LECTURE22-23:WEYL'SLAW1.FUNCTIONAL CALCULUS OF PSEUDODIFFERENTIAL OPERATORSInsections1and2wewillalwaysassume. m ≥ 1 is an order function,.pS(m)isa real-valued symbol.p+iis elliptic in S(m).Under these assumptions we know thatP =pW : Hn(m) C L?(R") -→ L?(Rn)is a (densely-defined) self-adjoint operator, and moreover, P±i-Id is invertible forh E (O,ho)and the inverse is apseudodifferential operator with symbol in S(1/m).Helffer-Sjostrand formula.InLecture8 wementioned thatforaself-adjoint operatorP onaHilbert spaceH and a Borel measurable function f on R, one can define a new self-adjoint linearoperator 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-valuedfunction h on X and a unitary isomorphism V : H → L?(X, μ) so thatVoPoV*= Mhon L?(X,μ). Then the operator f(P) is defined to bef(P) = V*o Mr(h(r) o V.We notice that by definition,(1)If IfI ≤C, then IIf(P)Ilc≤C.We want to answer the following natural question:Question:Is f(P)a semiclassical pseudodifferential operator if P isa semiclassical pseudodifferential operator and f is a (nice) function?Unfortunately the construction of f(P) above is a bit too abstract to work withHowever, if f E is a Schwartzfunction, then by using the so-called almost analyticertensionf off,Helffer-Sjostrand gaveamore concreteformula for f(P),namely(P) =-元 [5. ()(2- P)-L(da)(2)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 E1
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 ∈ S(m) is a real-valued symbol, • p + i is elliptic in S(m). Under these assumptions we know that P = pb W : H~(m) ⊂ L 2 (R n ) → L 2 (R n ) is a (densely-defined) self-adjoint operator, and moreover, P ± i · Id is invertible for ~ ∈ (0, ~0) and the inverse is a pseudodifferential operator with symbol in S(1/m) . ¶Helffer-Sj¨ostrand formula. 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 → L 2 (X, µ) so that V ◦ P ◦ V ∗ = Mh on L 2 (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 |f| ≤ 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 ∈ S is a Schwartz function, then by using the so-called almost analytic extension ˜f of f, Helffer-Sj¨ostrand gave a more concrete formula for f(P), namely (2) f(P) = − 1 π Z C ¯∂z ˜f(z)(z − P) −1L(dz) 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 ∈ 1
2LECTURE22-23:WEYL'SLAWCo(C) of a Schwartz function f E (R) is by definition a smooth function on Csuch thatFlr= f, suppf c (z: [Im(z)|≤1)and such that as |Im(z)l→ 0,0,f(2) = O(Im(z)/~),where as usual, = (Or + iou)/2 for z = r + iy, and L(dz) denotes the Lebesguemeasure on C (wedon't use drdy since rhas different meaningbelow.)To proveformula (2) it is enough to notice the following relation between f and f (c.f.PSet1):f(t) = -元 [ 5. (2)(2 -t)-(d)爪Jand use the spectral theorem (multiplication form) for both (z -P)-1 and f(P).In literature there areat least two different ways to construct an almost analyticextension: thefirst construction is due to Hormander (who proposed the conceptionofalmostanalyticextensionin1968)whoadaptedtheconstructioninBorel'sLemmabyputting(+ i) -一((i) (A),k!where A is a sequence of real numbers that is chosen so that they tends to +oosufficiently fast, and x is a cut-off function. The other way is due to Mather whomake use of the Fourier transform (PSet 1):/ x(yE)f(E)eis(r+i) dE. (+ i) =2x() /Of course the almost analytic extension is not unique in general. Also note thatif f is compactly supported on R,then we can take f to be compactly-supportedin C[For thefirst construction this is obvious, for the second construction wemaymultiply theformula bya cut-off function which is identically one on supp(f))Symboloftheresolvent(z-P)-1.We want to prove that f(P) is a pseudodifferential operator if f is Schwartzfunction on R and P is a semiclassical pseudodifferential operator, and calculate itssymbol.In view of the Helffer-Sjostrant formula (2),we start with the resolventoperator (z-P)-1.Lemma 1.1. Under the previous assumptions, for any zE C R, the operator z-Pisinvertible, andthereeristsr.ES(1/m)suchthatrw= (z-P)-1Proof. This is just a consequence of ellipticity: for any fixed z = a +bi (b o), wehaveinfle-t> 0.tERIt+i
2 LECTURE 22-23: WEYL’S LAW C ∞(C) of a Schwartz function f ∈ S (R) is by definition a smooth function on C such that ˜f|R = f, supp ˜f ⊂ {z : |Im(z)| ≤ 1} and such that as |Im(z)| → 0, ¯∂z ˜f(z) = O(|Im(z)| ∞), 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): f(t) = − 1 π Z C ¯∂z ˜f(z)(z − t) −1L(dz) 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 ˜f(x + iy) = X k f (k) (x) k! (iy) kχ(λky), where λk is a sequence of real numbers that is chosen so that they tends to +∞ 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): ˜f(x + iy) := 1 2π χ(y) Z R χ(yξ) ˆf(ξ)e iξ(x+iy) dξ. 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 ∈ C\R, the operator z−P is invertible, and there exists rz ∈ S(1/m) such that rbz W = (z − P) −1 . Proof. This is just a consequence of ellipticity: for any fixed z = a + bi (b 6= 0), we have inf t∈R |z − t| |t + i| > 0
3LECTURE22-23:WEYL'SLAWWe need a more explicit lower bound below.So let's try tofind aconstantC>0 suchthat[2 - t]≥bC,(3)VtER.[t+iThis is equivalent to(a-t)?+b2-(bC)(t?+1)≥0,VtER.We will take C small enough so that bC < 1.By calculating thediscriminant and simplifying it, we get the condition on C:1-C2(1 - C2b2) ≥ a2.C2As a consequence, if we assume |zl < Co, then we can find a constantC so that (3) holds for all t E R.It follows that[z-pl≥Clp+i≥cm口and thus z- p is elliptic in S(m).To apply Helffer-Sjostrand formula, we need to study the dependence of rz onz.For this purpose, we introduce the following variation of Beals's theorem.Recall that in Lecture 14 we have seen that any continuous linearoperatorA:y→gcanbewrittenasA=awforsomea:a(r,s,h) e i, and Beals's theorem tells us a e S(1) if and only ifIladaw o o adw Allc(L2) =O(N)for any N and any linear functions li,..:,l on IR2n. Of course themainpart in the proof of Beals'stheorem isto provethe conditionabove implies a E S(1), namely, to prove oa.al≤ Co.Proposition 1.2 (Beals's Estimate with Parameter). Suppose a =a(r,E,z;h), i.e. a depends on a parameter z. Let s = S(z) be afunction valued in (o,1] such thatIladgw o ... 0 adwAllc(L2) = O(8-N)holds for any N and any linear functions li,..:,ln on R2n. Thenthere erists a universal constant M such that for any Q,[oe,ea(r,5,z; h)/ ≤ Camax(1, Vh/8)Ms-lal.We will leave the proof as an exercise. [c.f. Helffer-Sjostrand,Spectral Asymptotics in the Semi-Classical Limit, Prop. 8.4.]Using this result, we will prove the following resolvent symbol estimate:
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 (3) |z − t| |t + i| ≥ bC, ∀t ∈ R. This is equivalent to (a − t) 2 + b 2 − (bC) 2 (t 2 + 1) ≥ 0, ∀t ∈ 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 − C 2 C2 (1 − C 2 b 2 ) ≥ a 2 . As a consequence, if we assume |z| < C0, then we can find a constant C so that (3) holds for all t ∈ R. It follows that |z − p| ≥ C|p + i| ≥ cm and thus z − p is elliptic in S(m). To apply Helffer-Sj¨ostrand formula, 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 operator A : S → S 0 can be written as A = ba W for some a = a(x, ξ, ~) ∈ S 0 , and Beals’s theorem tells us a ∈ S(1) if and only if kadlb1 W ◦ · · · ◦ ad lcN W AkL(L2) = O(~ N ) for any N and any linear functions l1, · · · , lN on R 2n . Of course the main part in the proof of Beals’s theorem is to prove the condition above implies a ∈ S(1), namely, to prove |∂ α x,ξa| ≤ 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 kadlb1 W ◦ · · · ◦ ad lcN W AkL(L2) = O(δ −N ~ N ) holds for any N and any linear functions l1, · · · , lN on R 2n . Then there exists a universal constant M such that for any α, |∂ α x,ξa(x, ξ, z; ~)| ≤ Cα max(1, √ ~/δ) Mδ −|α| . 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:
4LECTURE22-23:WEYL'SLAWTheorem 1.3. Fir Co > 0. For any z E CR with |zl <Co, we have[0,er-/ ≤ Ca max(1, /2|m(2)]-1)[Im(2)]-1-lal,where r E S(1/m) is the symbol of the resolvent of P, namely (z - P)-1 = rWProof.ByProposition1.2,we only need toprove(4)ladew o... 0 adw(z -P)-illc(L2) = O(Im(z)-1-NN)Using the formulaeada(B-1) = -B-1(adAB)B-1andadA(BC) = (ad^B)C + Bad^Cwe can write ad.ww o ... o adw(z - P)-1 as a summation of terms of the form±(z- P)-lada(P)(z-P)-lad(P) ... (z- P)-lad(P)(z-P)-1,where aj = [aj1,*-,ain,} such that (aj, / Vj, ] = [1,.*, N]. Note thatpE S(m) = [l,p) E S(m) = ad(P)=pW for some p, EniS(m),Thus in view of the fact P +i has symbol in S(1/m), we get thatIlad(P)(z - P)-illc(L2) ≤ Iad(P)(P + i)-Ilc(L2) Il(P +i)(z - P)-illc(L2)≤O(ni[Im(z)/-1),where in the last step we used Calderon-Vaillancourt theorem for the first term, andII(P+i)(z - P)-1llc(L2(Rn) ≤C|Im(z)-1for the second term, which is a consequence of.The fact () at the beginning of this lecture, which is a consequence of thespectraltheorem,.The argument in theproof of Lemma 1.1,i.e.if weassume[zl< Co,thenthere exists a universal constant C that is independent of z such that (3)holds. In other words长二≤CIm(2)]-1, Vte R.[t+iSimilarly we haveI(z - P)-1llc(L2(R") ≤ [Im(z)]-1,口so the estimate (4) holds,which completes the proof
4 LECTURE 22-23: WEYL’S LAW Theorem 1.3. Fix C0 > 0. For any z ∈ C \ R with |z| < C0, we have |∂ α x,ξrz| ≤ Cα max(1, ~ 1/2 |Im(z)| −1 ) M|Im(z)| −1−|α| . where rz ∈ S(1/m) is the symbol of the resolvent of P, namely (z − P) −1 = rbz W . Proof. By Proposition 1.2, we only need to prove (4) kadlb1 W ◦ · · · ◦ ad lcN W (z − P) −1 kL(L2) = O(|Im(z)| −1−N ~ N ). Using the formulae adA(B −1 ) = −B −1 (adAB)B −1 and adA(BC) = (adAB)C + BadAC we can write adlb1 W ◦ · · · ◦ ad lcN W (z − P) −1 as a summation of terms of the form ±(z − P) −1 adα1 blW (P)(z − P) −1 adα2 blW (P)· · ·(z − P) −1 adαk blW (P)(z − P) −1 , where αj = {αj,1, · · · , αj,nj } such that {αj,l | ∀j, l} = {1, · · · , N}. Note that p ∈ S(m) =⇒ {l, p} ∈ S(m) =⇒ adαj blW (P) = pbj W for some pj ∈ ~ njS(m). Thus in view of the fact P + i has symbol in S(1/m), we get that kadαj blW (P)(z − P) −1 kL(L2) ≤ kadαj blW (P)(P + i) −1 kL(L2) · k(P + i)(z − P) −1 kL(L2) ≤ O(~ nj |Im(z)| −1 ), where in the last step we used Calderon-Vaillancourt theorem for the first term, and k(P + i)(z − P) −1 kL(L2(Rn)) ≤ C|Im(z)| −1 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 |z| < C0, then there exists a universal constant C that is independent of z such that (3) holds. In other words, |z − t| |t + i| ≤ C|Im(z)| −1 , ∀t ∈ R. Similarly we have k(z − P) −1 kL(L2(Rn)) ≤ |Im(z)| −1 , so the estimate (4) holds, which completes the proof.
LECTURE22-23:WEYL'SLAW5The functional calculus.Now we are ready to prove that the operator f(P)is also a semiclassical pseu-dodifferential operator:Theorem 1.4. If f e J, then f(P) = aw, where a E S(m-k) for any k E N.Moreover,wehave anasymptoticerpansiona(r,) ~hax(r,E),k≥0where ao(r, $)= f(p(r,), and in general,as(r,s) =(2)()*[()(,,)()Proof. Using Helffer-Sjostrand formula, we see f(P) = aW, where[ (0f(2)r2(r,3)L(dz),a(r,s) =-1where is an almost analytic extension of f.Although the (r,)-derivatives of rzis unbounded as Im(z) → 0, the unboundedness is controlled by Theorem 1.3. Soby using the fact ,f(z) = O(|Im(z)[) we see a E S(1). More generally, for anykeN, ifwe applythe above arguments to f(t)=f(t)(t+i),we can provethatfu(P) = (P + i)*f(P) is a pseudodifferential operator with symbol in S(1), whichimplies a e S(m-k).We alsoneed an asymptotic expansionof a.For thispurposewe startwith theasymptotic expansion of rz.Recall that by construction (Lecture14), rz=r*(1-u)for some u E hS(1), and r can be solved form the equation(z-p)*r,- 1=0(h)inductively,which hastheform (exercise)qk(r,s,2)~k(5)(2 - p(r, $)2k+1k=0where qk is a degree 2k polynomial in z (and thus is holomorphic in z):2kqk(r,E,z) =dk,3(r, )2ij=0with qo = 1, q1 = 0 and in general, qkj E S(m2k-j)Again the expansion (5) is an expansion for each fixed z, and is not a goodone as Im(z) → 0. However, we may resolve this problem by fixing a E (0, 1/2
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 pseudodifferential operator: Theorem 1.4. If f ∈ S , then f(P) = ba W , where a ∈ S(m−k ) for any k ∈ N. Moreover, we have an asymptotic expansion a(x, ξ) ∼ X k≥0 ~ k ak(x, ξ), where a0(x, ξ) = f(p(x, ξ), and in general, ak(x, ξ) = 1 (2k)! (∂t) 2k [f(t)qk(x, ξ, t)] t=p(x,ξ) . Proof. Using Helffer-Sj¨ostrand formula, we see f(P) = ba W , where a(x, ξ) = − 1 π Z C ( ¯∂z ˜f(z))rz(x, ξ)L(dz), where ˜f is an almost analytic extension of f. Although the (x, ξ)-derivatives of rz is unbounded as Im(z) → 0, the unboundedness is controlled by Theorem 1.3. So by using the fact ¯∂z ˜f(z) = O(|Im(z)| ∞) we see a ∈ S(1). More generally, for any k ∈ N, if we apply the above arguments to fk(t) = f(t)(t + i) k , we can prove that fk(P) = (P + i) k f(P) is a pseudodifferential operator with symbol in S(1), which implies a ∈ S(m−k ). We also need an asymptotic expansion of a. For this purpose we start with the asymptotic expansion of rz. Recall that by construction (Lecture 14), rz = rez ?(1−u) for some u ∈ ~ ∞S(1), and rez can be solved form the equation (z − p) ? rez − 1 = O(~ ∞) inductively, which has the form (exercise) (5) rz ∼ X∞ k=0 ~ k qk(x, ξ, z) (z − p(x, ξ))2k+1 where qk is a degree 2k polynomial in z (and thus is holomorphic in z): qk(x, ξ, z) = X 2k j=0 qk,j (x, ξ)z j with q0 = 1, q1 = 0 and in general, qk,j ∈ S(m2k−j ). Again the expansion (5) is an expansion for each fixed z, and is not a good one as Im(z) → 0. However, we may resolve this problem by fixing a δ ∈ (0, 1/2)
6LECTURE22-23:WEYL'SLAWand considering the tworegionIm(z)/≤hand Im(z)/≥hseparately.Since, f(z2) = O(|Im(z)/0), we see1.f(2)r (r, E)drdy E h S(1/m).JIm(z) ho, Theorem 1.3 implies that rz E h° Ss(1/m). So the expansion (5) isan expansion in hSs(1/m) in this case, and we thus we get from Hellfer-Sjostrandformulatheasymptoticexpansiona(r,E) ~hax(r,s)k≥0inS(1/m),whereqk(c,E,2)1a.f(2)as(a,s):(2 - p(z, 3)2k+1 L(d2),元JiIm(≥)/>hgModulohooS(1/m),wemayreplaceakbyqk(r,s,z)[,f(2)ar(r,s) =(z -p(z, )2k+1L(d2)元JC11. (F(2)g(r,E,2)) (-0.)2kL(dz元(2k)！(z-p(r,))111[5(02)2 (F(2)d(r,5,2))L(dz)) (z-p(r,E)元(2k)！J1()2* (f(t)g(,,t)t=p(s)(2k)!where we used the fact that qk is a polynomial and thus is analytic in z, and thefact f(2)q(r, s, z) is an almost analytic extension of f(t)qk(r, E,t). In particular,ao(r,)=f(p(r,))口Remark. Suppose p~pi+hp2+..., then using qi =0 we easily geta~f(pi)+hf'(pi)p2+
6 LECTURE 22-23: WEYL’S LAW and considering the two region |Im(z)| ≤ ~ δ and |Im(z)| ≥ ~ δ separately. Since ¯∂z ˜f(z) = O(|Im(z)| ∞), we see − 1 π Z |Im(z)| ~ δ , Theorem 1.3 implies that rz ∈ ~ δSδ(1/m). So the expansion (5) is an expansion in ~ δSδ(1/m) in this case, and we thus we get from Hellfer-Sj¨ostrand formula the asymptotic expansion a(x, ξ) ∼ X k≥0 ~ k a˜k(x, ξ) in ~ δSδ(1/m), where a˜k(x, ξ) = − 1 π Z |Im(z)|>~ δ ¯∂z ˜f(z) qk(x, ξ, z) (z − p(x, ξ))2k+1L(dz). Modulo ~ ∞S(1/m), we may replace ˜ak by ak(x, ξ) = − 1 π Z C ¯∂z ˜f(z) qk(x, ξ, z) (z − p(x, ξ))2k+1L(dz) = − 1 π 1 (2k)! Z C ¯∂z ˜f(z)qk(x, ξ, z) (−∂z) 2k 1 (z − p(x, ξ))L(dz) = − 1 π 1 (2k)! Z C ¯∂z(∂z) 2k ˜f(z)qk(x, ξ, z) 1 (z − p(x, ξ))L(dz) = 1 (2k)! (∂t) 2k (f(t)qk(x, ξ, t)) t=p(x,ξ) , where we used the fact that qk is a polynomial and thus is analytic in z, and the fact ˜f(z)qk(x, ξ, z) is an almost analytic extension of f(t)qk(x, ξ, t). In particular, a0(x, ξ) = f(p(x, ξ)). Remark. Suppose p ∼ p1 + ~p2 + · · · , then using q1 = 0 we easily get a ∼ f(p1) + ~f 0 (p1)p2 + · · · .
LECTURE22-23:WEYL'SLAW72.WEYLS'LAWFORh-PSEUDODIFFERENTIAL OPERATORSA trace formula.NowwearereadytoproveTheorem 2.1. Suppose I = (a,b) is a finite interval and supposelim inf p(r, s) > b.(,)→0Then for any f e Co(I), the operator f(P) is a trace class operator on L2(Rn) withtrf(P) ~ (2元h)-nhk(6)a;(r,s)drde,kwhere the leading term ao(r,s)= f(p(r,$))Proof. Let pi E S(m) be a real-valued symbol such thatp-pi eC(R2n)and infpi>b.Then pi + i is also elliptic in S(m). As a consequence, both P = pW and P = pWare densely defined self-adjoint operator mapping Hr(m) L?(Rn) into L?(Rn).Moreover, there is an open neighborhood of 1 =[a,b] such that (z-P)-1 isholomorphic for z in 2.For any f e Co(1), we let f be an almost holomorphic extension of f such thatsupp(f) C 2. Since Spec(R)nI =0, we have f(P)=0.1For Imz + O, from z - P = z - P + P - P we get the following resolventidentity(z- P)-1 = (z - P)-1 + (z - P)-1(P- P)(z- P)-1Itfollows from the Hellfer-Sjostrand formula that[ (z)[(z - P)-1(P- P)(z - PI)-1JL(dz).f(P) =元JSincep-pr is compactly supported,theoperatorP-Pis traceclass.Itfollowsthat f(P) has finite trace norm and thus is also trace class. It follows from Lecture13 thata(r, s)drdetrf(P) :(2元h)n口Nowtheconclusion follows.1This is also a consequence of Hellfer-Sjostrand formula
LECTURE 22-23: WEYL’S LAW 7 2. Weyls’ law for ~-Pseudodifferential Operators ¶A trace formula. Now we are ready to prove Theorem 2.1. Suppose I = (a, b) is a finite interval and suppose lim inf (x,ξ)→∞ p(x, ξ) > b. Then for any f ∈ C ∞ 0 (I), the operator f(P) is a trace class operator on L 2 (R n ) with (6) trf(P) ∼ (2π~) −nX∞ k=0 ~ k Z R2n aj (x, ξ)dxdξ, where the leading term a0(x, ξ) = f(p(x, ξ)). Proof. Let p1 ∈ S(m) be a real-valued symbol such that p − p1 ∈ C ∞ c (R 2n ) and inf p1 > b. Then p1 + i is also elliptic in S(m). As a consequence, both P = pb W and P1 = pb1 W are densely defined self-adjoint operator mapping H~(m) ⊂ L 2 (R n ) into L 2 (R n ). Moreover, there is an open neighborhood Ω of I = [a, b] such that (z − P1) −1 is holomorphic for z in Ω. For any f ∈ C ∞ 0 (I), we let fe be an almost holomorphic extension of f such that supp(fe) ⊂ Ω. Since Spec(P1) ∩ I = ∅, we have fe(P1) = 0.1 For Imz 6= 0, from z − P1 = z − P + P − P1 we get the following resolvent identity (z − P) −1 = (z − P1) −1 + (z − P) −1 (P − P1)(z − P1) −1 . It follows from the Hellfer-Sj¨ostrand formula that f(P) = − 1 π Z C ¯∂z ˜f(z)[(z − P) −1 (P − P1)(z − P1) −1 ]L(dz). Since p − p1 is compactly supported, the operator P − P1 is trace class. It follows that f(P) has finite trace norm and thus is also trace class. It follows from Lecture 13 that trf(P) = 1 (2π~) n Z a(x, ξ)dxdξ. Now the conclusion follows. 1This is also a consequence of Hellfer-Sj¨ostrand formula.
8LECTURE22-23:WEYL'SLAWIWeyl's law.DenoteNn(P,[a,b])= #(Spec(P)n[a,b])be the number of eigenvalues of Ph in the interval [a,b]. To estimate Nr(P,[a,b]),we approximate the characteristic function of the interval [a,b] by smooth functionsboth from below and from above.Thus it is natural to introduceV([a,b) = lim Vol(p-1([a +=,b -=))andV([a,b]) = lim Vol(p-1([a -e,b + s]),As a direct consequence of the trace formula, we getTheorem 2.2 (Weyl's law). For any a< b, as h → 0 we have(7)(2h) (a, ) (1) ≤ N(P,[0, ) ≤(2h)(g, b)+ (1),Proof. Pick two sequence of compactly supported smooth functions fe, fe (e.g. byregularization via convolution) that approaches the characteristic function X[a,b] ofthe interval [a, b] from below and from above, namely1(a+e,b) ≤ fe≤1(a,b) ≤ ≤1[a-e,b+eThen we havetrf(P) ≤ Nn(P,[a, b) ≤trfe(P),So the conclusion follows from the trace formula we just proved.口In particular, for Schrodinger operator P =-h? +V, where V satisfying the“polynomial growth"and “almost elliptic"conditions that we mentioned last time,we have the following Weyl's law for Schrodinger operators:Theorem2.3(Weyl'slawfor Schrodingeroperator onR").Foranya<b,1(8)Nn(P, [a, b]) =(2h) (Vol(r,) a≤[SP + V(a) ≤b) + (1),Note that a special case of this theorem (namely P is the harmonic oscillator)has been proven in Lecture 3
8 LECTURE 22-23: WEYL’S LAW ¶Weyl’s law. Denote N~(P, [a, b]) = #(Spec(P) ∩ [a, b]) be the number of eigenvalues of P~ in the interval [a, b]. To estimate N~(P, [a, b]), we approximate the characteristic function of the interval [a, b] by smooth functions both from below and from above. Thus it is natural to introduce V ([a, b]) = limε→0+ Vol(p −1 ([a + ε, b − ε])) and V ([a, b]) = limε→0+ Vol(p −1 ([a − ε, b + ε])). As a direct consequence of the trace formula, we get Theorem 2.2 (Weyl’s law). For any a < b, as ~ → 0 we have (7) 1 (2π~) n (V ([a, b]) + o(1)) ≤ N~(P, [a, b]) ≤ 1 (2π~) n (V ([a, b]) + o(1)). Proof. Pick two sequence of compactly supported smooth functions fε, ¯fε (e.g. by regularization via convolution) that approaches the characteristic function χ[a,b] of the interval [a, b] from below and from above, namely 1[a+ε,b−ε] ≤ fε ≤ 1[a,b] ≤ ¯fε ≤ 1[a−ε,b+ε] . Then we have trfε(P) ≤ N~(P, [a, b]) ≤ tr ¯fε(P). So the conclusion follows from the trace formula we just proved. In particular, for Schr¨odinger operator P = −~ 2∆ + V , where V satisfying the “polynomial growth” and “almost elliptic” conditions that we mentioned last time, we have the following Weyl’s law for Schr¨odinger operators: Theorem 2.3 (Weyl’s law for Schr¨odinger operator on R n ). For any a < b, (8) N~(P, [a, b]) = 1 (2π~) n Vol{(x, ξ) | a ≤ |ξ| 2 + V (x) ≤ b} + o(1) . Note that a special case of this theorem (namely P is the harmonic oscillator) has been proven in Lecture 3.
9LECTURE22-23:WEYL'SLAW3.WEYL'sLAWFOR(M)In this section we always assume: (M,g) is a compact Riemannian manifold,.m > 0 is a positive integer,. P : Hm(M) → L?(M) is a self-adjoint pseudodifferential operator in m(M),. the principal symbol p = m(P) is real-valued and almost elliptic in Sm(T*M).Moreover, for any a E R, limg→o Vol[p-1(a - e, a + e)] = 0.Basicproperties of eigenvalues/eigenfunctions.By adapting the proofs of Theorem 1.1 and Proposition 1.2 in Lecture 21 to thesetting of compact Riemannian manifolds, we haveProposition 3.1.Under the above assumptions,(1) P has discrete real spectrumSpec(P) : >i(h) ≤ 入2(h) ≤.≤^n(h) ≤ ...→ 00(2) Each eigenfunction ;(r) is a smooth function, and (;(a)) can be taken tobe an L?-orthonormal basis.The functional calculus.As we have seen, to prove Weyl's law, the crucial ingredient is the followingTheorem 3.2. Suppose f E (R). Then f(P) E ↓-(M) with principal symbol 2o(f(P)) = f(p(r,S).Proof.Idea:We firstprove f(P)o(M).According toProposition 2.2in Lecture20,it is enough to prove(a) For any coordinate patch (a,Ua, Va), there exists x E C(U) such that(-)*Mxf(P)M(a)*E(R")(b) For any X1,X2 EC(M)with supp(xi) n supp(x2) = 0, wewant to proveMxf(P)Mx Eh-(M).The passing from o(M) to -o is standard: one only need to apply the previousresultto (P+i)*f(P)=gk(P),whereg(t)= (t+i)*f(t)E.Finallywe calculatethe principal symbol of f(P)via the Helffer-Sjostrand formula.Step 1We first prove (b), namely for any Xi,X2 E C(M) with supp(xi) nsupp(x2)=0, we want to prove Mxif(P)Mx2 EhN-N(M) for any N. Accordingto Beals's theorem, it is enough to proveIMx (P)Mxll(HN,HN) = O(hN).2Note that in our definition, -oo(M) is not negligible, and the principal symbol of an elementin -(M)is an elementin S-(T*M).Onlyelementsinh-(M)arenegligibleand haszero principal symbol of any order
LECTURE 22-23: WEYL’S LAW 9 3. Weyl’s law for Ψ(M) In this section we always assume • (M, g) is a compact Riemannian manifold, • m > 0 is a positive integer, • P : Hm ~ (M) → L 2 (M) is a self-adjoint pseudodifferential operator in Ψm(M), • the principal symbol p = σm(P) is real-valued and almost elliptic in S m(T ∗M). Moreover, for any a ∈ R, limε→0 Vol[p −1 (a − ε, a + ε)] = 0. ¶Basic properties of eigenvalues/eigenfunctions. By adapting the proofs of Theorem 1.1 and Proposition 1.2 in Lecture 21 to the setting of compact Riemannian manifolds, we have Proposition 3.1. Under the above assumptions, (1) P has discrete real spectrum Spec(P) : λ1(~) ≤ λ2(~) ≤ · · · ≤ λn(~) ≤ · · · → ∞. (2) Each eigenfunction ϕj (x) is a smooth function, and {ϕj (x)} can be taken to be an L 2 -orthonormal basis. ¶The functional calculus. As we have seen, to prove Weyl’s law, the crucial ingredient is the following Theorem 3.2. Suppose f ∈ S (R). Then f(P) ∈ Ψ−∞(M) with principal symbol 2 σ(f(P)) = f(p(x, ξ)). Proof. Idea: We first prove f(P) ∈ Ψ0 (M). According to Proposition 2.2 in Lecture 20, it is enough to prove (a) For any coordinate patch (ϕα, Uα, Vα), there exists χ ∈ C ∞ 0 (Uα) such that (ϕ −1 α ) ∗Mχf(P)Mχ(ϕα) ∗ ∈ Ψ0 (R n ). (b) For any χ1, χ2 ∈ C ∞(M) with supp(χ1) ∩ supp(χ2) = ∅, we want to prove Mχ1 f(P)Mχ2 ∈ ~ ∞Ψ−∞(M). The passing from Ψ0 (M) to Ψ−∞ is standard: one only need to apply the previous result to (P + i) k f(P) = gk(P), where g(t) = (t + i) k f(t) ∈ S . Finally we calculate the principal symbol of f(P) via the Helffer-Sj¨ostrand formula. Step 1. We first prove (b), namely for any χ1, χ2 ∈ C ∞(M) with supp(χ1) ∩ supp(χ2) = ∅, we want to prove Mχ1 f(P)Mχ2 ∈ ~ N Ψ−N (M) for any N. According to Beals’s theorem, it is enough to prove kMχ1 f(P)Mχ2 kL(H −N ~ ,HN ~ ) = O(~ N ). 2Note that in our definition, Ψ−∞(M) is not negligible, and the principal symbol of an element in Ψ−∞(M) is an element in S −∞(T ∗M). Only elements in ~∞Ψ−∞(M) are negligible and has zero principal symbol of any order
10LECTURE22-23:WEYL'SLAWAccording to the Hellfer-Sjostrand formula1 [ a,f(z)(z - P)-C(dz),f(P) = -元it is enough to prove(9)IMx (z - P)-1Mxllc(H=N,HN) = O([Im(2)K)forsomeKn>o.[We can'tconclude(9)directly since wedon'tknowwheretheresolvent(z-P)-lisasemiclassicalpseudodifferentialoperatorornot.Sotheideais:approximate(z-P)-1byasemiclassicalpseudodifferentialoperator sothat wecan control both the norm of the expression (9)with (z-P)-1 replaced bythesemiclassical pseudodifferential operator, and the norm of the remainder.]Forz EC/R,welet Qo(z)=Op((z-p)-1) E-m(M).AccordingtoPropo-sition 2.5 in Lecture 20,0o(Id - (z-P)Qo) = 1 -m(z-P)g-m(Qo)= 0and thus(z - P)Qo = Id - Rifor some Ri(z) h-1(M). According to the Calderon-Vailancourt theorem, theoperator norm of Ri(z)llc(H-N,H-N+1) is controlled by finitely many derivatives ofri(z), which, by using the Moyal product formula in local charts, together with theresolvent estimate, namely Theorem 1.3, is controlled by Im(z)-Ki for some Ki > 0:IR(2)l(HN,HN+1) = O([Im(2)-K1).If we replaceQo(z) byQi(z) =Qo+QoRi+...+QoRI -m(M)and denote RL = (Ri)L EhL-L(M) we will get(z - P)QL= (Id - Ri)(Id + Ri +... + RI) = Id - RL+1(z)withIIR (2)c(HN,HN+L) = O([Im(2)-KL)for some K' > 0. and similarly, since the estimates for Qz(z) blow up as Im(z) → 0only polynomially,IMxQLMx ll(HN,HM) = O([Im(2)-KL)for some K' > 0. As a consequence,(z- P)-1 = QL +(z-P)-1RL+1(z)and if we take L large enough, we will getIMx(2 P)-IMxll(H=N,HN) =0([Im(2)K)as desired
10 LECTURE 22-23: WEYL’S LAW According to the Hellfer-Sj¨ostrand formula f(P) = − 1 π Z C ¯∂z ˜f(z)(z − P) −1L(dz), it is enough to prove (9) kMχ1 (z − P) −1Mχ2 kL(H −N ~ ,HN ~ ) = O(~ N |Im(z)| −KN ) for some KN > 0. [We can’t conclude (9) directly since we don’t know where the resolvent (z − P) −1 is a semiclassical pseudodifferential operator or not. So the idea is: approximate (z − P) −1 by a semiclassical pseudodifferential operator so that we can control both the norm of the expression (9) with (z − P) −1 replaced by the semiclassical pseudodifferential operator, and the norm of the remainder.] For z ∈ C \ R, we let Q0(z) = Op((z − p) −1 ) ∈ Ψ−m(M). According to Proposition 2.5 in Lecture 20, σ0(Id − (z − P)Q0) = 1 − σm(z − P)σ−m(Q0) = 0 and thus (z − P)Q0 = Id − R1 for some R1(z) ∈ ~Ψ−1 (M). According to the Calderon-Vailancourt theorem, the operator norm of kR1(z)kL(H −N ~ ,H−N+1 ~ ) is controlled by finitely many derivatives of r1(z), which, by using the Moyal product formula in local charts, together with the resolvent estimate, namely Theorem 1.3, is controlled by Im(z) −K1 for some K1 > 0: kR1(z)kL(H −N ~ ,H−N+1 ~ ) = O(~|Im(z)| −K1 ). If we replace Q0(z) by QL(z) = Q0 + Q0R1 + · · · + Q0R L 1 ∈ Ψ −m(M), and denote RL = (R1) L ∈ ~ LΨ−L (M) we will get (z − P)QL = (Id − R1)(Id + R1 + · · · + R L 1 ) = Id − RL+1(z) with kRL(z)kL(H −N ~ ,H−N+L ~ ) = O(~ L |Im(z)| −K0 L ) for some K0 L > 0. and similarly, since the estimates for QL(z) blow up as Im(z) → 0 only polynomially, kMχ1QLMχ2 kL(H −N ~ ,HN ~ ) = O(~ N |Im(z)| −K00 L ) for some K00 L > 0. As a consequence, (z − P) −1 = QL + (z − P) −1RL+1(z), and if we take L large enough, we will get kMχ1 (z − P) −1Mχ2 kL(H −N ~ ,HN ~ ) = O(~ N |Im(z)| −KN ) as desired