LECTURE 22-23: WEYL'S LAW 1. Functional calculus of pseudodifferential operators In sections 1 and 2 we will always assume • m ≥ 1 is an order function, • p 2 S(m) is a real-valued symbol, • p + i is elliptic in S(m). Under these assumptions we know that W 2 n 2 n P = pb : H~(m) ⊂ L (R ) ! L (R ) is a (densely-defined) self-adjoint operator, and moreover, P ± i · Id is invertible for ~ 2 (0; ~0) and the inverse is a pseudodifferential operator with symbol in S(1=m). {Helffer-Sj¨ostrandformula. In Lecture 8 we mentioned that for a self-adjoint operator P on a Hilbert space H and a Borel measurable function f on R, one can define a new self-adjoint linear operator f(P ) on H using the spectral theorem as follows: By spectral theorem (multiplication form) there is a measurable space (X; µ), a measurable real-valued function h on X and a unitary isomorphism V : H ! L2(X; µ) so that ∗ V ◦ P ◦ V = Mh on L2(X; µ). Then the operator f(P ) is defined to be ∗ f(P ) = V ◦ Mf(h(x)) ◦ V: We notice that by definition, (1) If jfj ≤ C; then kf(P )kL ≤ C: We want to answer the following natural question: Question: Is f(P ) a semiclassical pseudodifferential operator if P is a semiclassical pseudodifferential operator and f is a (nice) function? Unfortunately the construction of f(P ) above is a bit too abstract to work with. However, if f 2 S is a Schwartz function, then by using the so-called almost analytic extension f~ of f, Helffer-Sj¨ostrandgave a more concrete formula for f(P ), namely Z 1 ¯ ~ −1 (2) f(P ) = − @zf(z)(z − P ) L(dz) π C using which we will prove that f(P ) is a semiclassical pseudodifferential operator, and calculate its symbol expansion. Recall that an almost analytic extension f~ 2 1 2 LECTURE 22-23: WEYL'S LAW C1(C) of a Schwartz function f 2 S (R) is by definition a smooth function on C such that ~ ~ fjR = f; suppf ⊂ fz : jIm(z)j ≤ 1g and such that as jIm(z)j ! 0, ¯ ~ 1 @zf(z) = O(jIm(z)j ); ¯ where as usual, @z = (@x + i@y)=2 for z = x + iy, and L(dz) denotes the Lebesgue measure on C (we don't use dxdy since x has different meaning below.) To prove formula (2) it is enough to notice the following relation between f and f~ (c.f. PSet 1): Z 1 ¯ ~ −1 f(t) = − @zf(z)(z − t) L(dz) π C and use the spectral theorem (multiplication form) for both (z − P )−1 and f(P ). In literature there are at least two different ways to construct an almost analytic extension: the first construction is due to H¨ormander(who proposed the conception of almost analytic extension in 1968) who adapted the construction in Borel's Lemma by putting X f (k)(x) f~(x + iy) = (iy)kχ(λ y); k! k k where λk is a sequence of real numbers that is chosen so that they tends to +1 sufficiently fast, and χ is a cut-off function. The other way is due to Mather who make use of the Fourier transform (PSet 1): 1 Z f~(x + iy) := χ(y) χ(yξ)f^(ξ)eiξ(x+iy)dξ: 2π R Of course the almost analytic extension is not unique in general. Also note that if f is compactly supported on R, then we can take f~ to be compactly-supported in C [For the first construction this is obvious, for the second construction we may multiply the formula by a cut-off function which is identically one on supp(f)]. {Symbol of the resolvent (z − P )−1. We want to prove that f(P ) is a pseudodifferential operator if f is Schwartz function on R and P is a semiclassical pseudodifferential operator, and calculate its symbol. In view of the Helffer-Sj¨ostrant formula (2), we start with the resolvent operator (z − P )−1. Lemma 1.1. Under the previous assumptions, for any z 2 CnR, the operator z −P W −1 is invertible, and there exists rz 2 S(1=m) such that rbz = (z − P ) . Proof. This is just a consequence of ellipticity: for any fixed z = a + bi (b 6= 0), we have jz − tj inf > 0: t2R jt + ij LECTURE 22-23: WEYL'S LAW 3 We need a more explicit lower bound below. So let's try to find a constant C > 0 such that jz − tj (3) ≥ bC; 8t 2 : jt + ij R This is equivalent to 2 2 2 2 (a − t) + b − (bC) (t + 1) ≥ 0; 8t 2 R: We will take C small enough so that bC < 1. By calculating the discriminant and simplifying it, we get the condition on C: 1 − C2 (1 − C2b2) ≥ a2: C2 As a consequence, if we assume jzj < C0, then we can find a constant C so that (3) holds for all t 2 R. It follows that jz − pj ≥ Cjp + ij ≥ cm and thus z − p is elliptic in S(m). To apply Helffer-Sj¨ostrandformula, we need to study the dependence of rz on z. For this purpose, we introduce the following variation of Beals's theorem. Recall that in Lecture 14 we have seen that any continuous linear 0 W operator A : S ! S can be written as A = ba for some a = a(x; ξ; ~) 2 S 0, and Beals's theorem tells us a 2 S(1) if and only if N kad W ◦ · · · ◦ ad W AkL(L2) = O(~ ) lb1 lcN 2n for any N and any linear functions l1; ··· ; lN on R . Of course the main part in the proof of Beals's theorem is to prove the condition α above implies a 2 S(1), namely, to prove j@x,ξaj ≤ Cα. Proposition 1.2 (Beals's Estimate with Parameter). Suppose a = a(x; ξ; z; ~), i.e. a depends on a parameter z. Let δ = δ(z) be a function valued in (0; 1] such that −N N kad W ◦ · · · ◦ ad W AkL(L2) = O(δ ~ ) lb1 lcN 2n holds for any N and any linear functions l1; ··· ; lN on R . Then there exists a universal constant M such that for any α, p α M −|αj j@x,ξa(x; ξ; z; ~)j ≤ Cα max(1; ~/δ) δ : We will leave the proof as an exercise. [c.f. Helffer-Sj¨ostrand, Spectral Asymptotics in the Semi-Classical Limit, Prop. 8.4.] Using this result, we will prove the following resolvent symbol estimate: 4 LECTURE 22-23: WEYL'S LAW Theorem 1.3. Fix C0 > 0. For any z 2 C n R with jzj < C0, we have α 1=2 −1 M −1−|αj j@x,ξrzj ≤ Cα max(1; ~ jIm(z)j ) jIm(z)j : −1 W where rz 2 S(1=m) is the symbol of the resolvent of P , namely (z − P ) = rbz . Proof. By Proposition 1.2, we only need to prove −1 −1−N N (4) kad W ◦ · · · ◦ ad W (z − P ) kL(L2) = O(jIm(z)j ~ ): lb1 lcN Using the formulae −1 −1 −1 adA(B ) = −B (adAB)B and adA(BC) = (adAB)C + BadAC −1 we can write ad W ◦ · · · ◦ ad W (z − P ) as a summation of terms of the form lb1 lcN ±(z − P )−1adα1 (P )(z − P )−1adα2 (P ) ··· (z − P )−1adαk (P )(z − P )−1; blW blW blW where αj = fαj;1; ··· ; αj;nj g such that fαj;l j 8j; lg = f1; ··· ;Ng. Note that αj W nj p 2 S(m) =) fl; pg 2 S(m) =) ad (P ) = pj for some pj 2 S(m): blW b ~ Thus in view of the fact P + i has symbol in S(1=m), we get that αj −1 αj −1 −1 kad (P )(z − P ) kL(L2) ≤ kad (P )(P + i) kL(L2) · k(P + i)(z − P ) kL(L2) blW blW n −1 ≤ O(~ j jIm(z)j ); where in the last step we used Calderon-Vaillancourt theorem for the first term, and −1 −1 k(P + i)(z − P ) kL(L2(Rn)) ≤ CjIm(z)j for the second term, which is a consequence of • The fact (1) at the beginning of this lecture, which is a consequence of the spectral theorem, • The argument in the proof of Lemma 1.1, i.e. if we assume jzj < C0, then there exists a universal constant C that is independent of z such that (3) holds. In other words, jz − tj ≤ CjIm(z)j−1; 8t 2 : jt + ij R Similarly we have −1 −1 k(z − P ) kL(L2(Rn)) ≤ jIm(z)j ; so the estimate (4) holds, which completes the proof. LECTURE 22-23: WEYL'S LAW 5 {The functional calculus. Now we are ready to prove that the operator f(P ) is also a semiclassical pseu- dodifferential operator: W −k Theorem 1.4. If f 2 S , then f(P ) = ba , where a 2 S(m ) for any k 2 N. Moreover, we have an asymptotic expansion X k a(x; ξ) ∼ ~ ak(x; ξ); k≥0 where a0(x; ξ) = f(p(x; ξ), and in general, 1 2k ak(x; ξ) = (@t) [f(t)qk(x; ξ; t)] : (2k)! t=p(x,ξ) W Proof. Using Helffer-Sj¨ostrandformula, we see f(P ) = ba , where Z 1 ¯ ~ a(x; ξ) = − (@zf(z))rz(x; ξ)L(dz); π C ~ where f is an almost analytic extension of f.