COLLOQUIUM MATHEMATICUM

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1 COLLOQUIUM MATHEMATICUM VOL NO. 1 TWO REMARKS ABOUT SPECTRAL ASYMPTOTICS OF PSEUDODIFFERENTIAL OPERATORS BY WOJCIECH CZAJA AND ZIEMOWIT RZESZOTNIK (ST.LOUIS,MISSOURI, ANDWROC LAW) Abstract. In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics. In [11] Shubin Tulovskiĭ discuss the class of pseudodifferential operators ( ) x+y Af(x) = e 2πi(x y)ξ a,ξ f(y)dydξ 2 on R N whose Weyl symbols are positive satisfy (i) (ii) (iii) c w δ a(w) C w p, α a(w) C α a(w) 1 α, a (w),w Ca(w) 1 κ, where δ > 0, 0 < κ < 1, p N are fixed constants, α N 2N w R 2N. Such an operator A is selfadjoint on L 2 (R N ), bounded from below, has a discrete spectrum. Shubin Tulovskiĭ prove the Weyl asymptotic formula N A (λ) = a λ dw +O(λ σ ), 0 < σ < κ, for the number of eigenvalues of A smaller than or equal to λ. This has been extended by Hörmer [5] to much more general classes with better estimates for the error term within the framework of his general Weyl calculus. Then a similar question was considered by G lowacki [4] for the class of positive symbols with the following properties: (a) (b) α a(w) C α a(w) 1, α > 0, α a C α, α 1, 1991 Mathematics Subject Classification: Primary 47G30. [131]

2 132 W. CZAJA AND Z. RZESZOTNIK (c) lim a(w) =. w This class is larger than that of Shubin Tulovskiĭ is not included in any of Hörmer s classes. However, the Weyl formula still holds. The reader is referred to the papers quoted above for comparison of the estimates of the error term. The first goal of this note is to show that a special case of G lowacki s theorem still holds even if the condition (b) is dropped. For this we will use a version of Beals theorem on fractional powers of pseudodifferential operators (see Beals [1]). This reduces the proof to the theorem obtained by G lowacki. The other part of the paper is devoted to the quasi-classical asymptotics N A(h) (λ) = h N (V(λ)+O(h 1/2 )), h 0, where A (h) = Op(a (h) ), a (h) (z) = a( hz), h > 0, for positive symbols satisfying (a ) (c ) α a(w) C α a(w) (1 α ) +, with t + = max(t,0), t R, lim a(w) =. w We use G lowacki s version of the Shubin Tulovskiĭ method of approximate spectral projectors based on a lemma of Hörmer [5]. This improves results of Roĭtburd [9] (see also [10]). We would like to express our thanks to Professor Pawe l G lowacki for the inspiration enormous help we received during work on this article. 1. Pseudodifferential operators. Let V be an N-dimensional vector space V its dual space. Fix a Euclidean norm on V the dual norm on V. The space W = V V with the product norm will be called the phase space. Let {e j } N j=1 be an orthonormal basis in V {e j} 2N j=n+1 the dual basis in V. For a multindex α N 2N a smooth function f on W we define α f = α α 2N 2N f, where jf(w) = 1 2πi d dt f(w+te j ). t=0 In the phase space W we define the following symplectic form: σ(w,v) = yξ xη, where w = (x,ξ) v = (y,η). Moreover, let v = (1+ v 2 ) 1/2 for v W. Then we have Peetre s inequality (1) v +w 2 v w.

3 SPECTRAL ASYMPTOTICS 133 A strictly positive, continuous function m on W is called a weight if m(w +v) Cm(w) v n for every w,v W some constants C,n > 0. In particular, for every weight m we have 1 C w n m(w) C w n. The set of weights is a group under multiplication. Moreover, if α R m is a weight, then m α is also a weight. For a fixed weight m we define S(m) to be the set of all functions a C (W) such that max α k m 1 α a < for k = 0,1,2,... Let S(V) denote the space of Schwartz functions on V. Every function a C (W) having derivatives with a common polynomial growth (e.g. a S(m)) defines a continuous endomorphism A = Op(a) : S(V) S(V) given by the Weyl formula Af(x) = e 2πi(x y)ξ a ( x+y 2 ),ξ f(y)dydξ. Such an operator A is called a pseudodifferential operator the function a is the symbol of A. We denote by L(m) the class of operators Op(a) with a S(m). We will use the following propositions: (1.1)Proposition (see [3]). If A L(m 1 ) B L(m 2 ), then AB L(m 1 m 2 ) for k N large enough, we have [ ] b(w +v) e 4πiσ(v,u) dudv, a b(w) = 4 N (1+ u ) k a(w +u) 2u 2k (1+ v ) k where a b denotes the symbol of AB. 2v 2k (1.2)Proposition(see [3]). If a S(m 1 ), b S(m 2 ) α a S(m 1 ), α b S(m 2 ) then r(a,b) = a b ab S(m 1 m 2 ). Moreover, N 1 r(a,b)(w) = 4N (1+t 2 u ) k j a(w +u) dt 2 j=1 0 2u 2k [ ] (1+ v ) k j+n b(w +v) 2v 2k e 4πiσ(v,u) dudv N 1 4N (1+t 2 u ) k j+n a(w +u) dt 2 j=1 0 2u 2k [ ] (1+ v ) k j b(w +v) 2v 2k e 4πiσ(v,u) dudv.

4 134 W. CZAJA AND Z. RZESZOTNIK A symbol a S(m) is called elliptic if for some constants C,K > 0, a(w) Cm(w) for w K α a S(m 1 ) for α > 0 some > 0, independent of α. (1.3)Proposition (see [3] [4]). Suppose m c > 0 is a weight a is an elliptic symbol in S(m). Let Op(a) : S(V) L 2 (V). (i) If a is real-valued then Op(a) is essentially selfadjoint. (ii) If a 0 then Op(a) is bounded from below. (iii) If lim w m(w) =, then the spectrum of Op(a) is discrete. (iv) If lim w a(w) = 0, then Op(a) is compact. (1.4)Proposition (see [4]). If a 0 max 0< α 2N+2 α a C, then Op(a) LC, where L is a constant independent of a. (1.5)Proposition (see [10]). If a S(m) a 0, then Op(H a) is positive, where H(w) = e ( 2π w 2). (Here denotes convolution.) We will also need the following technical estimates: (1.6) Proposition (see [4]). Let a b be symbols. Then α r(a,b) 1 C max 0< β α +2k+1 β a 1 max 0< β α +2k+1 β b. Moreover, if j a S(m 1 ) j b S(m 2 ) for j = 1,...,2N, then α r(a,b) C α r(a,b) C max 0< β α +2k+1 m 1 β b max m 2 β a. 0< β α +2k+1 (1.7)Proposition (see [4]). Let e,f S(1), a S(m) j a S(n), for j = 1,...,2N. Then for all α k sufficiently large α r(e,a,f) C( max 0< β α +k n β e f + max 0< β α +k β (ea) max 0< β α +k β f + max 0< β α +k β e max 0< β α +k n β f ). Let H be a Hilbert space with an inner product,. An operator T L(H) is called a trace operator if there is an orthonormal basis {e α } such that Te α,e α <. α

5 SPECTRAL ASYMPTOTICS 135 Then the number TrT = α Te α,e α does not depend on the choice of basis is called the trace of T. We let T Tr = Tr T. (1.8)Proposition (see [4]). If a L 1 (W) Op(a) is a trace operator then TrOp(a) = a(x,ξ)dxdξ. (1.9)Proposition (see [4]). Let a C (W). If α a L 1 (W) for α 2N +1, then Op(a) extends uniquely to a trace operator A on L 2 (V) A Tr C max α 2N+1 α a L 1. Let us also state a lemma which connects the condition (a ) with the definition of weight (the proof of the lemma relies only on Taylor s formula). (1.10)Lemma. If a C > 0 there exists > 0 such that α a C α a (1 α ) + for all α, then 1+a is a weight. 2. Classical asymptotics. Let A be a selfadjoint operator bounded from below with a discrete spectrum. We denote by N A (λ) the number of eigenvalues of A less than or equal to λ (counting with multiplicities). For a positive function a C (W) we define V a (λ) = a λ For λ,m > 0, we also define dw ψ a (t) = inf a(w)=t a (w),w. ν(λ,m) = λ+m λ ψ a (t) 1 dt. (2.1)Proposition. If for t λ we have ψ a (t) > 0 ν(λ,m) log2, then V a (λ+m) V a (λ) 2ν(λ,m)V a (λ). Proof. Note that V is increasing, hence differentiable almost everywhere on R +. By formula (28.41) of Shubin [10] for t λ we have V a(t)/v a (t) ψ a (t) 1.

6 136 W. CZAJA AND Z. RZESZOTNIK Using a simple estimate e x 1 xe x valid for x R we obtain ( λ+m V a (λ+m) V a (λ) V a = exp (t) ) V a (λ) V a (t) dt 1 λ+m λ λ 2ν(λ,m). V a(t) V a (t) dt exp ( λ+m λ V a(t) ) V a (t) dt The following theorem is a particular case of the theorem due to G lowacki (Theorem (2.3) of [4]). (2.2)Theorem. Let m be a weight such that lim w m(w) = let b be a positive elliptic symbol in S(m). If α b(w) C α for α > 0, ψ b (t) Ct r, where 0 < r 1, t > 0, then for λ large enough N Op(b) (λ) 1 V b (λ) Cλ r. We will show that a similar theorem is true with a weaker assumption α b(w) C α b 1 (w) for α > 0, where 0 < < 1. For this purpose we will use results due to Beals [1]. In his paper Beals defines pseudodifferential operators by the Kohn Nirenberg formula, buthismethodissogeneralthatonecaneasilyapplyittooperators defined by the Weyl formula. The next theorem is a special case of Theorem (4.2) of [1]. (2.3)Theorem. Let m c > 0 be a weight, A > 0 have an elliptic symbol A L(m). Then A s L(m s ) for s R. (2.4)Corollary. Let m c > 0 be a weight a be a positive elliptic symbol in S(m) such that Op(a) > 0 α a S(m 1 ) for α > 0. Let a s denote the symbol of Op(a) s. Then where r s S(m s ). a s = a s +r s, Proof. Let us consider the case when 1 < s < 0. Then for A = Op(a), A s = sinπs π 0 t s (t+a) 1 dt

7 SPECTRAL ASYMPTOTICS 137 (see [7]). Hence a s = sinπs π 0 t s a t dt, wherea t denotes thesymbol of the operator (t+a) 1. In this way weobtain We claim that r s = sinπs π 0 t s (a t (a+t) 1 )dt. a t (a+t) 1 S((m+t) 1 ) uniformly with respect to t. In fact we have a t (a+t) 1 = (a+t) 1 (a t (a+t) 1) = r(a t,a)(a+t) 1, but β a S(m 1 ) for β > 0 a t S((m+t) 1 ) uniformlywith respect to t, so also r(a t,a) S((m+t) ) uniformly with respect to t (Proposition (1.2)). Hence our claim is true. In this way we obtain β r s sinπs π 0 t s C s (m+t) 1 dt C s m s, so that r s S(m s ). To prove the corollary for s R we only have to remark that r s S(m s ) implies r 2s S(m 2s ) r s S(m s ). (2.5)Corollary. Let m c > 0 be a weight, B > 0 have an elliptic symbol B L(m). Then for all s R R L(m s ) there exists a constant L such that Rf L B s f for f S(V). Proof. By Theorem (2.3) we have B s L(m s ), so the Calderón Vaillancourt theorem [2] shows that the operator RB s is bounded. Hence RB s f L f, which implies that Rf L B s f. (2.6)Corollary. Let m c > 0 be a weight, B > 0 have an elliptic symbol B L(m). Then for all s R R L(m s ) there exists a constant L such that Rf,f L B s f,f for f S(V). Proof. We have R = B s/2 B s/2 R, so Rf,f = B s/2 B s/2 Rf,f = B s/2 Rf,B s/2 f B s/2 Rf B s/2 f. Note that by Theorem (2.3) the operator B s/2 is in L(m s/2 ), therefore B s/2 R L(m s/2 ); so by Corollary (2.5) we obtain B s/2 Rf

8 138 W. CZAJA AND Z. RZESZOTNIK L B s/2 f. In this way Rf,f L B s/2 f 2 = L B s f,f. (2.7)Lemma. Let m be a weight such that lim w m(w) = let b be a positive elliptic symbol in S(m). If α b(w) C α for α > 0 ψ b (t) Ct r, where 0 < r 1, then for s 1 λ large enough N Op(bs )(λ) 1 V b s(λ) Cλ r/s. Proof. First let us study the functions V b s N Op(bs ) (our assumptions imply that Op(b s ) is bounded from below has a discrete spectrum (Proposition (1.3))). Denoting the symbol b s by a we obtain V a (λ) = V b (λ 1/s ). Note that by Corollary (2.4) we have b s = b s +r, where r S(m s 1 ). Let A denote Op(a) B denote Op(b). We know that b > 0 so there is a constant M such that B + M > 0 (Proposition (1.3)). We also have r S(m s 1 ) S((m+M) s 1 ), so by Corollary (2.6), Hence L(B +M) s 1 Op(r) L(B +M) s 1. B s L(B +M) s 1 A B s +L(B +M) s 1 therefore, using the mini-max principle [8], we obtain b s n L(b n +M) s 1 a n b s n +L(b n +M) s 1 where a n b n denote the nth eigenvalues of the operators A B, respectively. In this way increasing the constants L M if necessary, we obtain so b s n Lbs 1 n M a n b s n +Lbs 1 n +M, {n N : b s n +Lbs 1 n +M λ} {n N : a n λ} {n N : b s n Lb s 1 n M λ}. But it is easy to see that for λ large enough b s n Lb s 1 n M λ implies b n λ 1/s +L. Infactwecanconsiderthefunctionf(x) = x s Lx s 1 M λ definedforx>0. Ithasonlyonezerowithpositivederivative forlargeλ, f(λ 1/s +L) = λ 1/s ((λ 1/s +L) s 1 λ (s 1)/s ) M 0.

9 SPECTRAL ASYMPTOTICS 139 Similarly b n λ 1/s L implies b s n +Lb s 1 n so +M λ. Therefore {n N : b n λ 1/s L} {n N : a n λ} {n N : b n λ 1/s +L} N B (λ 1/s L) N A (λ) N B (λ 1/s +L). Denote λ 1/s L by µ 1 λ 1/s +L by µ 2. Then So using Proposition (2.1) we obtain N B (µ 1 ) V b (µ 1 +L) N A(λ) V a (λ) N B(µ 2 ) V b (µ 2 L). N B (µ 1 ) V b (µ 1 )(2ν(µ 1,L)+1) N A(λ) V a (λ) N B(µ 2 ) V b (µ 2 ) (2ν(λ1/s,L)+1). Note that N B (µ 1 ) V b (µ 1 )(2ν(µ 1,L)+1) 1 N B (µ 1 ) V b (µ 1 ) 1 +2ν(µ 1,L) N B (µ 2 ) V b (µ 2 ) (2ν(λ1/s,L)+1) 1 N B (µ 2 ) V b (µ 2 ) 1 +3ν(λ1/s,L). Obviously the symbol b satisfies the assumptions of Theorem (2.2), therefore N B (µ i ) V b (µ i ) 1 Cµ r i for i = 1,2. Now we only need to note that µ r 1, µ r 2, ν(µ 1,L) ν(λ 1/s,L) are estimated by Mλ r/s, where M is a constant. Finally, we conclude that there is a constant C such that for λ large enough N A (λ) V a (λ) 1 Cλ r/s. (2.8)Theorem. Let m be a weight such that lim w m(w) = let a be a positive elliptic symbol in S(m). If α a(w) C α a 1 (w) for α > 0, ψ a (t) Ct r, where 0 < 1, 0 < r 1 r + > 1, then for λ large enough N Op(a) (λ) 1 V a (λ) Cλ (r+ 1). Proof. Note that the symbol a satisfies the assumptions of Lemma (2.7). In fact, denoting a by b, we obtain α b(w) C α ψ b (t) Ct r/ +1 1/.

10 140 W. CZAJA AND Z. RZESZOTNIK So from Lemma (2.7) it follows that N Op(a) (λ) 1 V a (λ) Cλ σ, where σ = (r/ +1 1/ ) = r + 1. (2.9) Example. Let us assume that a symbol a satisfies conditions analogous to those of Tulovskiĭ Shubin: (i ) (ii ) (iii) 0 a(w) S(m), a elliptic lim m(w) =, w Then (iii) implies that so by Theorem (2.8), α a(w) C α a(w) 1 for α > 0, a (w),w Ca(w) 1 κ. ψ a (t) Ct 1 κ, N Op(a) (λ) = V a (λ)+o(λ ( κ) ), which is stronger than the result of Tulovskiĭ Shubin. Moreover, by Lemma (1.10) the conditions (i) (ii) imply that a + 1 is a weight also that a is an elliptic symbol in S(a+1). This means that the condition (i ) is satisfied with m = a + 1. An easy observation that (ii) implies (ii ) proves that the conditions (i ), (ii ), (iii) are more general than (i) (iii). 3. Quasiclassical asymptotics. As before let A be a selfadjoint operator defined on a dense subspace of a Hilbert space, with spectrum SpA discrete bounded from below. Let N A (λ) be the spectral function of A. The following result is due to L. Hörmer [5]. (3.1)Lemma. Let E be a selfadjoint trace operator such that AE is bounded. If (I E)(A λ)(i E) l, then If E(λ A)E k, then N A (λ 4l) TrE +2 E E 2 Tr. N A (λ+4k) TrE 2 E E 2 Tr. Forδ > 0letχ(t,λ,δ) beasmoothfunctionwiththefollowingproperties: { 1 if t λ, χ(t,λ,δ) = 0 if t λ+2δ, ( / t) k χ(t,λ,δ) C k δ k. For any h > 0 any λ we can define a smooth function e h (z) = χ(a(z),λ,h 1/2 ),

11 SPECTRAL ASYMPTOTICS 141 where a 0 is the symbol of some pseudodifferential operator A with the property that lim w a(w) =. Let a (h) (w) = a(h 1/2 w). Define E h to be the operator with symbol e (h) h. The function e(h) h satisfies γ e (h) h (z) = γ k=0 m m k = γ e (h) h (z) = { 1 if a(h 1/2 z) λ, 0 if a(h 1/2 z) λ+2h 1/2, C m,k h γ /2 k χ z k(a(h1/2 z),λ,h 1/2 ) m 1 1 a(h 1/2 z)... m k k a(h1/2 z). If there is a > 0 such that for every γ we have C γ satisfying γ a(z) C γ a (1 γ ) + (z), then we can summarize all the above results to get (2) γ e (h) h (z) C γ,λ. With the above assumptions we have the following lemma: (3.2)Lemma. E h is a trace operator E 2 h E h Tr Ch N+1/2. Proof. The symbol of the operator E h Eh 2 can be written as e(h) h (z) e (h) h e(h) h (z) = e(h) h (z) e(h) h e(h) h (z) r(e(h) h,e(h) h )(z). By definition, Proposition (1.6) (2), for h small enough, we get γ (e (h) h (e(h) h )2 ) 1 Ch N (V(λ+2h 1/2 ) V(λ)), γ (r(e (h) h,e(h) h )) 1 Ch N (V(λ+2h 1/2 ) V(λ)). By Proposition (1.9) Lebesgue s differentiation theorem ([8], Theorem 9.2, p. 226), for almost every λ we have E h E 2 h Tr C max α 2N+2 α (e (h) h (z) e(h) h e(h) h (z)) 1 Ch N (V(λ+2h 1/2 ) V(λ)) Ch N h 1/2. The main purpose of this section is to prove the following theorem. (3.3)Theorem. Let m be a weight such that lim z m(z) =. Let a C > 0 be an elliptic symbol in the class S(m) such that (3) γ a(z) C γ a (1 γ ) + (z) for some > 0. Let A (h) be the operator with symbol a (h). Then for almost all λ, N A(h) (λ) = h N (V(λ)+O(h 1/2 )).

12 142 W. CZAJA AND Z. RZESZOTNIK Moreover, to obtain the above asymptotics it is enough to assume only that a C > 0 is a smooth function satisfying the inequality (3) such that lim z a(z) =. where (3.4)Lemma. For h small enough we have the estimate E h (λ A (h) )(I E h ) Ch 1/2. Proof. The symbol of the operator E h (λ A (h) )(I E h ) is e (h) h (λ a(h) ) (1 e (h) h )(z) = a λ,h(z)+r λ,h (z), a λ,h (z) = e (h) h (λ a(h) )(1 e (h) h )(z), r λ,h (z) = r(e (h) h,λ a(h),1 e (h) h )(z). By (2) Proposition (1.7), max α 2N+2 α a λ,h Ch 1/2, max α 2n+2 α r λ,h Ch 1/2. The proof can now be completed by using the Calderón Vaillancourt theorem (for references see [2]). (3.5) Proposition. For h small enough Proof. We can write E h (λ A (h) )E h Ch 1/2. E h (λ A (h) )E h = E h (λ A (h) ) E h (λ A (h) )(I E h ). The symbol of the first operator on the right h side is a λ,h (z) r λ,h (z), where now a λ,h (z) = e (h) h (λ a(h) )(z) r λ,h (z) = r(e (h) h,a(h) h )(z). Moreover, a λ,h (z) 2h 1/2. By (2), for α > 0 we have α a λ,h Ch 1/2. Therefore Proposition (1.4) gives us Op(a λ,h ) Lh 1/2. Now by Proposition (1.6) we get α r λ,h Ch 1/2, so by the Calderón Vaillancourt theorem Op(r λ,h ) Ch 1/2. Finally, we can prove the proposition by combining all the above with Lemma (3.4). (3.6)Proposition. For h > 0 small enough (I E h )(A (h) λ)(i E h ) Ch 1/2. Proof. We have a similar decomposition to the one before: (I E h )(A (h) λ)(i E h ) = (A (h) λ)(i E h )+E h (λ A (h) )(I E h ).

13 SPECTRAL ASYMPTOTICS 143 The symbol of the operator (A(h) λ)(i E h ) is a λ,h (z) r λ,h (z), where now a λ,h (z) = (a (h) λ)(1 e (h) h )(z) 0 r λ,h(z) = r(a (h),e (h) h )(z). Notice, therefore, that we only need to estimate Op(a λ,h ). It is easy to see that α a λ,h C α,h h 1/2 a λ,h for α > 0. Moreover, for every b S(m) k N, (δ H) k b(w) therefore = [0,1] k W k b (k) ( w k t j v j )(v 1,...,v k )H(v 1 )...H(v k )dvdt, j=1 α (δ H) k a λ,h C λ,h,k h 1/2 a λ,h. Notice also that for k + α > [ 1 ], where [ ] denotes the greatest integer function, we get α (δ H) k a λ,h C λ,α,k h 1/2. Since we can decompose a λ,h as n a λ,h = H (δ H) k a λ,h +(δ H) (n+1) a λ,h, k=0 where n = [ 1 ], the above calculations Propositions (1.4) (1.5) give us Op(a λ,h ) Ch 1/2. (3.7)Fact. TrE h = h N V(λ)(1+O(h 1/2 )). Proof. By Proposition (1.9), E h is a trace operator. Moreover, by Proposition (1.8), TrE h = e (h) h (z)dz. So according to the definition of e h, which is equivalent to h N V(λ) TrE h Ch N+1/2 +h N V(λ), TrE h h N V(λ) 1 = O(h1/2 ). (3.8)Remark. The operator A (h) E h is bounded. Proof. The symbol of this operator is a (h) (z)e (h) h (z) +r(a(h),e (h) h )(z). The derivatives of the first summ are bounded because it is a smooth, compactly supported function, of the other one by Proposition (1.6). Thus our claim follows from the Calderón Vaillancourt theorem (see Proposition (1.15) of [4]).

14 144 W. CZAJA AND Z. RZESZOTNIK Proof of Theorem (3.3). By Proposition (1.3) the operator A (h) is essentially selfadjoint bounded from below. Therefore we can apply Lemma (3.1) to its closure, which we will also denote by A (h). By Lemma (3.1) Propositions (3.5) (3.6) we have N A(h) (λ Ch 1/2 ) TrE h +2 E 2 h E h Tr N A(h) (λ+ch 1/2 ) TrE h 2 E 2 h E h Tr, for any h > 0 λ, parameters of the symbol e (h) h. So where E h,λ + h N A(h) (λ) TrE h,λ + +2 E 2 h h,λ + h N A(h) (λ) TrE h,λ h 2 E 2 h,λ h E h,λ + Tr h E h,λ Tr, h is the operator with symbol e (h) h,λ+ch 1/2 E h,λ h has symbol e (h) h,λ Ch 1/2. By the proofs of Lemma (3.2) Fact (3.7) it is easy to see that E 2 h,λ h E h,λh Tr Ch N+1/2 TrE h,λh h N V(λ) Ch N+1/2. Therefore the first part of our theorem is established. To prove the second part it is enough to notice by Lemma (1.10) that a+1 is a weight a is an elliptic symbol in the class S(a+1). (3.9)Example. Let p(w 1,...,w 2N ) c > 0 be a hypoelliptic polynomial (i.e. α p(ξ)/p(ξ) C ξ c α, α > 0, see [6], Theorem ). In particular lim w p(w) =. By the hypoellipticity of p, condition (3) is satisfied. Therefore Theorem (3.3) gives us the spectral asymptotics for Op(p (h) ). (3.10)Example. Let l(t) = ln(1+t) l n (t) = l... l(t) for n N. Let c n (w 1,...,w 2N ) = l n ( w 1,...,w 2N ). Since γ ( x b ) C γ x b γ, we have, for α > 0, α c n (w) C α w α C α. Since lim w c n (w) =, we can apply Theorem (3.3) to Op(c (h) n ). (3.11)Example. Finally, we mention that in our theory we can also consider Schrödinger operators with potential of logarithmic growth. Thus, as before, we can obtain our spectral asymptotics for the operator with symbol x 1 ξ 2 +log( x ).

15 SPECTRAL ASYMPTOTICS 145 REFERENCES [1] R. B e a l s, Characterization of pseudodifferential operators applications, Duke Math. J. 44(1977), [2]G.B.Foll,HarmonicAnalysisinPhaseSpace,PrincetonUniv.Press,Princeton, N.J.,1989. [3]P.G lowacki,stablesemi-groupsofmeasuresontheheisenberggroup,studiamath. 79(1984), [4], The Weyl asymptotic formula for a class of pseudodifferential operators, ibid. 127 (1998), [5]L.Hörmer,Ontheasymptoticdistributionofthepseudodifferentialoperatorsin R n,ark.mat.17(1979), [6], The Analysis of Linear Partial Differential Operators, Springer, Berlin, [7] A. P a z y, Semigroups of Linear Operators Applications to Partial Differential Equations, Springer, New York, [8]M.ReedB.Simon,MethodsofModernMathematicalPhysics,Vol. IV,Academic Press, New York, [9]V.L.Roĭtburd,Thequasiclassicalasymptoticbehaviorofthespectrumofapseudodifferential operator, Uspekhi Mat. Nauk 31(1976), no. 4, (in Russian). [10]M.A.Shubin,PseudodifferentialOperatorsSpectralTheory,Nauka,Moscow, 1978(in Russian). [11]M.A.ShubinV.N.Tulovskiĭ,Ontheasymptoticdistributionoftheeigenvaluesofpseudodifferentialoperatorsin R n,mat. Sb. 92(134)(1973), (in Russian). Department of Mathematics Washington University Campus Box 1146 St. Louis, Missouri U.S.A. wczaja@artsci.wustl.edu zrzeszot@artsci.wustl.edu Mathematical Institute University of Wroc law Pl. Grunwaldzki 2/ Wroc law, Pol Received 2 March 1998; revised12june199827november1998