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A. Ziggioto

QUASICLASSICAL ANALYSIS OF HYPOELLIPTIC

OPERATORS

Abstract. In this paper we take into account the hypoelliptic operators introduced by H¨ormander in [3] and we develop their quasiclassical anal-ysis, obtaining an asymptotic formula for their counting function, as the Planck constant ǫ goes to 0, under somewhat more general hypotheses than H¨ormander’s ones.

1. Introduction

At the beginning of our work we give a brief sketch of the physical motivations under-lying the mathematical branch known as quasiclassical analysis, without having the pretension of being totally exhaustive. For further details we refer to [7], [8] and [9] among others.

In Classical Mechanics it is well known that a particle of mass m can be described by a curve t _→ (x(t), ξ(t))(called the classical trajectory) such that, in the case the particle is subject to a force field F _{= −}grad V , the so called Hamiltonian equations are fulfilled:

( ˙

ξ (t)_{= −}grad V(x(t)) ˙

x(t)₌_m1ξ(t),

where x(t)is the position of the particle at any time t andξ(t)₌mx_˙(t)is the momen-tum of the particle.

A classical observable is any real smooth function h(x, ξ )defined on the phase spaceRn_×Rn: its value at the point(x(t), ξ(t))gives an information on the position of the particle at time t. An example of classical observable is the total energy_2m1ξ2₊

V(x).

In Quantum Mechanics a particle is instead described by a functionψ(t,x)(called the wave function) such that for all t _∈R

Rn|

ψ(t,x)_|2d x₌1,

that isψt : x _7→ψ(t,x)belongs to L2(Rn)and_kψt_kL2₍_Rn₎ =1. The functionψt is

called the state of the particle at time t.

The classical quantities of position x and momentumξcan be investigated in Quan-tum Mechanics by considering the following two operators:

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1. the multiplication operator

x :ψ→xψ

for the position; 2. the derivation operator

ǫDx :ψ→ −iǫgradψ

for the momentum, whereǫis the Planck constant.∗

Note that these operators are selfadjoint with respect to the L2scalar product. More generally, any selfadjoint operator on L2is called a quantum observable.

The question about the correspondence between classical observable h(x, ξ )and quantum observable h(x, ǫDx)is called the problem of quantization and one of the

purposes of pseudodifferential calculus is just to provide a correspondence between the space of all classical observables endowed with the usual multiplication and the space of quantum observables endowed with the composition of operators.

For example, to the classical total energy_2m1 ξ2₊V(x)we associate the operator

−ǫ

2m1+V(x)

(where1is the standard Laplacian) which is the Schr ¨odinger operator. Actually, it is convenient to consider the so called Weyl quantization: (1) hw₍_x_{, ǫ}2_{ξ )}_u₍_x₎₌

ei(x−y)·ξh _x

+y 2 , ǫ

2_ξ_u₍_y₎_{d y d}−_ξ

or the unitarily equivalent form (2) hw_(ǫ_x_{, ǫξ )}_u₍_x₎₌

ei(x−y)·ξh

ǫx+y

2 , ǫξ

u(y)d y d−ξ.

(for suitable functions u and h) since these operators are symplectically invariant, as remarked at the end of the paper, and are selfadjoint if and only if the symbol h is real valued.

Since Classical Mechanics (which is much simpler than Quantum Mechanics) de-scribes quite well most of common elementary physical phenomena, we expect that Quantum Mechanics is a kind of generalization of Classical Mechanics, in the sense that one could recapture the classical properties of a system by making some approxi-mation of its quantum properties. This is the so called Bohr correspondence principle: Classical Mechanics is nothing but the limits of Quantum Mechanics as the Planck constantǫtends to 0. This motivates an interest in the asymptotic properties of Weyl ∗_{In order to be consistent with the notations used in [13] here we denote the Planck constant by}ǫ_and

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operators (1) or (2) whenǫ→0. The corresponding asymptotic analysis is called the quasiclassical analysis.

In our work we consider the Weyl operators already considered by H¨ormander in [3] for which he obtained an asymptotic formula for their counting functionN(τ )as

τ _{→ +∞} (see Section 2). Then we consider the corresponding quantum observ-ables hw_(ǫ_x_{, ǫξ )}_{and we obtain a quasiclassical asymptotic formula for the associated} counting functionN_ǫ(τ ) asǫ _→ 0, under somewhat more general hypotheses than H¨ormander’s ones.

We employ the following notation: given two functions f,g : X _→R, and a subset A_⊂X , we write

f(x)_≺g(x), _∀x_∈ A,

if there exists a constant C such that

f(x)_≤Cg(x), _∀x_∈ A.

I would like to acknowledge professor Buzano for his precious suggestions while writing this paper.

2. Basic definitions and results

First we recall some definitions and results from [2], [4] and [13]. Let us begin with a brief review of Weyl-H¨ormander calculus. Let

φ(x, ξ;y, η)= n X

j=1

ξjyj−xjηj

be the standard symplectic form onRn_×Rn.

DEFINITION1. A Riemannian metric gx,ξ(y, η)onRn×Rn is slowly varying if

there exist a positive real number r such that

gx,ξ(y, η)≺gt,τ(y, η)≺gx,ξ(y, η),

for all(x, ξ ), (t, τ ), (y, η)_∈Rn_×Rnsuch that gx,ξ(x−t, ξ−τ )≤r.

DEFINITION2. A positive function m(x, ξ )is said to be g continuous if there is a positive real number r such that

m(y, η)_≺m(x, ξ )_≺m(y, η),

for all(x, ξ ), (y, η)_∈Rn_×Rnsuch that

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DEFINITION3. A Riemannian metric gx,ξ(y, η)onRn×Rnis locallyφtemperate

if it is slowly varying and there exist two positive real numbers r and N and a slowly varying metric Gx(y)onRnsuch that

Gx(y)≤gx,ξ(y, η),

for all(x, ξ ), (y, η)_∈Rn_×Rn, and

(3) gx,ξ(z, ζ )≺gy,η(z, ζ )(1+gφx,ξ(x−y, ξ−η))

N_,

for all(x, ξ ), (y, η), (z, ζ )_∈Rn_×Rnsuch that Gx(x−y)≤r.

The quadratic form gφ_x_,ξ(y, η)appearing in (3) is the dual metric g_xφ_,ξ(y, η)₌sup_{(φ(y, η_;z, ζ ))2: gx,ξ(z, ζ )=1}.

DEFINITION4. A positive function m(x, ξ )is locallyφ,g temperate if it is g con-tinuous and there exist two positive real numbers r and N such that

m(x, ξ )_≺m(y, η)(1₊gφ_x_,ξ(x₋y, ξ₋η))N,

for all(x, ξ ), (y, η)_∈Rn_×Rnsuch that

Gx(x−y)≤r.

Now we recall the definition of the class of symbols of Weyl-H¨ormander S(m,g). DEFINITION5. Let g be a slowly varying Riemannian metric. Let m be a g con-tinuous function. Then we say that a function h_∈S(m,g)if it is smooth and

sup

x,ξ

|h_|_kg(x, ξ )

m(x, ξ ) <+∞, ∀k∈N,

where

|h_|g_k(x, ξ )₌ sup

yj,ηj

|h(k)((x, ξ );(y1, η1), . . . , (yk, ηk))| Qk

j=1gx,ξ(yj, ηj)

1/2

and h(k)((x, ξ )_;(y1, η1), . . . , (yk, ηk))is the k-th differential of h at(x, ξ ). We can introduce here the operators we deal with in the next section.

DEFINITION6. A differential operator hw_{is formally hypoelliptic}†_{if its Weyl} sym-bol h(x, ξ )satisfies the following conditions:

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1. h is a smooth function such that

h(x, ξ )₆₌0, _∀(x, ξ )_∈Rn_×Rn_;

2. there exists a locallyφtemperate metric gx,ξ(y, η)onRn×Rnsuch that|h|is

locallyφ,g temperate and

h_∈ S(|h_|,g).

In the proof of our main theorem (see Theorem 5) we will make use of the following two results.

THEOREM 1. Consider a positive and formally hypoelliptic symbol h _∈ S(h,g)

and assume that there exists a positive real numberγsuch that gx,ξ(y, η)≺h(x, ξ )−γg_xφ_,ξ(y, η),

for all(x, ξ ), (y, η)_∈Rn_×Rn, then hw_{is semi-bounded from below and essentially} self-adjoint in L2(Rn).

Moreover, if

(4) h(x, ξ )→ +∞, as|x| + |ξ| → +∞,

then the closure H of hw _{in L}2₍_Rn₎_{has discrete spectrum diverging to} +∞.

Proof. This is Theorem 1 of [13]. Actually we must observe that in the proof of Theo-rem 1 of [13] we did not assume that the metric g should satisfy the so called principle of indetermination, that is

sup

x,ξ

gx,ξ

gφ_x_,ξ ≤1. In our case, from the fact that there existsγ >0 such that

gx,ξ(y, η)≺h(x, ξ )−γgφx,ξ(y, η)

and that h(x, ξ )_{→ +∞}as_|x_{| + |}ξ_{| → +∞}, it follows that sup

x,ξ

gx,ξ

gφ_x_,ξ ≤ C

for a suitable constant C >0. If 0 <C _≤ 1 then the principle of indetermination is trivially satisfied. If C>1 it is sufficient to replace g with the following new metric

g= √g

C. It is easy to show thatg_˜φ₌√C gφ. Therefore

sup

x,ξ

gx,ξ

g_xφ_,ξ = sup

x,ξ

gx,ξ √

C √

C gφ_x_,ξ = 1 Csup_x_,ξ

gx,ξ

gφ_x_,ξ ≤ 1.

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Thanks to Theorem 1, we can define the counting function of the operator H : (5) N(τ )₌number of eigenfunctions of H , corresponding to

eigen-values less than or equal toτ.

THEOREM2. Under the hypotheses of Theorem 1 assume there existsκ >0 such that

(6) h−κ _∈L1.

Then for all 0< δ <γ₃ we have

(7) N_{(τ )}₌W_{(τ )}

1₊O R_{(τ )} _, _as_τ _{→ +∞}_,

where

(8) W(τ )₌(2π )−n

h_≤τ

d x dξ,

and

(9) R(τ )₌W τ +τ

1−δ

−W τ ₋τ1−δ

W(τ ) .

REMARK1. Estimate (7) is known as Weyl formula. Proof. This is Theorem 2 of [13].

3. Quasiclassical Analysis of Hypoelliptic Operators

Consider a formally hypoelliptic operator hw_. Let us introduce the operator hw

ǫ whose Weyl symbol is

hǫ(x, ξ )=h(ǫx, ǫξ ),

whereǫis a real parameter, such that 0< ǫ_≤1.

Then we define a new Riemannian metricǫgx,ξ(y, η)in this way: ǫ_g

x,ξ(y, η)=gǫx,ǫξ(ǫy, ǫη)=ǫ2gǫx,ǫξ(y, η),

where gx,ξ(y, η)is the one appearing in Definition 6. We start by giving some results

concerning this new metricǫg.

PROPOSITION1. The metricǫgx,ξ(y, η)is slowly varying and locallyφtemperate,

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Proof. We know that g is slowly varying. Then there exists a positive real number r such that

gx,ξ(t, τ )≺gy,η(t, τ )≺gx,ξ(t, τ )

for all(x, ξ ), (y, η), (t, τ )∈Rn_×_Rn_{such that g}

x,ξ(x−y, ξ−η)≤r . Therefore

gǫx,ǫξ(ǫt, ǫτ )≺gǫy,ǫη(ǫt, ǫτ )≺gǫx,ǫξ(ǫt, ǫτ )

when gǫx,ǫξ(ǫ(x−y), ǫ(ξ −η))≤r , that is ǫ_g

x,ξ(t, τ )≺ǫgy,η(t, τ )≺ǫgx,ξ(t, τ )

whenǫ_g

x,ξ(x−y, ξ−η)≤r and so we have proved thatǫg is slowly varying, for all

0< ǫ _≤1.

Let us define a new metric

ǫ_G

x(y)=Gǫx(ǫy).

Therefore it is obvious that

ǫ_G

x ≤ǫgx,ξ.

By hypothesis we know that the metric g is locallyφtemperate. Therefore there exist two positive real numbers r >0 and N such that

gx,ξ(t, τ )≺gy,η(t, τ )(1+gxφ,ξ(x−y, ξ−η))

N_,

when Gx(x−y)≤r . Therefore

gǫx,ǫξ(t, τ )≺gǫy,ǫη(t, τ )(1+gφ_ǫ_x_,ǫξ(ǫ(x−y), ǫ(ξ −η)))N,

when Gǫx(ǫ(x−y))≤r , for all 0< ǫ ≤1, that is whenǫGx(x−y)≤r . Now, we

have that

g_ǫφ_x_,ǫξ(ǫt, ǫτ )₌ sup

(y,η)6=0

φ(ǫt, ǫτ_;y, η)2

gǫx,ǫξ(y, η) =

= sup

(y,η)6=0

ǫ2φ(t, τ;y, η)2

gǫx,ǫξ(y, η) =

=ǫ4 sup

(y,η)6=0

φ(t, τ_;y, η)2 ǫ2_g

ǫx,ǫξ(y, η) =

=ǫ4 sup

(y,η)6=0

φ(t, τ_;y, η)2

gǫx,ǫξ(ǫy, ǫη) =

=ǫ4(ǫg)φ_x_,ξ(t, τ ).

Thus we can conclude that

ǫ2gǫx,ǫξ(t, τ )≺ǫ2gǫy,ǫη(t, τ )(1+gφǫx,ǫξ(ǫx−ǫy, ǫξ−ǫη))

N ≺ ≺ǫ2gǫy,ǫη(t, τ )(1+

ǫ4·g

ǫx,ǫξ(ǫx−ǫy, ǫξ−ǫη))

N = =ǫ2gǫy,ǫη(t, τ )(1+

ǫ4 ·ǫ 4₍ǫ_g₎φ

x,ξ(x−y, ξ−η))

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whenǫGx(x−y)≤r , that is

ǫ_g

x,ξ(t, τ )≺ǫgy,η(t, τ )(1+(ǫg)φx,ξ(x−y, ξ−η))

N_,

whenǫGx(x−y)≤r , for all 0< ǫ≤1.

PROPOSITION2. We have that

hǫ ∈S(|hǫ|,ǫg),

uniformly with respect to 0< ǫ _≤1.

Proof. Using Definition 5 we have that, for all k∈N,

sup

x,ξ

|hǫ| ǫ_g k (x, ξ ) |hǫ(x, ξ )| =

sup

x,ξ

sup

yj,ηj

|h(ǫk)_Q((x,ξ );(y1,η1),...,(yk,ηk))| k

j=1 ǫ_g

x,ξ(yj,ηj)1/2

|hǫ(x, ξ )| =

=sup

x,ξ

sup

yj,ηj

|h(k)_Q((ǫx,ǫξ );(y1,η1),...,(yk,ηk))| k

j=1gǫx,ǫξ(ǫyj,ǫηj)

1/2

|hǫ(x, ξ )| =

=sup

x,ξ

sup

yj,ηj

ǫk|h(k)((ǫx,ǫξ );(y1,η1),...,(yk,ηk))|

ǫkQk

j=1gǫx,ǫξ(yj,ηj)

1/2

|hǫ(x, ξ )| =

=sup

x,ξ

|h_|g_k(ǫx, ǫξ ) |hǫ(x, ξ )| =

=sup

x,ξ

|h|gk(ǫx, ǫξ ) |h(ǫx, ǫξ )_| =supx,ξ

|h|gk(x, ξ )

|h(x, ξ )_| <+∞,

uniformly with respect to 0< ǫ _≤1, since h_∈S(_|h_|,g)by hypothesis.

We can now state the following proposition:

PROPOSITION3. Let us suppose that there exists a real numberγ >0 such that (10) gx,ξ(y, η)≺h(x, ξ )−γg_xφ_,ξ(y, η),

for all(x, ξ ), (y, η)_∈Rn_×Rn. Then we have that

ǫ_g

x,ξ(y, η)≺(hǫ(x, ξ )ǫ−

γ₎−γ₍ǫ_g₎φ x,ξ(y, η),

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Proof. Recalling that

(ǫg)φ_x_,ξ(y, η)₌ǫ−2gφ_ǫ_x_,ǫξ(y, η),

and using (10) we have:

ǫ_g

x,ξ(y, η)

(ǫ_g₎φ

x,ξ(y, η)

= ǫ

2_g

ǫx,ǫξ(y, η)

ǫ−2_gφ

ǫx,ǫξ(y, η)

≺ ≺h(ǫx, ǫξ )−γǫ4₌ =(hǫ(x, ξ )·ǫ−

γ₎−γ_.

Due to this proposition, from now on we will work with the following symbol: Hǫ(x, ξ )=ǫ−

γ_h ǫ(x, ξ ).

Now we formulate Theorem 1 in this new context:

THEOREM 3. Consider a positive and formally hypoelliptic symbol h _∈ S(h,g)

and assume that there exists a positive real numberγ such that (10) is satisfied and h(x, ξ )_{→ +∞}as_|x_{| + |}ξ_{| → +∞}.

Then the operator Hw

ǫ (x, ξ ), corresponding to the new symbol Hǫ(x, ξ ), is

semi-bounded from below and essentially self-adjoint in L2(Rn)and its closure has discrete spectrum diverging to_+∞.

Proof. This is an immediate consequence of Theorem 1 .

REMARK2. Thanks to Theorem 3 we can define the counting function of the clo-sure of the operator Hw

ǫ : N_H

ǫ(τ )=number of eigenfunctions of the closure of Hǫw,

corre-sponding to eigenvalues less than or equal toτ.

Before claiming our results, we have to state the following theorem, which is a direct consequence of Theorem 2.

THEOREM4. Under the hypotheses of Theorem 1, assume that there exists k >0 such that

h−k ∈L1(R2n).

Then for all 0< δ <γ₃ we have

N_H

(10)

asτ → +∞, uniformly with respect to 0< ǫ≤1, where

W_ǫ_{(τ )}₌₍₂_{π )}−n Z Z Hǫ(x,ξ )≤τ

d x dξ

and

R_ǫ(τ )₌ Wǫ(τ+τ

1−δ₎₋_W

ǫ(τ−τ1−δ)

W_ǫ_{(τ )} .

Proof. By means of a change of coordinates we immediately obtain that

kH_ǫ−k_kL1 =ǫ 4k

|hǫ(x, ξ )|−kd x dξ =

=ǫ4kγ

h(ǫx, ǫξ )−kd x dξ ₌ =ǫ4kγ−2n

h(x, ξ )−kd x dξ _≤ ≤

h(x, ξ )−kd x dξ = kh−k_kL1, if we take4k_γ ₋2n_≥0, that is

γ ≤ 2k

n .

Therefore we obtain that the integrability of the symbol h(x, ξ )implies the integrability of the new symbol Hǫ(x, ξ )and that the L1norm of Hǫ−k is uniformly bounded with

respect to 0< ǫ _≤1. The rest of the proof is an immediate consequence of Theorem 2, Proposition 1 and Proposition 2.

Now we can state and prove our main result.

THEOREM5. Let N_ǫ_{(τ )}_{be the counting function associated to the operator h}w

ǫ .

Let

W_{(τ )}₌₍₂_{π )}−n Z Z h(x,ξ )≤τ

d x dξ.

Letτ be not a critical value of h(x, ξ ), that is grad h(x, ξ )₆₌0 on the surface_{(x, ξ ): h(x, ξ )₌τ_}. Under the hypotheses of Theorem 4, we have the following asymptotic formula forN_ǫ(τ ):

(11) N_ǫ(τ )₌ǫ−2n(W(τ )₊O(ǫθ)),

asǫ_→0, for all 0< θ < 4₃.

COROLLARY1. The asymptotic formula (11) is valid whenτ belongs to the com-plementary of a set of zero measure.

(11)

Proof of Corollary 1. We know from Theorem 5 that in the asymptotic formula (11) we must exclude the critical values of h(x, ξ ). Corollary 1 is an immediate consequence of Sard Theorem, according to which the set of the critical values of h(x, ξ )has zero measure.

Proof of Theorem 5. SinceN_ǫ(τ )is the counting function associated to the operator hw

ǫ , then it is clear that Hǫw has exactlyNǫ(ǫ

γ_{τ )}_{eigenvalues less than or equal to}_τ

and that

(2π )−n Z Z

h(ǫx,ǫξ )ǫ−γ4_≤_τ

d x dξ ₌ǫ−2nW(ǫγ4_{τ ).}

Thanks to Theorem 4, we obtain that for all 0 < δ < γ₃ there exists a real number Cδ>0 such that

|N_ǫ(ǫ4γ_{τ )}₋_ǫ−2n_W_(ǫ4γ_{τ )}_{| ≤} ≤Cδǫ−2n(W(ǫ

γ_(τ ₊_τ1−δ₎₋W_(ǫγ4_(τ₋_τ1−δ_)),

asτ _{→ +∞}, for all 0< ǫ_≤1. Lettingǫγ4_τ ₌_λ_{we obtain:}

(12) |

N_ǫ_(λ)₋_ǫ−2n_W_(λ)

| ≤ ≤Cδǫ−2n(W(λ(1+ǫ

4δ

γ_λ−δ₎₎₋W(λ(1₋ǫ4γδ_λ−δ₎₎

for 0< δ < γ₃ andǫ_→0.

From [9], 28.7, we know that whenτ is not a critical value of h(x, ξ )thenW(τ )is differentiable with

W′_{(τ )}₌₍₂_{π )}−n Z V(τ )

d S

|grad h_| where

V(τ )_{= {}(x, ξ ): h(x, ξ )₌τ_}

and d S is the area element of the surface V(τ ). Therefore whenλis not a critical value of h, using Taylor’s formula, we can state that

W_(λ₊_λ1−δ_ǫ4γδ₎₌W_(λ)₊_λ1−δ_ǫ4γδ₍W′_(λ)₊_o₍₁₎₎

and

W_(λ₋_λ1−δ_ǫ4γδ₎₌W_(λ)₋_λ1−δ_ǫ4γδ₍W′_(λ)₊_o₍₁_)),

asǫ→0. Thus

(12)

asǫ→0, and then

(13) W_(λ(₁₊_ǫ4γδ_λ−δ₎₎₋W_(λ(₁₋_ǫ4γδ_λ−δ₎₎₌_O_(ǫ4γδ_),

asǫ_→0.

Since 0< δ <γ₃, using (12) and (13) we conclude that

N_ǫ_(λ)₌_ǫ−2n₍_W_(λ)

+O(ǫθ)),

asǫ_→0, for all 0< θ < 4₃.

REMARK3. As pointed out in the introduction, quasiclassical operators could be also in the form hw₍_x_{, ǫ}2_{ξ )}_{. In this case they are also called semiclassical operators} andǫ2 is the Planck constant. Everything we obtained about quasiclassical opera-tors is true also for semiclassical operaopera-tors. Indeed the property of beingφ temper-ate is invariant under symplectic changes of coordintemper-ates and the two transformations

(x, ξ )_→ (ǫx, ǫξ )and(x, ξ )_→ (x, ǫ2ξ )are symplectically equivalent. It is enough to consider the map

T :R2n_→R2n

(x, ξ )7→T(x, ξ )=(ǫx, ǫ−1ξ ).

For further details see [6], Lemma 18.5.8. In the case of semiclassical operators we can actually keep the old metric:ǫGx(y)=Gx(y).

4. An example

At the end of our paper we give an example of hypoelliptic operator satisfying our hypotheses.

PROPOSITION4. Let us consider

• a real polynomial p(ξ )vanishing at the origin and hypoelliptic, i. e. such that lim

|ξ|→+∞p(ξ )= +∞

and

∇p(ξ )_≺ p(ξ )1−ρ, for p(ξ )_≥1,

with 0< ρ_≤1,

• a positive smooth function q(x)such that

– h(x, ξ )= p(ξ )+q(x) >0,∀(x, ξ )∈Rn_×_Rn_,

– there exists 0 ≤ ν < ρ such that for allα ∈ Nn we have Dαq(x) ≺

(13)

– there exists k1>0 such that q(x)−k1 _∈ _L1₍_Rn₎_. Then hw

ǫ(x, ξ )= pwǫ (ξ )+q(ǫx)satisfies the hypotheses of Theorem 5.

Proof. We give a brief sketch of the proof. First we prove that there exists k>0 such that

(14) h−k _∈L1(R2n_).

From hypotheses of Proposition 4, it is clear that there exists R>0 such that p

p(ξ )q(x)_≤ p(ξ )₊q(x), _∀ξ /_∈B(R), _∀x _∈Rn,

where B(R) is the open ball with center at the origin and radius R. Therefore, in order to verify (14), it suffices to show that there exists k2 > 0 such that p−k2 _∈ L1(Rn _\ B(R)). We have that p _{→ +∞}, as _|ξ_{| → +∞}. Then, because p is a polynomial, it follows from Tarski-Seidenberg Theorem that there exists k0>0 such that

(15) p(ξ )_{≻ |}ξ_|k0_, _{∀ |}_ξ_{| ≥}_R_. In fact,

(s,t, ξ )∈R_×R_×Rn: s= |ξ|2, p(ξ )=t is algebraic inR2+n, and therefore

f(s)= inf

|ξ|2₌_s p(ξ )=inf

t :∃s∃ξ (s,t, ξ )∈ E <+∞

is semi-algebraic. Moreover f(s) → +∞, as s → +∞; hence (15) follows from Corollary A.2.6 of [5].

It is clear that estimate (15) implies the existence ofκ2.

Then it suffices to verify the hypotheses of Theorem 3. Consider the metric gx,ξ(y, η)=h(x, ξ )2ν|y|2+h(x, ξ )−2ρ|η|2.

Then one verifies that g and h are locallyφtemperate, for example see [12]. Moreover one can show that h_{→ +∞}, as_|x_{| + |}ξ_{| → +∞}, h_∈S(h,g)and (10) is verified. We omit the details.

For example we can take (see [1], section 1.1) p(ξ )₌ξ₁2k₋ξ₂2k−1

+ξ₁2kξ₂2k−2,

q(x)=expx₁2m1+x₂2m2,

(14)

References

[1] BOGGIATTOP., BUZANOE.ANDRODINOL., Global hypoellipticity and spec-tral theory, vol. 92, Akademie Verlag, Berlin 1996.

[2] DENCKER N., The Weyl calculus with locally temperate metrics and weights, Archiv f¨or Math. 24 (1986), 59–79.

[3] H ¨ORMANDER L., On the asymptotic distribution of the eigenvalues of pseudod-ifferential operators inRn, Archiv f¨or Math. 17 (1979), 297–313.

[4] H ¨ORMANDER L., The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359–443.

[5] H ¨ORMANDERL., The analysis of linear partial differential operators II, Grundle-heren der mathematischen Wissenschaften, vol. 257, Springer-Verlag, Berlin 1983.

[6] H ¨ORMANDER L., The analysis of linear partial differential operators III, Grundleheren der mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin 1985.

[7] MARTINEZ A., An Introduction to Semiclassical Analysis, 102 pages, January 1998.

[8] ROBERTD., Autour de l’Approximation Semi-Classique, Progress in Mathemat-ics, vol. 68, Birkh¨auser, Boston,Basel,Stuttgart 1987.

[9] SHUBINM.A., Pseudodifferential operators and spectral theory, second edition, Springer-Verlag, Berlin 2001.

[10] TAYLORM.E., Pseudodifferential Operators, vol. 34, Princeton University Press, Princeton, New Jersey, 1981.

[11] ZIGGIOTOA., Comportamento asintotico dello spettro di operatori multi-quasi-ellittici di tipo Schr¨odinger, Boll. U.M.I. IV-A (2001), no. 8, 567–569.

[12] ZIGGIOTOA.AND BUZANOE., Weyl formula for multi-quasi-elliptic operators of Schr¨odinger type, Annali di Matematica Pura e Applicata (2001), no. 180, 223–243.

[13] ZIGGIOTO A. AND BUZANO E., Weyl formula for hypoelliptic operators of Schr¨odinger type, Proc. Amer. Math. Soc., 131 (1), 265–274.

(15)

AMS Subject Classification: 35P20, 47B06.

ANDREA ZIGGIOTO Dipartimento di Matematica Universit`a di Torino Via Carlo Alberto 10 10123 Torino, ITALY

e-mail:[email protected]

(16)

(1)

Proof of Theorem 5. SinceN_ǫ(τ )is the counting function associated to the operator hw

ǫ , then it is clear that Hǫw has exactlyNǫ(ǫ

γ_{τ )}_{eigenvalues less than or equal to}_τ

and that

(2π )−n

Z Z

h(ǫx,ǫξ )ǫ−γ4_≤_τ

d x dξ ₌ǫ−2nW(ǫγ4_{τ ).}

Thanks to Theorem 4, we obtain that for all 0 < δ < γ₃ there exists a real number Cδ>0 such that

|N_ǫ(ǫ4γ_{τ )}₋_ǫ−2n_W_(ǫ4γ_{τ )}_{| ≤}

≤Cδǫ−2n(W_(ǫγ4_(τ ₊_τ1−δ₎₋W_(ǫγ4_(τ₋_τ1−δ_)),

asτ _{→ +∞}, for all 0< ǫ_≤1. Lettingǫγ4_τ ₌_λ_{we obtain:}

(12) |

N_ǫ_(λ)₋_ǫ−2n_W_(λ) | ≤ ≤Cδǫ−2n(W(λ(1+ǫ

4δ

γ_λ−δ₎₎₋W(λ(1₋ǫ4δγ_λ−δ₎₎

for 0< δ < γ₃ andǫ_→0.

From [9], 28.7, we know that whenτ is not a critical value of h(x, ξ )thenW(τ )is differentiable with

W′_{(τ )}₌₍₂_{π )}−n Z V(τ )

d S |grad h_| where

V(τ )_{= {}(x, ξ ): h(x, ξ )₌τ_}

and d S is the area element of the surface V(τ ). Therefore whenλis not a critical value of h, using Taylor’s formula, we can state that

W_(λ₊_λ1−δ_ǫ4δγ₎₌W_(λ)₊_λ1−δ_ǫ4δγ₍W′_(λ)₊_o₍₁₎₎

and

W_(λ₋_λ1−δ_ǫ4δγ₎₌W_(λ)₋_λ1−δ_ǫ4δγ₍W′_(λ)₊_o₍₁_)),

asǫ→0. Thus

(2)

asǫ→0, and then

(13) W_(λ(₁₊_ǫ4δγ_λ−δ₎₎₋W_(λ(₁₋_ǫ4δγ_λ−δ₎₎₌_O_(ǫ4δγ_),

asǫ_→0.

Since 0< δ <γ₃, using (12) and (13) we conclude that N_ǫ_(λ)₌_ǫ−2n₍_W_(λ)

+O(ǫθ)),

asǫ_→0, for all 0< θ < 4₃.

(x, ξ )_→ (ǫx, ǫξ )and(x, ξ )_→ (x, ǫ2ξ )are symplectically equivalent. It is enough to consider the map

T :R2n_→R2n

(x, ξ )7→T(x, ξ )=(ǫx, ǫ−1ξ ).

For further details see [6], Lemma 18.5.8. In the case of semiclassical operators we can actually keep the old metric:ǫGx(y)=Gx(y).

4. An example

At the end of our paper we give an example of hypoelliptic operator satisfying our hypotheses.

PROPOSITION4. Let us consider

• a real polynomial p(ξ )vanishing at the origin and hypoelliptic, i. e. such that lim

|ξ|→+∞p(ξ )= +∞

and

∇p(ξ )_≺ p(ξ )1−ρ, for p(ξ )_≥1,

with 0< ρ_≤1,

• a positive smooth function q(x)such that

– h(x, ξ )= p(ξ )+q(x) >0,∀(x, ξ )∈Rn_×_Rn_,

– there exists 0 ≤ ν < ρ such that for allα ∈ Nn we have Dαq(x) ≺ q(x)1+ν|α|,∀x∈Rn,

(3)

– there exists k1>0 such that q(x)−k1 ∈ L1(Rn). Then hw

ǫ(x, ξ )= pwǫ (ξ )+q(ǫx)satisfies the hypotheses of Theorem 5.

Proof. We give a brief sketch of the proof. First we prove that there exists k>0 such that

(14) h−k _∈L1(R2n_).

From hypotheses of Proposition 4, it is clear that there exists R>0 such that p

p(ξ )q(x)_≤ p(ξ )₊q(x), _∀ξ /_∈B(R), _∀x _∈Rn,

where B(R) is the open ball with center at the origin and radius R. Therefore, in order to verify (14), it suffices to show that there exists k2 > 0 such that p−k2 _∈

L1(Rn _\ B(R)). We have that p _{→ +∞}, as _|ξ_{| → +∞}. Then, because p is a polynomial, it follows from Tarski-Seidenberg Theorem that there exists k0>0 such that

(15) p(ξ )_{≻ |}ξ_|k0_, _{∀ |}_ξ_{| ≥}_R_.

In fact,

(s,t, ξ )∈R_×R_×Rn: s= |ξ|2, p(ξ )=t is algebraic inR2+n, and therefore

f(s)= inf

|ξ|2₌_s p(ξ )=inf

t :∃s∃ξ (s,t, ξ )∈ E <+∞

is semi-algebraic. Moreover f(s) → +∞, as s → +∞; hence (15) follows from Corollary A.2.6 of [5].

It is clear that estimate (15) implies the existence ofκ2.

Then it suffices to verify the hypotheses of Theorem 3. Consider the metric gx,ξ(y, η)=h(x, ξ )2ν|y|2+h(x, ξ )−2ρ|η|2.

For example we can take (see [1], section 1.1)

p(ξ )₌ξ₁2k₋ξ₂2k−1

+ξ₁2kξ₂2k−2,

q(x)=expx2m1

1 +x 2m2