The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q, p R d R d define U(q, p by U(q, pϕ(x e ix p ϕ(x q, say for ϕ in the Schwartz space S(R d. Fix then a function η such that η S(R d, η L2 (R d 1, (1.1 and define the family of Schwartz functions (η q,p (q,p R 2d by η q,p U(q, pη. Denote the L 2 inner product by (ψ, ϕ L 2 ψϕ. R d Lemma 1.1. For all ϕ S(R d, the function T ϕ defined by T ϕ(q, p : (η q,p, ϕ L 2 belongs to S(R 2d. Furthermore, the linear map T : S(R d S(R 2d is continuous. Proof. For all α, β N d, we have β p α q T ϕ(q, p ( 1 α ( i β e ix p ( α η(x qx β ϕ(xdx. If we multiply this expression by q δ, with δ N d, and write q δ (q x + x δ δ +δ δ δ! δ!δ (q xδ x δ,! we see that q δ β p α q T ϕ is a linear combination of e ix p (q x δ ( α η(x qx β+δ ϕ(xdx. 1
Finally, by integrations by part we obtain that p µ q δ p β q α T ϕ is a linear combination of ( e ix p x µ (q x δ ( α η(x qx β+δ ϕ(x dx and the result follows easily. The formal adjoint T which maps a priori S (R 2d to S (R d is defined by (Ψ, T ϕ L 2 (R 2d (T Ψ, ϕ L 2 (R d, where the inner product in the right hand has to be taken in the distributions sense. However, if Ψ S(R 2d, this is actually a standard integral for Fubini s Theorem easily shows that ( (Ψ, T ϕ L2 (R 2d Ψ(q, p η q,p (xϕ(xdx dqdp ( Ψ(q, pη q,p (xdqdp ϕ(xdx, hence that (T Ψ(x Ψ(q, pη q,p (xdqdp. Lemma 1.2. The map T is continuous from S(R 2d to S(R d. The proof of this lemma is elementary and uses the same techniques as Lemma 1.1. It is left to the reader as an exercise. The important result is the following inversion formula. Proposition 1.3. For all ϕ S(R d, T T ϕ (2π d ϕ. Proof. Fix x R d and χ C 0 (R d such that χ 1 near 0. The result then follows by dominated convergence and Fubini s Theorem since T T ϕ(x lim ɛ 0 lim ɛ 0 lim ɛ 0 e ip y η(y qϕ(ye ip x η(x qχ(ɛpdydqdp ( y x ɛ d χ η(y qη(x qϕ(ydydq ɛ χ(zη(x + ɛz qη(x qϕ(x + ɛzdzdq χ η 2 L 2 ϕ(x (2πd ϕ(x, using the second assumption in (1.1. Corollary 1.4. For all ϕ, ψ S(R d, (ψ, ϕ L 2 (R d (2π d (T ψ, T ϕ L 2 (R 2d (2π d (η q,p, ψ L 2(η q,p, ϕ L 2dqdp. In particular ϕ 2 L 2 (R d (2π d (η q,p, ϕ L 2 2 dqdp. 2
Proof. The result is obtained either by polarization from ϕ 2 L 2 (R d (2π d T ϕ 2 L 2 (R 2d or by inserting the decomposition ϕ (η q,p, ϕ L 2η q,p dqdp into (ψ, ϕ L 2 (R d and using Fubini s Theorem. Now for all a L (R 2d, B a (ψ, ϕ : (2π d a(q, p(η q,p, ψ L 2(η q,p, ϕdqdp is a well defined sesquilinear form on S(R d S(R d and Corollary 1.4 together with the Cauchy- Schwarz inequality show that B a (ψ, ϕ a L (R 2d ϕ L 2 (R d ψ L 2 (R d. (1.2 Therefore there exists a unique bounded operator A on L 2 (R d such that for all ϕ, ψ S(R d. B a (ψ, ϕ (ψ, Aϕ L2 (R d, Definition 1.5. The operator A is the anti-wick quantization of a and is denoted by A Op aw (a. As a straightforward consequence of (1.2, we see that Note that Corollary 1.4 also shows that Next, by writting we get Op aw (a L 2 L 2 a L. (1.3 Op aw (1 I. (ψ, Op aw (aϕ L 2 B a (ψ, ϕ B a (ϕ, ψ (ϕ, Op aw (aψ L 2 The following last property is then straightforward Op aw (a Op aw (a. (1.4 a 0 a.e. Op aw (a 0, (1.5 and is the most important one. Note that all these properties are independent of η. 2 Wigner s function Definition 2.1 (Pseudo-differential operator. Given a S(R 2d, the pseudo-differential operator of symbol a, Op(a, is the operator defined by Op(aϕ(x (2π d e ix ξ a(x, ξ ϕ(ξdξ, for ϕ S(R d. 3
Pseudo-differential operators are actually defined for wider classes of symbols than S(R 2d but the latter will be sufficient here. Fix now ϕ, ψ S(R d. For any a S(R 2d, a straightforward application of Fubini s Theorem shows that where (ψ, Op(aϕ L 2 (R d R 2d aw ψ,ϕ, W ψ,ϕ (x, ξ (2π d e ix ξ ψ(x ϕ(ξ. (2.1 Definition 2.2. W ψ,ϕ is the Wigner function associated to ϕ and ψ. In the sequel we fix a single η satisfying (1.1 and set We further assume that which implies the easily verified property that W (x, ξ W η,η (x, ξ. (2.2 η is even and real valued, (2.3 W is even. (2.4 The important relationship between anti-wick quantization and the Wigner function is the following one. Proposition 2.3. For all a S(R 2d, we have Proof. By (2.1, (2.3 and (2.4, we have a W (x, ξ (2π d Op aw (a Op(a W. a(q + x, p + ξe iq p η(q η(pdqdp so that Op(a W ϕ(x (2π 2d e ix ξ iq p a(q + x, p + ξη(q η(p ϕ(ξdqdpdξ. On the other hand, using the fact that (η q,p, ϕ L 2 (2π d e iq (ξ p η(ξ p ϕ(ξdξ, where we also used (2.3, we have Op aw (aϕ(x (2π d (η q,p, ϕ L 2a(q, pη q,p (xdx (2π 2d e iq (ξ p η(ξ p ϕ(ξa(q, pe ip x η(x qdqdpdξ (2π 2d e i(q+x p η( p ϕ(ξa(q + x, p + ξe i(p+ξ x η( qdqdpdξ (2π 2d e iq p+ix ξ η(p ϕ(ξa(q + x, p + ξη(qdqdpdξ by changing q into q + x and p into p + ξ to get the third line and using (2.3 for the last one. This completes the proof. 4
3 Semiclassical scaling For h (0, 1], the semiclassical pseudo-differential quantization is given by Op h (aϕ(x (2π d e ix ξ a(x, hξ ϕ(ξdξ. In other words, if we set we have, according to Definition 2.1, a h (x, ξ : a(x, hξ Next, if η satisfies (1.1 and (2.3 then so does and the corresponding Wigner function is Op h (a Op(a h. η h (x : h d/4 η W ηh,η h (x, ξ (2π d e ix ξ η Lemma 3.1. For all a S(R 2d and all h (0, 1] ( x h 1/2, ( x h 1/2 η(h 1/2 ξ. (a h W ηh,η h (x, ξ (a W h (x, hξ, with where W is defined by (2.2. W h (x, ξ h d W ( x h, x. 1/2 h 1/2 We omit the very simple proof of this lemma which follows from an elementary change of variable. If we finally define Op aw h (a to be the anti-wick quantization of a associated to η h namely Op aw h (aϕ (2π d a(q, p ((η h q,p, ϕ L 2 (η h q,p (xdqdp, then Proposition 2.3 and Lemma 3.1 show that 4 The semiclassical Garding inequality Op aw h (a Op h ( a W h. (3.1 Theorem 4.1. There exists C, N 0 such that, for all a S(R 2d satisfying we have for all ϕ S(R d and all h (0, 1]. a 0 Re (ϕ, Op h (aϕ L 2 Ch max α+β N α x β ξ a L ϕ 2 L 2, 5
The non elementary tool for the proof is the Calderon-Vaillancourt Theorem which we recall for completeness. Theorem 4.2. There exists C, N CV 0 such that, for all a S(R 2d, we have for all ϕ S(R d and all h (0, 1]. We now prove Theorem 4.1. Proposition 4.3. There exists C 0 such that for all a S(R 2d and all h (0, 1]. Op h (aϕ L 2 C max α+β N CV α x β ξ a L ϕ L 2, a W h a L Ch max α+β 2 α x β ξ a L, Proof. Set for simplicity W αβ (x, ξ : x α ξ β W (x, ξ and consider its scaled version. Note that W αβ h (x, ξ : h d W αβ ( x h 1/2, ξ h 1/2 x α ξ β W h h α + β 2 W αβ h. (4.2 Recall next that W h 1 and observe that, since W h is even, x j W h ξ j W h 0 j 1,..., d. Thus, by using the Taylor formula d a(q + x, p + ξ a(q, p + x j xj a(q, p + ξ j ξj a(q, p + j1 α+β 2 where r αβ L (R 4d C x α β ξ a L (R, we see that 2d (a W h a(q, p h W αβ h (x, ξr αβ (x, ξ, q, pdxdξ, α+β 2 x α ξ β r αβ (x, ξ, q, p, using (4.2. The result follows since W αβ h is a bounded family in L 1 (R 2d (as h varies in (0, 1] because W αβ S(R 2d. Proof of Theorem 4.1. It suffices to write that Op h (a Op h ( a a W h + Oph (a W h ( Op h a a W h + Op aw h (a by (3.1. The second operator is non negative (and self-adjoint whereas the first one satisfies the estimate Op h ( a a W h ϕ L 2 C max α+β N CV α x β ξ (a a W h L ϕ L 2 Ch max α+β N CV+2 α x β ξ a L ϕ L 2 by the Calderon-Vaillancourt Theorem and Proposition 4.3. This completes the proof. 6