Basic definitions and results

Quasiclassical analysis for hypoelliptic operators 87 operators 1 or 2 when ǫ → 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 function N τ as τ → +∞ see Section 2. Then we consider the corresponding quantum observ- ables h w ǫ x , ǫξ and we obtain a quasiclassical asymptotic formula for the associated counting function N ǫ τ as ǫ → 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 ≺ gx, ∀x ∈ A, if there exists a constant C such that f x ≤ Cgx, ∀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 ξ j y j − x j η j be the standard symplectic form on R n × R n . D EFINITION 1. A Riemannian metric g x ,ξ y, η on R n × R n is slowly varying if there exist a positive real number r such that g x ,ξ y, η ≺ g t,τ y, η ≺ g x ,ξ y, η, for all x , ξ , t, τ , y, η ∈ R n × R n such that g x ,ξ x − t, ξ − τ ≤ r. D EFINITION 2. A positive function mx , ξ is said to be g continuous if there is a positive real number r such that my, η ≺ mx, ξ ≺ my, η, for all x , ξ , y, η ∈ R n × R n such that g x ,ξ x − y, ξ − η ≤ r. 88 A. Ziggioto D EFINITION 3. A Riemannian metric g x ,ξ y, η on R n × R n is locally φ temperate if it is slowly varying and there exist two positive real numbers r and N and a slowly varying metric G x y on R n such that G x y ≤ g x ,ξ y, η, for all x , ξ , y, η ∈ R n × R n , and 3 g x ,ξ z, ζ ≺ g y,η z, ζ 1 + g φ x ,ξ x − y, ξ − η N , for all x , ξ , y, η, z, ζ ∈ R n × R n such that G x x − y ≤ r. The quadratic form g φ x ,ξ y, η appearing in 3 is the dual metric g φ x ,ξ y, η = sup{φy, η; z, ζ 2 : g x ,ξ z, ζ = 1}. D EFINITION 4. A positive function mx , ξ is locally φ, g temperate if it is g con- tinuous and there exist two positive real numbers r and N such that mx , ξ ≺ my, η1 + g φ x ,ξ x − y, ξ − η N , for all x , ξ , y, η ∈ R n × R n such that G x x − y ≤ r. Now we recall the definition of the class of symbols of Weyl-H¨ormander Sm, g. D EFINITION 5. Let g be a slowly varying Riemannian metric. Let m be a g con- tinuous function. Then we say that a function h ∈ Sm, g if it is smooth and sup x ,ξ |h| g k x , ξ mx , ξ +∞, ∀k ∈ N, where |h| g k x , ξ = sup y j ,η j |h k x , ξ ; y 1 , η 1 , . . . , y k , η k | Q k j = 1 g x ,ξ y j , η j 12 and h k x , ξ ; y 1 , η 1 , . . . , y k , η k is the k-th differential of h at x , ξ . We can introduce here the operators we deal with in the next section. D EFINITION 6. A differential operator h w is formally hypoelliptic † if its Weyl sym- bol hx , ξ satisfies the following conditions: † This defi nition is not to be confused with Defi nition 2.3 of Chapter III of [10] Quasiclassical analysis for hypoelliptic operators 89 1. h is a smooth function such that hx , ξ 6= 0, ∀x, ξ ∈ R n × R n ; 2. there exists a locally φ temperate metric g x ,ξ y, η on R n × R n such that |h| is locally φ, 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. T HEOREM 1. Consider a positive and formally hypoelliptic symbol h ∈ Sh, g and assume that there exists a positive real number γ such that g x ,ξ y, η ≺ hx, ξ −γ g φ x ,ξ y, η, for all x , ξ , y, η ∈ R n × R n , then h w is semi-bounded from below and essentially self-adjoint in L 2 R n . Moreover, if 4 hx , ξ → +∞, as |x| + |ξ| → +∞, then the closure H of h w in L 2 R n 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 g x ,ξ g φ x ,ξ ≤ 1. In our case, from the fact that there exists γ 0 such that g x ,ξ y, η ≺ hx, ξ −γ g φ x ,ξ y, η and that hx , ξ → +∞ as |x| + |ξ| → +∞, it follows that sup x ,ξ g x ,ξ g φ 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 that ˜g φ = √ C g φ . Therefore sup x ,ξ ˜g x ,ξ ˜g φ x ,ξ = sup x ,ξ g x ,ξ √ C √ C g φ x ,ξ = 1 C sup x ,ξ g x ,ξ g φ x ,ξ ≤ 1. 90 A. Ziggioto 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 to τ . T HEOREM 2. Under the hypotheses of Theorem 1 assume there exists κ 0 such that 6 h −κ ∈ L 1 . Then for all 0 δ γ 3 we have 7 N τ = Wτ 1 + O Rτ , as τ → +∞, where 8 W τ = 2π −n Z h≤τ d x dξ, and 9 Rτ = W τ + τ 1−δ − W τ − τ 1−δ W τ . R EMARK 1. Estimate 7 is known as Weyl formula. Proof. This is Theorem 2 of [13].

3. Quasiclassical Analysis of Hypoelliptic Operators