-adjoint Operators in Hilbert Space,’’ Interscience, New YorkrLondon, 1963. 16. L. Hormander, ‘‘The Analysis of Linear Partial Differential Operators,’’ Vol. 2, Springer¨ Verlag, Berlin, 1983, 1985.
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17. A. Jensen, Scattering theory for Stark Hamiltonians, Proc. Indian Acad. Sci. Math. Sci. 104, No. 4 Ž1994., 599]651. 18. E. Mourre, Operateurs conjugues ´ ´ et proprietes ´ ´ de propagation, Comm. Math. Phys. 91 Ž1983., 279]300. 19. S. N. Naboko and A. B. Pushnitskii, Point spectrum on a continuous spectrum for weakly perturbed Stark type operators, Functional Anal. Appl. 29, No. 4 Ž1995., 248]257. 20. J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory, in press. 21. J. Sahbani, ‘‘Theoremes de Propagation, Hamiltoniens Localement Reguliers et Applica´ ` ´ tions,’’ These, ` Universite´ Paris 7, Juillet 1996. 22. T. Umeda, Radiation conditions and resolvent estimates for relativistic Schrodinger ¨ operators, Ann. Inst. H. Poincare´ 63 Ž1995., 277]296.
211, 481]497 Ž1997.
AY975481
Propagation Theorems for Some Classes of Pseudo-Differential Operators Jaouad Sahbani Laboratoire de Physique mathematique et Geometrie, ´ ´ ´ Uni¨ ersite´ Paris 7, UMR 9994, Case 7012, 2, Place Jussieu, Paris Cedex 05, F-75251, France Submitted by James S. Howland Received August 26, 1996
We study continuity properties of the boundary values of the resolvent of perturbations of certain pseudo-differential operators by using recent versions of the conjugate operator method. Our results are optimal on the Holder]Zygmund ¨ scale. In particular, three physical situations are included, namely relativistic and non-relativistic Schrodinger operators and the Stark effect hamiltonian. We allow a ¨ large class of perturbations by giving an ‘‘optimal’’ compromise between regularity and decay at infinity. Q 1997 Academic Press
1. INTRODUCTION Recently considerable effort has been expended on the conjugate operator method in w5, 8, 9, 20x Žsee also w21x.. Our purpose here is to give an idea on the kind of results that can be obtained by using the theorems established in these works. More precisely, we shall study spectral and propagation properties of self-adjoint operators of the form H s H0 q V where H0 is a pseudo-differential operator with constant coefficients and V is an operator that tends to zero at infinity in some sense. Let Ž E, 5 ? 5. be a Banach space and f : R ª E a bounded continuous function. For each integer l G 1 we denote by w l the modulus of continuity of order l of f defined by w l Ž e . s sup x g R 5Ýlks0 Žy1. k Ž kl . f Ž x q k e .5. 481 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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JAOUAD SAHBANI
We say that f is of class La , p for some a ) 0, 1 F p F ` if there is an integer l ) a such that the function e ¬ ey a w l Ž e . belongs to L p ŽŽ0, 1., ey1 d e .; and we set La s La , ` . It is not difficult to show that this condition is independent of l. We denote C kq 0 , for an integer k G 0, the space of k times continuously differentiable functions whose kth order derivative is Dini continuous. Let A be a self-adjoint operator in a Hilbert space H . For each real s we denote by Hs A the Sobolev space associated to A and defined by the norm 5 f 5 s s 5² A:s f 5, with ² x :2 s 1 q x 2 . By interpolation we obtain the Besov space Hs,Ap associated to A, namely Hs,Ap s Ž Hs1A , Hs A2 .u , p if s1 - s2 , 0 - u - 1, 1 F p F `, and s s u s1 q Ž1 y u . s2 . If s ) 0 one can define Hs,Ap as the set of vectors f g H such that the function t ¬ e i At f g H is of class Ls, p. A self-adjoint operator H in H is of class C k Ž A. Žresp. C kq 0 Ž A. or s, p Ž .. C A if there is a number z g C _ s Ž H . such that the function yi t A Ž . yi t A t¬e R z e g B Ž H . is strongly of class C k Žresp. C kq 0 or Ls, p .. We have denoted RŽ z . s Ž H y z .y1 the resolvent of H. A detailed presentation of these notions can be found in w3x. If H is of class C 1 Ž A. then one may introduce the real open set m A Ž H . of points l such that EŽ J .w H, iA x EŽ J . G aEŽ J . for some number a ) 0 and neighborhood J of l; here E is the spectral measure of H. Denote by Pq the spectral projector of A associated to w0, `. and set Pys 1 y Pq. The regularity properties of the maps l ¬ RŽ l " i0., l ¬ P . RŽ l " i m ., considered as B Ž Hs,Ap , Ht,Aq .-valued functions on m A Ž H ., were studied in w9x Žsee also w5, 8, 21x.. The results of these works are optimal on the scales Hs,Ap 4 , C s, p Ž A.4 , and La , p 4 . We use these results in Section 2 for the treatment of the case when H0 s hŽ P . and h is a hypoelliptic function on X s R n, P is the momentum operator in H s L2 Ž X ., and V is in a class of symmetric operators which is described more precisely in our theorems. The simplest version of this situation is when h is an elliptic symbol of strictly positive degree. In particular, non-relativistic and relativistic Schrodinger operators with non¨ local perturbations are covered and we obtain for them sharp local decay and propagation properties. In the relativistic case one may compare our results with those of Umeda w22x and more recently Ben-Artzi and Nemirovsky w13x Žnote that in these references the order of Holder ¨ continuity of the boundary values of the hamiltonians is not specified.. In Section 3 we make a similar study of the perturbations of the Dirac hamiltonian. In both cases we give an ‘‘optimal’’ compromise between regularity and decay at infinity by allowing several components of the perturbations. More explicitly, we consider a perturbation V s Ý Vk such that: the first
PROPAGATION THEOREMS
483
component can be very singular but it must have a faster decay at infinity, this is the so-called ‘‘short range part’’; the kth component may have a weaker decay at infinity but is supposed to be more regular and its kth derivative has to decay at infinity as a short range component. In particular, we recover the usual decomposition of the perturbation in two components: the short and long range parts. We stress the fact that our results are best possible on the Holder]Zygmund scale La 4 . ¨ The main results of w9x are obtained under the condition that H has a spectral gap Ži.e., the spectrum of H is not equal to R.. This is rather annoying in some applications, e.g., it excludes the Stark effect hamiltonians Žsee w14x or w17x. or the simply characteristic operators Žsee w2x.. We have eliminated this condition in w20x for ‘‘small’’ orders of A-regularity classes. We prove the power of these results by studying hamiltonians of the form H s hŽ P . q V where h is a simply characteristic function and V is a pseudo-differential operator with admissible Weyl symbol Žin a sense to be explained.. We allow short and long range bounded perturbations with ‘‘optimal’’ decay at infinity conditions. This is the subject of Subsection 2.2. Similarly, in Section 4 we treat the Stark effect hamiltonian with ‘‘singular’’ perturbations. Indeed, our limiting absorption principle is better than those obtained, for example, in w17x or w4x. In fact, under their regularity condition on the potential V we obtain not only the existence of the boundary values of the resolvent of H but we also give the optimal Žon the Holder]Zygmund scale. order of regularity of these boundary values. ¨
2. PSEUDO-DIFFERENTIAL OPERATORS Let X s R n with n G 2 and H s L2 Ž X . the Hilbert space of all square integrable functions on X. Let Q s Ž Q1 , . . . , Q n . and P s Ž P1 , . . . , Pn . be the position and momentum operators on H given by Ž Q j f .Ž x . s x j f Ž x . and Ž Pf .Ž x . s yiŽ j f .Ž x .. For each multi-index a s Ž a 1 , . . . , a n . we set a s xa1 1 ??? xan n , < a < s a 1 q ??? qa n , Q a [ Q1a 1 ??? Q na n , and ad Qa [ ad Qa 11 ??? ad Qa nn, where ad 0AŽ B . s B, ad 1AŽ B . s w A, B x, and ad kAŽ B . s 1 Ž .. ad 1AŽad ky B ; and similarly we define ad aP . The Schwartz’s test functions A space will be denoted S Ž X .. This section is devoted to the study of the continuity properties of the boundary values of the resolvent of the self-adjoint operators of the form H s hŽ P . q V, when h: X ª R is a simply characteristic function and V belongs to some perturbation classes which are described in our statements.
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2.1. Hypoelliptic Operators Let h: X ª R be a function of class C m , with m G 2, such that lim h Ž x . s q`,
< x <ªq`
Ý
a hŽ x . F C ;
Ž 2.1.
< a
Ý
a h Ž x . F C Ž1 q h Ž x . . .
Ž 2.2.
< a
These conditions are satisfied by all hypoelliptic polynomials, and for this reason we say that h is a function of hypoelliptic type. The self-adjoint operator H0 s hŽ P . in H is bounded from below Žin particular it has a spectral gap.. Let G s DŽ< H0 < 1r2 . be the form domain of H0 and G * be its adjoint space. We have S Ž X . ; G ; H ' H * ; G * ; S *Ž X . continuously and densely. From Ž2.1. and Ž2.2. it follows that the group e iQ x leaves invariant the spaces G and G * Žsee Lemma 7.6.7 of w3x.. Hence one may consider the Sobolev scale Gs 4 associated to this group in G : if s G 0 then Gs is the Hilbert space defined by the norm 5 u 5 Gs s 5² Q :s u 5 G ; if s - 0 then Gs is the completion of G for the norm 5 ? 5 Gs defined by the same formula. We have G0 s G and Gs ; Gt for t - s. By interpolation we obtain the weighted Besov space Gr, p s Ž Gs , Gt .u , p for 0 - u - 1, 1 F p F `, and r s Ž1 y u . s q u t. Note that Gs, 2 s Gs . Similarly we define Ž G *.s ' GsU , U Ž G U .s, p ' Gs,U p related to the preceding spaces by the formula Ž Gs .* s Gys U and Ž Gs, p .* s Gys, p9 for each s g R, 1 F p - `, and p9 s prŽ p y 1.. Let V: G ª G * be a symmetric operator such that H0 q V q i: G ª G * is an isomorphism and denote by H the associated self-adjoint operator in H . Suppose that Ž H q i .y1 y Ž H0 q i .y1 is a compact operator in H . Assumption 2.1. Let u g C0`Ž X . with u Ž x . ) 0 in an annulus 0 - a < x < - b - ` and u Ž x . s 0 otherwise. Let s ) 1 and suppose that V [ Ý0 F k - s Vk where Vk g X ' B Ž G , G *. is a symmetric operator such that
Ý
sup r sykq< g < u Ž Qrr . ad gP ad Qb Ž Vk .
< g
X
- `.
Ž 2.3.
In the particular case where V s Ý k - s Vk Ž Q . is a multiplication operator in H our assumption is quite simple and natural, as the next examples show: Ž1. If 1 - s - 2 then V s V0 Ž Q . q V1Ž Q . such that V0 Ž x . s O Ž< x
PROPAGATION THEOREMS
485
Ž2. If 2 - s - 3 then V s V0 Ž Q . q V1Ž Q . q V2 Ž Q . such that V0 , V1 are as above and V2 Ž x . s O Ž< x
a a Ž x . F Ca² x :y< a < .
Ž 2.4.
Let c " be two functions of class C0`ŽR n . such that < h9Ž j .< G a ) 0 on the supports of c " and let w " be two symbols of order zero such that "x ? h9 Ž j . G b < x < ? h9 Ž j .
on supp w "= supp c " Ž with b ) 0 . .
Ž 2.5. The next theorem shows that the pseudo-differential operators a " Ž Q, P . s w " Ž Q . c " Ž P . localize in non-propagation regions of the phase space with respect to the flow e i H t generated by H. THEOREM 2.2. In addition to the assumptions of Theorem 2.1 let the function h be of class C`Ž X .. Then a " Ž Q, P . RŽ l . i0. GsU ; Gsy1y a if 0 - a - s y 1r2 and l g m Ž H .. Moreo¨ er, the maps l ¬ a " Ž Q, P . RŽ l . i0. g B Ž GsU , Gsy1y a . are locally of class La on m Ž H .. Here again the orders of Holder continuity are optimal in the same ¨ sense as above. The case a s 0 is treated in the next theorem.
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JAOUAD SAHBANI
Assumption 2.2. With the notations of Assumption 2.1, let `
Ý
< g
H1 r
sykq< g <
dr
u Ž Qrr . ad gP ad Qb Ž Vk .
X
r
- `.
Ž 2.6.
THEOREM 2.3. Suppose that the function h is of class C` and that Assumption 2.2 is ¨ alid for s s s q 1r2. Then a " Ž Q, P . RŽ l . i0. GsU ; Gsy1 if l g m Ž H .. Moreo¨ er, the maps l ¬ a " Ž Q, P . RŽ l . i0. g B Ž GsU , Gsy1 . are weakly continuous on m Ž H .. The weak continuity cannot be replaced by norm-continuity in general Žsee again w8, 9x or w21x.. We emphasize the fact that from the theorems stated above one can deduce, via a Fourier transformation, important information concerning the dispersion and propagation properties of the flow e i H t generated by H as in w8, 9x Žsee also w21x.. For example, we have the following theorem: THEOREM 2.4. Under the assumptions of Theorem 2.2, for each w g C0`Ž m Ž H .. there exists a finite constant C such that t ² Q :ys eyi H w Ž H . ² Q:ys
yŽ sy1r2.
G *ª G F C ² t :
t ² Q :sy 1y a a . Ž Q, P . eyi H w Ž H . ² Q:ys
ya
G *ª G F C ² t :
;
Ž 2.7. if "t G 0.
Ž 2.8. In fact, by using a complex interpolation argument one can obtain more precise estimates, but we shall not state them here. Proof of Theorem 2.1. Ži. Let J be a bounded real open set such that J l Ž k Ž h. j sp Ž H .. s B. Then from Ž2.1., V [ hy1 Ž J . is a bounded open subset of X on which < h9Ž x .< ) c ) 0. So, one can construct a vector field F of class C0`Ž X . such that ŽŽ F ? = . h.Ž x . G c ) 0 in V. Let us set f s div F. It is clear that H0 is of class C m Ž A., where A is the self-adjoint operator As
1 2
n
n
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Q j Fj Ž P . y 2 f Ž P . . Ž 2.9.
js1
js1
In particular, A is strictly conjugate to H0 on J. Žii. We introduce now some notations which will be needed below. It is clear that e i A t leaves invariant the spaces G and G *. Then e i A t induces two strongly continuous groups of bounded operators on these spaces, and so the automorphism group W t on X defined by W t w T x s W t T s eyi A t Te i A t for all T g X . For s s 0 and p s 1, we say that T is of class
487
PROPAGATION THEOREMS
C 0, 1 Ž A; G , G *. ' Cq0 Ž A; G , G *. if the function t ¬ W t T g X is Dinicontinuous on R. For s ) 0 we say that T is of class C s , p Ž A; G , G *. Žresp. C k Ž A; G , G *., k g N. if the preceding function is of class Ls , p on R Žresp. strongly of class C k .. Similarly we define the regularity classes C s , p Ž A; G *, G ., C k Ž A; G *, G ., C kq 0 Ž A; G *, G .. Remark that if H g X is of class C s , p Ž A; G , G *. then Ž H q i .y1 is of class C s , p Ž A; G *, G . Žbecause H q i: G ª G * is an isomorphism., and this is better than H being of class C s , p Ž A.. So, in order to prove that H is of class C sq1r2 Ž A; G , G *., it suffices to prove that V is of class C sq 1r2 Ž A; G , G *. Ž H0 being of class C m Ž A; G , G *.., and this assertion will be shown with the help of the next criterion. THEOREM 2.5. Let L be a self-adjoint operator in H bounded from below by a strictly positi¨ e constant and such that: Ž1. e iLt G ; G and 5 e iLt 5 BŽ G . F C²t :N with N - `, and Ž2. there is an integer l G 1 such that AlLyl extends to a continuous operator in G *. Let 0 F s - l. Then a symmetric operator T g X is of class C s , p Ž A; G , G *. if there is a function u g C0`ŽR. with u Ž x . ) 0 if 0 - a - < x < - b - ` such that Ž with the usual con¨ ention if p s `. `
H1
r su Ž Lrr . T
p
dr
X
1rp
- `.
r
Ž 2.10.
In particular, if p s 1 or ` and if the operator T is of class C k Ž A; G , G *. for an integer 0 F k F s and `
H1
r
u Ž Lrr . A w T x
s yk
k
p
X
dr r
1rp
- `,
Ž 2.11.
then T is of class C s , p Ž A; G , G *.. The proof of this result is very similar to the proof of Theorem 7.5.8 of w3x up to minor modifications Žthe details can be found in w21x.. Žiii. By applying the preceding theorem to the operators L s ² Q : and A defined by Ž2.8., it is not difficult to establish the following proposition. PROPOSITION 2.1. If the operator V satisfies Assumption 2.1 or 2.2 then V is of class C s Ž A; G , G *. or C s , 1 Ž A; G , G *., respecti¨ ely. Živ. We then conclude that under the conditions of Theorem 2.1 the operator H is of class C sq 1r2 Ž A. and in particular, A is strictly conjugate to H on J. But H has a spectral gap Žbecause H0 is bounded from below and sessŽ H . s s Ž H0 ... Then the main theorem of w6x implies that the A . maps l ¬ RŽ l " i0. g B Ž Hs A , Hys are locally of class Lsy 1r2 .
488
JAOUAD SAHBANI
Žv. Let Gs Ž A. and GsU Ž A. be the spaces of vectors f in G , resp. G *, such that the functions t ¬ e i At f with values in G , resp. G *, are of class Ls, 2 . Let l0 g R _ s Ž H .. Then by iterating the first resolvent equation we get 2
2
R Ž z . s R Ž l0 . q Ž l0 y z . R Ž l0 . q Ž l0 y z . R Ž l0 . R Ž z . R Ž l0 . . On the other hand, we have RŽ l 0 . GsU Ž A. ; Gs Ž A. Žbecause RŽ l 0 . is of class C s Ž A; G *, G .., so RŽ l0 . GsU Ž A. ; Hs A. Finally, from Theorem 3.7.8 in w3x, Gs Ž A. coincides Žas a topological vector space. with the space u g G N ² A:s u g G 4 endowed with the norm 5 u 5 Gs Ž A. s 5² A:s u 5 G ; similarly for GsU . So by taking into account the fact that ² A:s² Q :ys is a bounded operator in G and G * we deduce that GsU ; GsU Ž A. and so the maps l ¬ RŽ l " i0. g B Ž GsU , Gys . are locally of class Lsy 1r2 on J, hence on m Ž H .. This finishes the proof of Theorem 2.1. Žvi. By combining the arguments used above and an adapted version of Mourre’s proof of Theorem 3.1 from w18x to our case, and then by applying Theorem 2 Žresp. Theorem 1. of w8x we obtain easily Theorem 2.2 Žresp. Theorem 2.3.. Detailed proofs of these theorems can be found in w21x. Elliptic Case. In order to illustrate the preceding considerations in two physically important situations, namely the non-relativistic and relativistic Schrodinger operators, we shall first discuss the following particular case. ¨ Let h be an elliptic symbol of degree 2 r ) 0, i.e., h g C`ŽR n ., < hŽ a . Ž x .< F Ca² x :2 ry< a < for each multi-index a and < hŽ x .< G C² x :2 r, for some C ) 0, outside a compact set. In this case G s H r ŽR n ., G * s H yr ŽR n . the usual Sobolev spaces, and Gt, p s Ht,r p ŽR n ., G * s Ht,yrp ŽR n . are the standard weighted Besov spaces. Let V: H r ª H yr be a symmetric operator. We say that V is small at infinity if there is t ) r and a function j g C`ŽR n . with j Ž x . s 0 if < x < F 1 and j Ž x . s 1 if < x < G 2 such that lim R ª` 5 j Ž QrR.V 5 H t ª H y r s 0. In this case it is easy to see that Ž H y i .y1 y Ž H0 y i .y1 is a compact operator in H , where we denote by H the self-adjoint operator in H associated to the sum H0 q V Žsee w3, p. 348x.. So, this implicit condition is expressed now in terms of the decay of V at infinity. It is clear that all the theorems stated above are valid in this situation. But the advantage now is that all spaces are standard. Ži. Non-Relati¨ istic Schrodinger Operator. A trivial example of the ¨ elliptic symbol is the quadratic function hŽ j . s < j < 2 which is of degree 2 Ži.e., r s 1.. In this case we obtain sharp results concerning local decay and propagation properties for the non-relativistic Schrodinger operator H s ¨ yD q V with a non-local perturbation V satisfying Assumption 2.1 or 2.2.
PROPAGATION THEOREMS
489
Žii. Relati¨ istic Schrodinger Operator. Another physically important ¨ situation is obtained when hŽ x . s < j < 2 q 1 which is an elliptic symbol of degree 1 Ži.e., r s 1r2.. In this case the hamiltonian is the relativistic Schrodinger operator H s 'yD q 1 q V. Again we get sharp results ¨ concerning spectral and propagation properties of H. For instance, we get
'
THEOREM 2.6. Let s ) 1r2 and V: H 1r2 ª H y1 r2 be a symmetric and small at infinity operator which satisfies Assumption 2.1 for s s s q 1r2. Denote by H the self-adjoint operator in H associated to the sum 'yD q 1 q V. Then Ži. sp Ž H . is a countable set and has no accumulation points in R _ 14 . Žii. The boundary ¨ alues Ž H y l . i0.y1 s w* ylim m ª " 0 Ž H y r2 Ž n . 1r2 Ž n .. locally uniformly in l g l . i m .y1 exist in B Ž H y1 1r2, 1 R , Hy1r2, ` R R _ w 14 j sp Ž H .x. In particular, H has no singularly continuous spectrum. 1r2 Ž n .. Žiii. The maps l ¬ Ž H y l . i0.y1 g B Ž Hsy1 r2 ŽR n ., Hys R are sy1r2 locally of class L on R _ w 14 j sp Ž H .x. Theorems 2.2]2.4 also give interesting results in this particular context, but we shall not state them explicitly. The limiting absorption principle follows from the main theorem in w7x and is better than the similar result obtained in Umeda w22x or more recently ŽMay 1996. by Ben-Artzi and Nemirovsky w13x. Moreover, the order of regularity of the boundary values Ž H y l . i0.y1 as a function of l is optimal on the Holder]Zygmund scale even if V s 0 Žin the paper ¨ cited above the order of Holder continuity is not specified.. On the other ¨ hand our perturbations are non-local, and even in the local case our class is larger. Note, however, that in w13x the limiting absorption principle is also established near the threshold point 1 Žsee also w10, 11x.. 2.2. Simply Characteristic Operators Let us now consider h: X ª R of class C m , m G 4, such that lim
< x <ªq`
Ý
Ž hŽ x .
q h9 Ž x .
. s q`,
hŽ a . Ž x . F C Ž h Ž x . q h9 Ž x . . .
Ž 2.12. Ž 2.13.
< a
These conditions are satisfied by simply characteristic polynomials Žsee w16x for the definition.. The spectrum of H0 s hŽ P . is equal, in general, to R, for example, if hŽ x 1 , x 2 . s x 1 x 2 . In such a situation the results of w8, 9x are not useful, but rather the results of w20x.
490
JAOUAD SAHBANI
Let V be a bounded symmetric operator in H such that the self-adjoint operator H s H0 q V satisfies that Ž H q i .y1 y Ž H0 q i .y1 is compact in H . We assume that V s VS q VL , with VS , VL bounded symmetric operators in H . The short range component VS of V will be submitted to the following type of condition sup r sx Ž Qrr . VS - `,
Ž 2.14.
r)1
where x is the characteristic function of the annulus 1 - < x < - 2. In order to introduce the long range assumptions we use the following DEFINITION 2.1. Let s be a real number such that 1 - s - 2. A function b: R 2 n ª C is a s-admissible symbol if its Fourier transform ˆ b is a measure satisfying
HH
ˆb Ž x, y . ? ² y :sy1
H< z <-1 ˆb Ž x q z, y . y ˆb Ž x, y .
q
? < z
If s s 1 then it is convenient to define the 1-admissibility by Žsee w6x.
HH
ˆb Ž x, y . log Ž 2 q < y < .
H< z <-1 ˆb Ž x q z, y . y ˆb Ž x, y .
q
? < z
We say that a pseudo-differential operator VL is s-admissible if its Weyl symbol is a distribution a: R 2 n ª R such that a j Ž x, y . s x j aŽ x, y . and a j, k Ž x, y . s Ž y k y ix k . a j Ž x, y . are s-admissible symbols for each j, k s 1, . . . , n. We consider the condition sup r sy1 x Ž Qrr . VL q x Ž Qrr . Q j , VL
4 - `.
Ž 2.17.
r)1
We know Žsee w6x. that if Ž2.14. and Ž2.17. are true for s s 1 with H1` dr in place of sup r ) 1 and if VL is a 1-admissible pseudo-differential operator then the boundary values RŽ l . i0. of H exist in optimal spaces. Let Hs 4 be the weighted Sobolev space defined by the norm 5 f 5 s 5² Q :s f 5. THEOREM 2.7. Let s be a real number such that 1r2 - s - 1. Assume that Ž2.14. and Ž2.17. are true for s s s q 1r2 and that VL is Ž s q 1r2.-
491
PROPAGATION THEOREMS
admissible. Then the maps l ¬ RŽ l . i0. g B Ž Hs , Hys . are locally of class Lsy 1r2 on m Ž H . [ R _ Ž sp Ž H . j k Ž h... Sketch of the Proof. Ži. It is clear that the function F: R n ª R n defined by FŽ j . s
h9 Ž j . 1 q hŽ j .
2
q h9 Ž j .
Ž 2.18.
2
is of class BC 1 ŽR n .. Let us set f s div F. Then As
1 2
n
n
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Fj Ž P . Q j q 2 f Ž P . Ž 2.19.
js1
js1
is a self-adjoint operator in H . It is easy to see that H0 is of class C 2 Ž A. and that A is locally strictly conjugate to H0 on R _ k Ž h. Že.g., see w1x or w6x.. Moreover, ² A:s² Q :ys is a bounded operator in H Žsee w6x., for each 0 - s F 2. Žii. As in the last section, from Theorem B in w20x, it suffices to show that H is of class C sq1r2 Ž A.. This assertion follows by mimicking the proof of w6x and by taking into account Theorem 2.5 of the last subsection. Let c " be two functions of class C0`ŽR n . such that supp c " are disjoint from the set of critical points of h; and let w " be two symbols of order zero such that for some b ) 0, "x ? h9 Ž j . G b < x < ? h9 Ž j .
if Ž x, j . g supp w "= supp c " . Ž 2.20.
If we set p "s w " Ž Q . c " Ž P . we then get THEOREM 2.8. Under the assumptions of Theorem 2.6 let a be a real number such that 0 - a - s y 1r2. Then p. Ž Q, P . RŽ l " i0. Hs ; Hsy1y a if l g m Ž H ., and the maps l ¬ p. Ž Q, P . RŽ l " i0. g B Ž Hs , Hsy1y a . are locally of class La on m Ž H ..
3. DIRAC OPERATOR Let E be a four dimensional Hilbert space endowed with 4 symmetric operators a 1 , a 2 , a 3 , b such that for j, k s 1, 2, 3 one has
¡a a
~
¢
j
k
q ak a j
a j b q ba j b
2
s 2 d jk s0 s1
Ž 3.1. .
492
JAOUAD SAHBANI
We set H s L2 ŽR 3 . m E ( L2 ŽR 3, E . and we denote the tensor products of the preceding matrices with the identity of L2 ŽR 3 . by the same symbols. Let Q, P be the position and momentum operators, respectively, in L2 ŽR 3 .. Let us consider the free Dirac operator in H defined by 3
H0 [
Ý a j Pj q m b ' a ? P q m b ,
m ) 0.
Ž 3.2.
js1
H0 is a self-adjoint operator in H and s Ž H0 . s Žy`, ym x j w m, q`.. Denote by m Ž P ., p " the pseudo-differential operators associated to the symbols
¡m Ž k . ~ ¢p Ž k . "
1r2
s Ž k 2 q m2 . , 1 1 s " Ž a ? k q mb . . 2 mŽ k .
Ž 3.3.
It is clear that pq pys py pqs 0 and pqq pys 1, in particular, the symmetric operator pqy py is unitary in H . One has H0 s m Ž P .Žpqy py .. Let us set H s ŽR 3 . s H s ŽR 3 . m E, for s g R, with H s ŽR 3 . the usual Sobolev space. Remark that H 0 s H and that the form domain of H0 is H 1r2 . Let Hs " 1r2 be the weighted Sobolev space defined by the norm 5 u 5 s," 1r2 [ 5² P :" 1r2² Q :s u 5. We say that a symmetric bounded operator T : H 1r2 ª H y1 r2 is small at infinity if there is a function u g C0`ŽR 3 . with u Ž x . s 0 near zero and u Ž x . s 1 near infinity such that lim r ª` 5 u Ž Qrr .T 5 X s 0, where X s B Ž H 1r2 , H y1r2 .. For a positive number s , an integer 0 F k - s , and an operator Vk g X , let us consider the condition 3
Ý
Ý
sup r sykq< n < j Ž Qrr . ad Qm ad nP ad ab nb Ž Vk .
ns1 < m
X
- `, Ž 3.4.
where j is a function of C0`ŽR 3 . with j Ž x . ) 0 if a - < x < - b and j Ž x . s 0 otherwise. We know that Žsee w12x. if V s V0 q V1 with Vk g X satisfying condition Ž3.4. with s s 1 and H1` dr in place of sup r ) 1 then the boundary values of the resolvent of H0 q V exist in a certain topology. THEOREM 3.1. Let a number s ) 1r2 and let V: H 1r2 ª H y1 r2 be a symmetric bounded operator such that H0 q V q i: H 1r2 ª H y1 r2 is an
493
PROPAGATION THEOREMS
isomorphism, and denote by H the associated self-adjoint operator in H . Assume that V is small at infinity and V [ Ý0 F k - s Vk with Vk g X satisfying condition Ž3.4. with s s s q 1r2. Then the maps l ¬ Ž H y l . i0.y1 1r2 . g B Ž Hsy1 r2 , Hys are locally of class Lsy1r2 on the open set m Ž H . [ R_w sp Ž H . j "m4x. This result is the best possible on the Lipschitz]Zygmund scale La 4 , even if V s 0. Sketch of the Proof. Let J be a bounded open interval with closure included in Ž m, `., and let u be a function of class C0`ŽŽ m, `.. equal to 1 on J. Then the function F defined on R 3 by F Ž j . [ m2 Ž j . Ž u ( m .Ž j .Ž jr< j < 2 . is of class C0`ŽR 3 _ 04. Žbecause m Ž0. s m f supp u .. We set f s div F and we consider the self-adjoint operator in L2 ŽR 3 . defined by
A˜ [
1 2
3
3
i
Ý Ž Fj Ž P . Q j q Q j Fj Ž P . . s Ý Fj Ž P . Q j q 2 f Ž P . . Ž 3.5.
js1
js1
It is easily shown that A s pq A˜pqq py A˜py is a self-adjoint operator in H , H0 is of class C`Ž A., and A is strictly conjugate to H0 on J. Moreover we have the following lemma: LEMMA 3.1. For each real positi¨ e number s, the operator Js s ² A:s² Q :ys is bounded in H . Indeed, it suffices Žinterpolation. to show it for s s 2 m an even integer. But this assertion is a simple consequence of the boundedness of A n ² Q :yn for each integer n G 1, which follows easily by writing A s Ý3js1 Fj Ž P . Q j q Ž ir2. f Ž P . q GŽ P . where the function G: R 3 ª E is of class C0` . Since V is small at infinity, the operator Ž H q i .y1 y Ž H0 q i .y1 is compact in H Žsee, for example, w12x.. As in Subsection 2.1, it suffices to show that H is of class C sq1r2 Ž A., and this is a simple consequence of the next proposition. PROPOSITION 3.1. Let Vk : H 1r2 ª H y1 r2 be a symmetric bounded operator. If Vk satisfies Ž3.4., then Vk is of class C s Ž A; H 1r2 , H y1r2 .. The proof is very similar to the proof of Proposition 2.1. It should be clear by now that one can easily obtain results similar to Theorems 2.2 and 2.3 of the last section, so we shall not state them explicitly here.
494
JAOUAD SAHBANI
¨ 4. SCHRODINGER OPERATORS WITH CONSTANT ELECTRIC FIELD We consider the free Stark effect hamiltonian H0 defined in H s L2 ŽR. by H0 [ P 2 q Q,
Ž 4.1.
where P s yiŽ drdx . and Q is the operator of multiplication by x. 3 Let us set U [ eyi P r3 . It is clear that w Q, U x s P 2 . It follows that y1 H0 s U QU. Then H0 is essentially self-adjoint on S ŽR. in H and s Ž H0 . s R. Moreover
Ž H0 y z .
y1
s Uy1 Ž Q y z .
y1
U,
Ž 4.2.
and U commutes with P Žhence with all functions of P .. On the other hand, yP is conjugate to Q and Q is of class C`Ž P .. Denote P " the spectral projectors of P associated to the semi-axis .x G 0, and H s, p the usual Besov space, and C "s z g C N "I z ) 04 ; we set C " s C "j R for the closure of C ". PROPOSITION 4.1. Ž1. The holomorphic maps C "2 z ¬ Ž H0 y z .y1 g Ž B H 1r2,1 , H y1r2, ` . extend to weakly* continuous functions on C " . Let us set Ž H0 y l " i0.y1 [ lim m ªq0 Ž H0 y l . i m .y1 . Ž2. For each s ) 1r2, the maps l ¬ Ž H0 y l " i0.y1 g B Ž H s, H ys . are locally of class Lsy 1r2 on R. Ž3. For each s ) 1r2, we ha¨ e P " Ž H0 y l " i0.y1 H s ; H sy1 and the functions l ¬ P " Ž H0 y l " i0.y1 g B Ž H s, H sy1 . are weakly continuous on R. Ž4. For each a ) 0 and s ) a q 1r2, the maps l ¬ P " Ž H0 y l " i0.y1 g B Ž H s, H sy1y a . are locally of class La on R. Note that we have also P " Ž H0 y l " i0.y1 P .s 0. Obviously, these results are simple consequences of the known properties on the operators Q and P Žsee w21x for a detailed treatment .. We shall describe a class of potentials V such that the last proposition remains valid for the hamiltonians of the form H s H0 q V. For a non-negative integer k we denote by BC k ŽR. the Banach space of the functions of class C k such that the derivatives f Ž j. are bounded for each j s 0, . . . , k; and BC kq 0 ŽR. the space of the functions f of class BC k such that f Ž k . is Dini continuous. THEOREM 4.1. Let V Ž Q . be the operator of multiplication by a function V and assume that V is of class BC 1q 0 ŽR.. Denote H the self-adjoint operator in
PROPAGATION THEOREMS
495
H associated to the sum H0 q V. Then sp Ž H . has no accumulation points in R. Moreo¨ er, the boundary ¨ alues Ž H y l " i0.y1 [ lim m ªq0 Ž H y l " i m .y1 of the resol¨ ent of H exist locally uniformly in l g R _ sp Ž H . in the weak*-topology of B Ž H 1r2, 1 , H y1r2, ` .. Proof. Let us set A s yP. Then we have
w H , iA x s w H0 q V , iA x s w H0 , iA x q w V , iA x s 1 q V 9 Ž Q . , Ž 4.3. which is obviously bounded in H . So, we can easily Žcf. w14x. see that A is locally conjugate to H on R. From Theorem A of w20x, the limiting absorption principle is valid if H is of class C 1q 0 Ž A.. The last property is true if the function V is of class BC 1q 0 . This finishes the proof. Remark that the eigenvalues of H Žif they exist. are simple. In w4x the potential V is supposed of class BC 2 , which is more restrictive than our condition, but their condition implies absence of eigenvalues. If V is more regular with respect to A, the boundary values Ž H y l " . i0 y1 are Holder continuous in the appropriate topologies. We give the ¨ optimal order Žon the Besov scale. of continuity, with an optimal Žon the Holder]Zygmund scale. regularity condition on V. ¨ THEOREM 4.2. With the notations of the last theorem, let s be a real number such that 1r2 - s - 1. Suppose that the function V is of class Lsq 1r2 ŽR.. Then: Ž1. The maps l ¬ Ž H y l " i0.y1 g B Ž H s, H ys . are locally of class L on R _ sp Ž H .. Ž2. We ha¨ e P " Ž H y l " i0.y1 H s ; H sy1, ` if l f sp Ž H . and the maps l ¬ P " Ž H y l " i0.y1 g B Ž H s, H sy1, ` . are weak* continuous on R _ sp Ž H .. Ž3. For each a q 1r2 - s the maps l ¬ P " Ž H y l " i0.y1 g B Ž H s, H sy1y a . are locally of class La on R _ sp Ž H .. sy 1r2
In particular, if V is of class BC 2 ŽR. then all the assertions of that last theorem are true for each s and each a such that 0 - a - s y 1r2 - 1r2, and sp Ž H . s B. We ask that V be of class BC 2 in order to apply w15, Corollary 22, p. 1414x Žsee w4x., which implies that sp Ž H . s B. Note that in w4x this condition is imposed by the regularity requirement of the version of the conjugate operator method which they use. It is reasonable to conjecture that if V is a Holder continuous function ¨ of order a - 1 then the singularly continuous spectrum of H can cover all the real axis; and so the Lipschitz class could be the optimal regularity class for H to have purely absolutely continuous spectrum Žsee also w19x..
496
JAOUAD SAHBANI
ACKNOWLEDGMENTS I am extremely grateful to Professors A. Boutet de Monvel and V. Georgescu for helpful discussions and stimulating suggestions.
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