Hypoellipticity, Spectral theory and Witten Laplacians

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1 Hypoellipticity, Spectral theory and Witten Laplacians Bernard Helffer March 3, 2003 Contents 1 Introduction 1 2 Kohn s proof of the hypoellipticity Vector fields and Hörmander condition Main results Kohn s proof Compactness criteria Helffer-Morame proof About Witten Laplacians and Schrödinger operators Maximal hypoellipticity and nilpotent groups Maximal hypooellipticity Nilpotent Lie algebras Representation theory Rockland s conjecture Spectral properties Rothschild-Stein lifting and towards a general criterion Folland s result Discussion on Rothschild-Stein and Helffer-Métivier-Nourrigat 25 1

2 5 Maximal hypoellipticity for systems and spectral theory for Witten Laplacians Introduction Microlocal hypoellipticity and semi-classical analysis Around the proof and spectral byproducts for the Witten Laplacians Semi-classical analysis for the Schrödinger operator Harmonic approximation Introduction The case of dimension Quadratic models The harmonic approximation, analysis in large dimension Decay of eigenfunctions Introduction Energy inequalities The Agmon distance Decay of eigenfunctions for the Schrödinger operator Splitting between the two first eigenvalues Interaction matrix Semi-classical analysis and Witten Laplacians Preliminaries The Morse inequalities The Witten complex Semi-classical analysis of the Witten Laplacian Semi-classical analysis of the Witten complex Spectral properties of the Witten-Laplacians in connexion with Poincaré inequalities for Laplace integrals Laplace integrals and Dirichlet Laplacians Links with the Witten-Laplacian Generalities and results of Helffer-Nier Spectral properties of the Witten Laplacian on R d Non negative potential near Main assumptions Sufficient conditions for the compactness of the resolvent and for the Poincaré inequality

3 8.5.3 Necessary conditions Non positive potential near infinity Abstract These preliminary notes correspond to the material prepared for a minicourse of 8 hours given at Rennes, for the Workshop équations cinétiques, hypoellipticité et Laplacien de Witten. We do not feel specialist in the first subject but we have tried to present in this course the strong links existing (at the technical level) between techniques developed originally for the hypoellipticity and the spectral techniques appearing in the analysis of the Witten Laplacian. 1 Introduction We are mostly interested in hypoellipticity problems concerning : the Hörmander s operator j X2 j but also on other operators of the type : j X2 j +i jk c jk[x j, X k ]. In parallel, we are interested in the spectral properties of the operator (0) Φ,h := h2 + Φ 2 h Φ, for fixed h > 0, or semi-classically as h 0. This operator can be recognized as a Witten Laplacian on 0-forms and we shall analyze the links betweeen its properties and the properties of its brother on the 1-forms : (1) Φ,h := (0) Φ,h I + 2h Hess Φ. These notes intend to give a flavor on various techniques due to Hörmander, Kohn, Helffer-Morame, Helffer-Nourrigat, Helffer-Sjöstrand, Witten and we will report on recent results obtained by Hérau-Nier, Helffer-Nier... New results are presented based on the use of Helffer-Nourrigat and Nourrigat techniques. These notes will hopefully be an introduction to more specialized talks given at this workshop by : M. Klein, O. Matte, M. Hairer, J.S. Moeller... Acknowledgements. I would like to thank M. Klein, F. Nier, J. Nourrigat and J. Moeller for discussions concerned with these notes. 3

4 2 Kohn s proof of the hypoellipticity. 2.1 Vector fields and Hörmander condition We consider p C real vector fields (X 1,, X p ) in a open set Ω of R n. If X and Y are two vector fields, the bracket of X and Y, denoted by [X, Y ] or ad XY, is defined by [X, Y ]f = X(Y f) Y (Xf). We note that [X, Y ] is a new vector field. We are interested in the case when the Hörmander condition is satisfied. Hörmander Condition. We say that the Hörmander condition is satisfied at x 0, if there exists r(x 0 ) 1 such that the vector space generated by the iterated brackets ( ad X) α X k at x 0 with α r 1 is R n. When r(x 0 ) = 1, we say that the system is elliptic and this imposes of course p n. Let us give typical examples. Heisenberg : Grushin : Nilpotent group G 4 : n = 3, p = 2, r = 2, X 1 = x, X 2 = x z + y, [X 1, X 2 ] = z. n = 2, r = 2, X 1 = x, X 2 = x y, [X 1, X 2 ] = y (2.1) n = 4, r = 3, X 1 = x, X 2 = 1 2 x2 t + x z + y [X 1, X 2 ] = x t + z, [X 1, [X 1, X 2 ]] = t. We say that the vector fields X j satisfy the Hörmander condition of rank r in an open set Ω if r min (x 0 ) r, for all x Ω. 4

5 2.2 Main results Theorem If the Hörmander condition is satisfied for some r in Ω, then the operator of type 1: L = X 2 j is hypoelliptic in Ω. This result is due to L. Hörmander. Remark This is a necessary condition in the case when the X j s are with analytic coefficients. The proof (due to Derridj) is based on Nagano s Theorem. Theorem If the Hörmander condition is satisfied for some r in Ω, then, for any K Ω, there exists C K such that ( ) u 2 1 r C j X j u u 2 0 where u s is the Sobolev norm corresponding to H s., u C 0 (Ω), with supp u K, (2.2) Remark One can actually show that this inequality implies the Hörmander condition of rank r. Remark There exists a microlocal version of this inequality which is due to Bolley- Camus-Nourrigat [BoCaNo]. Note also that we have the Hörmander s operator of type 2, corresponding to : p L = Xj 2 + X 0, j=1 where the vector fields (X 0, X 1,..., X p ) satisfy the Hörmander condition. The simplest example is the heat equation: x1,...,x n 1 + xn A more typical result is : 2 x + x y. 5

6 2.3 Kohn s proof The starting point is to get p X j u 2 C ( Re Lu u + u 0) 2. (2.3) j=1 This inequality is immediate by integration by parts if one observes that for a C function c j and that : X j = X j + c j, (2.4) 2 Re X 0 u u = c 0 u u C u 2 0. A Cauchy-Schwarz argument permits then to conclude. We observe that this inequality of course implies : and p X j u 2 C ( Lu u + u 0) 2, (2.5) j=1 p X j u 2 C ( Lu u 0) 2. (2.6) j=1 Note that there is a lost of information in (2.6) in comparison with (2.3). There is a general proof establishing that the subelliptic estimate (or some weaker subelliptic estimate) joint with this inequality gives the hypoellipticity. The critical point in this part of the proof that we omit is the control of commutators of L with pseudo-differential operators. Note that the two inequalities (2.2) and (2.6) imply, that for some ɛ > 0 : u 2 ɛ C ( Lu u 2 0), (2.7) but this inequality alone is not enough for proving hypoellipticity. We will concentrate our lecture on Kohn s proof of some subelliptic (not optimal) subelliptic estimate whose proof is simpler than the initial proof of Hörmander and permits other extensions.. 6

7 We now concentrate on the proof of the subelliptic estimate which is written in the form : u 2 ɛ C( Lu u 2 0, u C 0 (V ), (2.8) where ɛ > 0 and V is a fixed open set containing the point in the neighborhood of which we want to show the hypoellipticity. Although the general theory of pseudo-differential operators is not completely necessary, let us briefly recall that the pseudo-differential operators are operators which are defined by u Op (a)u with : Op (a)u(x) := 1 (2π) n exp ix ξ a(x, ξ)û(ξ)dξ. Here (x, ξ) a(x, ξ) is a symbol which admits an expansion in homogeneous terms with respect to the ξ variables : a(x, ξ) j 0 a m j (x, ξ), with a l (x, λξ) = λ l a l (x, ξ). The real m is called the degree of the symbol (or of the corresponding pseudodifferential). Actually we need uniquely a composition of operators which are multiplications by C functions, differentiations and the family of convolution operators Λ s where Λ s corresponds to the symbol ξ s = (1 + ξ 2 ) s 2. When s = 1, we simply write Λ. The important point is that the composition of a pseudo-differential operators of order m 1 and m 2 is a pseudodifferential operator of order (m 1 + m 2 ) whose main symbol is the product of the two main symbols. One important point is that the class of pseudodifferential operators of order 0 is an algebra. Now let P be the set of all pseudo-differential operators of order 0 such that if P P, then there exists ɛ > 0 and C > 0 such that : P u 2 ɛ C ( Lu u 2 0), u C 0 (V ). (2.9) This set satisfies the following properties : 7

8 Property 1.. (P1) P is a left and right ideal in the set of all pseudo-differential operators of order 0. Property 2.. (P2) P is stable by taking the adjoint. Property 3.. (P3) If P P, then X j Λ 1 P for j = 0,, p. Property 4.. (P4) If P P then [X j, P ] P for j = 0,, p. Let us first observe that one can prove inductively starting from (P 4) that Y i1 i p Λ 1 P. Let us for example show that [X j, X k ]Λ 1 belongs to P. We know that [X j, X k Λ 1 ] has the property. But Now [X j, X k Λ 1 ] = [X j, X k ]Λ 1 + X k [X j, Λ 1 ]. X k [X j, Λ 1 ] = X k Λ 1 (Λ[X j, Λ 1 ]), and (Λ[X j, Λ 1 ]) is a pseudo-differential operator of order 0. Using (P 1), we get that X k Λ 1 (Λ[X j, Λ 1 ]) belongs to P. Hence, using Hörmander condition of rank r, we obtain that P contains any pseudodifferential operator of order 0. It remains to prove the properties (P j ). Proof of (P 1). That it is a left ideal is trivial. That it is a a right ideal can then be deduced using property (P 2 ). Proof of (P 2 ). It is enough to observe that if P is a pseudo-differential operator of order 0, then Λ ɛ P u 2 = P Λ 2ɛ P u u = Λ ɛ P u 2 + ( P Λ 2ɛ P P Λ 2ɛ P ) u u. It is then enough to observe that (P Λ 2ɛ P P Λ 2ɛ P ) is a pseudodifferential operator of order 1 + 2ɛ 0 if ɛ

9 Proof of (P 3). For j > 1, we have Λ 1 X j u 2 ɛ C( X j u 2 + u 2 ), if ɛ 1. One can then conclude that Λ 1 X j P. Now we observe that X j Λ 1 = (Λ 1 X j ) + c j Λ 1. We then use (P 2) for getting that X j Λ 1 is in P. The treatment of the case of X 0 is a little more delicate. We start from : Λ 1 X 0 u X 0 u T u, (2.10) where T is a pseudodifferential operator of order 0. The second step is to write that X 0 u = Lu j X2 j u, which leads to the estimate : Λ 1 X 0 u Lu T u + j>0 X 2 j u T u. (2.11) The first term of the right hand side is OK. Let us show how to treat the second one. We have : X 2 j u T u = X j u X j T u C X j u ( X j T u + u ). Then we observe that X j T u ( X j u + [X j, T ]u ) C ( X j u + u ). So we have shown that Λ 1 X 0 belongs to P with ɛ = 1. Taking the adjoint 2 and observing that a pseudodifferential operator of strictly negative order belongs to P we get the result. Proof of (P 4). Let us start from a P such that (2.8) holds for some ɛ > 0. The case j > 0. Now consider : [X j, P ]u 2 δ = [X j, P ]u Λ 2δ [X j, P ]u = X j P u T 2δ u P X j u T 2δ u, 9

10 where T 2δ is a pseudo-differental operator of order 2δ. It then follows that : P X j u T 2δ u X j u P T 2δ u X j u 2 + P T 2δ u 2 X j u 2 + T 2δ P u 2 + C u 2 2δ 1. Similarly : X j P u T 2δ u P u X j T 2δ u + C P u 2δ u 0 C P u 2δ X j u 0 + P u, [X j, T 2δ ]u + C P u 2δ u 0. It remains to observe that : P u [X j, T 2δ ]u C P u 2δ u. So if δ min( 1, ɛ ), we are done for the j s such that j > The case j = 0 It is a little more delicate. We write X 0 P u, T 2δ u P u, T 2δ X 0 u + C P u 2 2δ + C u 2 0 P u T 2δ Lu + j>0 P u, T 2δ Xj 2 u + C P u 2 2δ + C u 2 0 Lu 2 + j>0 X jp u T 2δ X j u +C P ( u 2 2δ + C u C P u 2δ X j u C Lu 2 + ) j>0 X jp u 2 2δ + P u 2 2δ + u 2 0. It remains to treat X j P u 2 2δ. We claim that We have indeed X j P u 2 2δ C ( Lu P u 2 4δ + u 2 0). (2.12) X j P u 2 2δ = Λ ( 2δ X j P u 2 ) C j>0 X jλ 2δ P u 2 + P u 2 2δ. Then using (2.5), we get (we cheat a little because we do not care of the compact support) : j>0 X jλ 2δ P u 2 C ( LΛ 2δ P u Λ 2δ P u + P u 2 2δ + ) u 2 0 C ( [L, Λ 2δ P ]u Λ 2δ P u + P u 2 4δ + ) + Lu u 2 0 C ( X j u Q 4δ P u + P u 2 4δ + ) Lu u 2 0 C ( P u 2 4δ + Lu u 2 0). 10

11 This proves (2.12). Taking δ min( ɛ, 1 ), this is OK. 4 4 The treatment of P X 0 u, T 2δ u is similar. Remark If p = n, the operator j X2 j + j Y j 2 + it [X j, Y j ] for t < 1 is also hypoelliptic. The problem is that this is the case t = ±1 which we would like to understand better. 3 Compactness criteria. One of course knows that for a Schrödinger operator in the form + V on R n, with V C and semi-bounded then the operator is essentially selfadjoint and if in addition the potential V tend to + then it is with compact resolvent. The aim is to analyze the case when V does not tend to. 3.1 Helffer-Morame proof. We will analyze the problem for the family of operators : P A (h) = n (D xj A j (x)) 2 + j=1 p V l (x) 2. (3.1) l=1 Here the magnetic potential A(x) = (A 1 (x), A 2 (x),, A n (x)) is supposed to be C and the electric potential V (x) = j V j(x) 2 is such tat V j C. Under these conditions, the operator is essentially selfadjoint on C 0 (R n ). We note also that it has the form: n+p P A (h) = Xj 2 = j=1 n Xj 2 + j=1 p l=1 Y 2 l, with X j = (D xj A j (x)), j = 1,, n, Y l = V l, l = 1, p. In particular, the magnetic field is recovered by observing that B jk = 1 i [X j, X k ] = j A k k A j, j, k = 1,, n. 11

12 We start by two trivial easy cases. First we consider the case when V +. In this case, it is well known that the operator is with compact resolvent.(see the argument below). On the opposite, we assume that V = 0 and consider the case when n = 2 and when V = 0. We assume moreover that B(x) = B Then one immediatly observe the following inequality : B(x) u(x) 2 dx X 1 u 2 + X 2 u 2 = P A u, u. (3.2) Under the condition that B(x) +, this implies that the operator is with compact resolvent. Example 3.1. A 1 (x 1, x 2 ) = x 2 x 2 1, A 2 (x 1, x 2 ) = x 1 x 2 2. A sufficient condition is indeed to show that the form domain of the operator D(q A ) which is defined by : satisfies : D(q A ) = {u L 2 (R n ), X j u L 2 (R n ), for j = 1,, n + p}. (3.3) D(q A ) L 2 ρ(r n ), (3.4) for some ρ tending to as x. For treating more general situations, we introduce the quantities : m q (x) = x α V l + x α B jk (x). (3.5) l j α =q α =q 1 It is easy to reinterpret this quantity in terms of commutators of the X j s. When q = 0, the convention is that m 0 (x) = l V l (x). (3.6) Let us also introduce r m r (x) = 1 + m q (x). (3.7) q=0 Then the criterion is 12

13 Theorem Let us assume that there exists r such that and Then P A (h) is with compact resolvent. m r+1 (x) Cm r (x), (3.8) m r (x) +, as x +. (3.9) Remark 3.3. It is shown in [Mef], that one can get the same result under the weaker assumption that m r+1 (x) Cm r (x) 1+δ, (3.10) where δ [0, 1[. This result is optimal according to a counterexample by A. Iwatsuka. Other generalizations are given in [She] (Corollary 0.11) (see also references inside) and [KonShu] for a quite recent contribution with other references. One can for example replace j V j 2 by V and the conditions on the m j can be reformulated in terms of the variation of V and B in suitable balls. Remark If p = n, the operator j X2 j + j Y j 2 + it [X j, Y j ] for t < 1 is also with compact resolvent. The problem is that this is the case t = ±1 which appears in the Witten Laplacian case. Before to enter in the core of the proof, we observe that we can replace m r (x) by an equivalent C function Ψ(x) which has the property that : D α x Ψ(x) C α Ψ(x). In the same spirit as in Kohn s proof, les us introduce for all s > 0 Definition We denote by M s the space of C functions T such that there exists C s such that : Ψ 1+s T u 2 C s ( PA u u + u 2), u C 0. (3.11) We observe that and we will show the V l M 1, (3.12) 13

14 Lemma 3.6. [X j, X k ] M 1 2, j, k = 1,, n. (3.13) Another claim is contained in Lemma If T is in M s and α x T C α Ψ then [X k, T ] M s 2, when α = 1 or α = 2. Assuming these two lemmas, then it is clear that We get indeed from (3.12) and Lemma 3.7 that and we get from Lemmas 3.6 and 3.7 that Ψ(x) M 2 r. (3.14) α x V l M 2 α, (3.15) α x B jk M 2 ( α +1). (3.16) The proof is then easy. Proof of Lemma 3.6. We start from the identity (and observing that X j = X j ) : Ψ 1 2 [X j, X k ]u 2 = (X j X k X k X j )u Ψ 1 [X j, X k ]u = X k u X j Ψ 1 [X j, X k ]u X j u X k Ψ 1 [X j, X k ]u = X j u Ψ 1 [X k, X j ]X k u X k u Ψ 1 [X k, X j ]X k u + X j u [X k, Ψ 1 [X k, X j ]]u X k u [X j, Ψ 1 [X k, X j ]]u. (3.17) If we observe that Ψ 1 [X k, X j ] and [X k, Ψ 1 [X k, X j ]] are bounded (look at the definition of Ψ), we obtain : Ψ 1 2 [Xj, X k ]u 2 C ( X k u 2 + X j u 2 + u 2). (3.18) This ends the proof of the lemma. 14

15 Proof of Lemma 3.7. Let T M s. For each k, we can write : Ψ 1+ s 2 [X k, T ]u 2 = Ψ 1+s (X k T T X k )u Ψ 1 [X k, T ]u = Ψ 1+s X k T u Ψ 1 [X k, T ]u Ψ 1+s T X k u Ψ 1 [X k, T ]u = Ψ 1+s T u Ψ 1 [X k, T ]X k u X k u Ψ 1 [X k, T ]Ψ 1+s T u + T u [X k, Ψ 2+s [X k, T ]]u = Ψ 1+s T u Ψ 1 [X k, T ]X k u X k u Ψ 1 [X k, T ]Ψ 1+s T u + Ψ 1+s T u Ψ 1 s [X k, Ψ 2+s [X k, T ]]u. (3.19) We now observe, according to the assumptions of the lemma and the properties of Ψ, that Ψ 1 s [X k, Ψ 2+s [X k, T ]] and Ψ 1 [X k, T ] are bounded. So finally we get : Ψ 1 2 [Xj, T ]u 2 C ( Ψ 1+s T u 2 + X k u 2 + u 2). (3.20) This ends the proof of the lemma. Example A classical example is the operator + x 2 1x 2 2 in two dimension. Another proof is as follow. Although the potential V = x 2 1x 2 2 is 0 along {x 1 = 0} or {x 2 = 0}, the estimate for the one-dimensional scaled harmonic oscillator gives + x 2 1x ( 2 x1 + x 2 2x1) 2 1 ( x2 + x 2 1x2) ( x 2 + x 1 ) and the operator + x 2 1x 2 2 has a compact resolvent. This example can actually treated by many approachs (see [Sim1] and [HelNo3], [HelMo]). Remark Helffer-Mohamed (=Morame) describe also in [HelMo] the essential spectrum when the compactness criterion is not satsified. We mention also the negative answer to the problem of finding magnetic bottles for Dirac due to Helffer-Nourrigat-Wang [HeNoWa] (see the book by B. Thaller on this question). 15

16 3.2 About Witten Laplacians and Schrödinger operators. Let us consider the Dirichlet Laplacian Of course, it is trivial to show that if (0) Φ := + Φ 2 φ. Φ(x) 2 Φ(x) +, as x +, (3.21) then the operator is with compact resolvent. But this is not optimal!! One can indeed play with the bracket argument. We have Xj u 2 + α Y j u 2 α [X j, Y j ]u u. (3.22) So interpolating, we get that : This gives : (0) Φ u u ((1 ɛ) Φ 2 + ( ɛ 1) Φ ) u(x) 2 dx. (3.23) (0) Φ u u (1 ɛ) ((1 + ɛ) Φ 2 Φ ) u(x) 2 dx, (3.24) for any ɛ ]0, 1[. So we have obtained the following proposition (see [BoDaHel], [Hel5]). Proposition Let us assume that there exists t ]0, 2[ such that Then the Witten Laplacian (0) Φ t Φ(x) 2 Φ(x) +, as x +. (3.25) is with compact resolvent. One should notice that for Φ = x 2 1x ε(x x 2 2), ε > 0, the total potential V = Φ 2 Φ goes to as x 1 + and x 2 = 0. Meanwhile the operator (0) Φ is positive by construction and Theorem 8.7 says that it has a compact resolvent if (and only if) ε > 0. The uncertainty principle plays a role in the lower bound of of (0) Φ. Remark One can also find criteria taking into account higher order brackets. [BoDaHel]. See 16

17 4 Maximal hypoellipticity and nilpotent groups 4.1 Maximal hypooellipticity We assume that we are given p vector fields satisfying the rank r Hörmander condition. If P is a non-commutative polynomial of degree m of vector fields with C coefficients, that is an operator of the form P := a α (x)x α ;, α m where α {1,, p} k, α = k, then we say that P is maximally hypoelliptic if X α u 2 C ( ) P u u 2 0 (4.1) α m One can show that this inequality (joint to the Hörmander condition) implies hypoellipticity. The fact that the operator L = j X2 j (or more generally L = j X2 j + X 0 ) is maximally hypoelliptic was proved for the first time by Rotschild-Stein [RoSt] using a technique of lifting of the problem on a nilpotent universal group. With a stronger property, there is a hope to characterize the maximally hypoelliptic operator (cf Helffer-Nourrigat book [HelNo3]). Our aim in this section is to give a flavor about the nature of the criteria and also to explain some byproducts of the proofs. These maximally hypoelliptic operators are more stable in the sense that this property depends only on the principal part P 0 := a α (x)x α. α =m 4.2 Nilpotent Lie algebras We refer to [No5] and [HelNo3] for a more comprehensive description of the theory. The first example of this type corresponds to homogeneous invariant operators on a stratified nilpotent Lie groups. 17

18 Let us explain this in the great lines. The starting point is an abstract Lie algebra G (think of a subalgebra of the up-triangular maps) admitting the following decomposition : G = r j=1g j, with the property that the G j s are vector subspaces in direct sum satisfying [G i, G j ] G i+j. Usually, we will consider the case when G is generated by G 1 but it could be necessary to consider other cases in order to treat the Hörmander operators of type 2. In the case of these algebras there exists a global diffeomorphism from G onto a simply connected group G via the exponential map (restriction of the exponential map on matrices). This gives a group structure on G denoted by. Typically when the algebra is of rank 2, we get a b = a + b + 1 [a, b]. 2 When the algebra is of rank 3, we have : a b = a + b [a, b] [a, [a, b]] + 1 [[a, b], b]. 12 One can then identify an element of G to a left invariant vector field on the group G (this will be done systematically later). In the same way, an element of the envelopping algebra U(G) can be identified to a left invariant operator on the group. On this lie algebra, we have a natural dilation δ t which has the property that δ t (X) = t j X, if X G j. It is then immmediate to define what is an homogeneous operator in the envelopping algebra. 4.3 Representation theory We now explain very briefly the Kirillov s theory. Starting from a unitary representation of the group G, we can always attach a representation of the 18

19 Lie algebra. Between these representations, the irreducible representations will play an important role. The Kirillov s theory permits to associate to any element of G an irreducible representation. Moreover the Kirillov theory says that any unitary representation can be represented in this way. Finally two irreducible representations are unitary equivalent if they belong to the same orbit. In order to be more precise, let us give the definition of induced representation (which is due to Mackey). We give here a rather pedestrian definition. The starting point is a subalgebra H G and a linear form in H such that : l([h, H]) = 0. Then we will associate a representation π l,h of the group G into L 2 (R k(l) ), which is uniquely defined modulo a unitary conjugation, where k(l) is the codimension of H in G. For this construction and using the nilpotent character, we can find an independent basis e 1,, e k such that any g can be written in the form and if g = h exp s k e k exp s k 1 e k 1 exp s 1 e 1, A j = H Re k Re k j+1, then A j 1 is an ideal of codimension 1 in A j. With this construction, one can get that g (s, h) is a global diffeomorphism from G onto R k H. The induced representation is given by : (π l,h (exp a)f)(t) = exp i l, h(t, a) f(σ(t, a)). Here h(t, a) and σ(t, a) are defined by : exp t k e k exp t 1 e 1 exp t k e k exp t 1 e 1 exp a = exp h(t, a) exp exp σ k (t, a)e k exp σ 1 (t, a)e 1. The corresponding representation of G is defined by : More explicitly, we get : (π l,h (a)f)(t) = d ds ((π l,h(exp sa)f)(t)) s=0. π l,h (a) = i l, h (t, a) + 19 k σ j(t, a) tj, (4.2) j=1

20 where h (t, a) = d ds (h(t, sa)) /s=0, σ (t, a) = d ds (σ(t, sa)) /s=0. There are two particular cases, which are interesting. When l = 0, we get the standard extension of the trivial representation of the subgroup H of G. It can be considered as a representation on L 2 (G/H). An intersting problem is to characterize the maximal hypoellipticity of π 0,H (P ) for P in U m (G) (elements of U(G) with degree m). We will see that Witten-Laplacians corresponding to a polynomial φ are of this type. The second point is when H is of maximal dimension, for a given l, with the above property. In this case, one can show that the representation is irreducible. This is the way one can construct all the irreducible representations. Starting this time from an l in G, one can construct a maximal subalgebra V l such that l([v l, V l ]) = 0. One can show that the codimension k(l) of V l is equal to 1 2 rank B l where B l is the two-form : G G (X, Y ) l[x, Y ]. For a G, we define by ( ad a) the adjoint of ad a : ( ad a) l(b) = l([a, b]). The group G acts naturally on G by : g gl = r k=0 1 ( ad a) kl k! with g = exp a, which is called the coadjoint action. So what we said of Kirillov s theory is that if l and l are on the same orbit, then the corresponding unitary representations are equivalent. Conversely two different orbits give two non equivalent irreducible representations, so Ĝ = G /G. 20

21 4.4 Rockland s conjecture The following theorem was conjectured by C. Rockland and proved by B.Helffer and J. Nourrigat [HelNo1, HelNo2] : Theorem An element P in U m (G) is hypoelliptic if and only if for any non trivial representation π of P π(p ) is injective in S π (defined below). For the basic example, the proof of the hypoellipticity of j Y j 2 becomes trivial. π(p )u = 0 implies π(y j )u = 0. This implies π(y )u = 0 for any Y G. A characterization of the irreducibility of π and the non-trivial assumption gives u = 0. An important ingredient is the proof of maximal inequalities of the type π(q)u C Q,π π(p )u, u S π (4.3) for all Q U m (G). A fundamental point for proving the maximal hypoellipticity is to show that the constant C Q,π can be chosen independent of π and to have it for Q belonging to a basis of U m (G). We can not explain here all the recursion argument, which is strongly related to Kirillov s theory, but we would like to emphasize on some ingredients of the proof which can give an interesting light to other problems. 4.5 Spectral properties The Sobolev spaces Hπ m are naturally defined. When π is irreducible, then we have compact injection of Hπ m into Hπ m if m > m. Moreover S π = m H m π. If P satisfies the Rockland s condition for all the degenerate π s, that is for irreducible representations associated to elements l G such that l r = l Gr = 0, then π(p ) is a Fredholm operator of H m π onto H 0 π = H π. If P is symmetric, π(p ) is essentially selfadjoint on S π, with domain H m π, and is consequently with compact resolvent. One of the step is to show an inequality with remainder : π(q)u C Q,π ( π(p )u + u ), u S π. (4.4) 21

22 There are actually many examples in physics which can be seen as π(p ). The most standard is the harmonic oscillator, which can be seen as a π(y1 2 + Y2 2 ) in the case when G is the three dimensional Heisenberg algebra and π is the irreducible representation attached to l 2 = 1. But let us also look at Y 2 1 +Y 2 2 i[y 1, Y 2 ]. This operator is not hypoelliptic. But it satisfies the degenerate Rockland condition. The corresponding π(p ) is 2 t + t 2 1. It is effectively with compact resolvent but not injective. The conclusion is that if we want get by this approach results on compact resolvent it is enough to consider the so called degenerate Rockland s condition. 4.6 Rothschild-Stein lifting and towards a general criterion L. Rothschild and E. Stein [RoSt] have shown that the analysis of the maximal hypoellipticity of the operator j X2 j or of X 0 + j X2 j can be deduced, when the vector fields satisfy the rank r conditions, from an analysis on a free nilpotent group of rank r. The free Lie algebra of rank r with p generators is the maximal Lie algebra with this property, that is the only relations existing are the necessary conditions satisfied by any Lie algebra. For example, the rank 2 free algebra with p generators has a basis : Y 1,, Y p ; [Y i, Y j ](i < j). Actually one can find a map λ such that : (X j u)(λ(x)) = Y j (u λ)(x). In particular, one will deduce that if P m (Y 1,, Y p ) is hypoelliptic then P (X 1,, X p ) is maximally hypoelliptic. This theorem is very strong at the level of the regularity, but still too weak in the sense that it gives only a sufficient condition of maximal hypoellipticity, which can be quite far from necessary. So it is natural to ask for the following question is there a subset Γ of G, having the property to be closed, stable by the coadjoint action and by dilation such that Conjecture P is maximally hypoelliptic at x 0 if and only if π l (P ) is injective for any l Γ \ {0}. 22

23 The set Γ = Γ x0 can be defined as follows: Definition 4.3. The set Γ x0 is attached to the vector fields X 1,, X p by : Here Γ x0 = {l G l = lim n + δ t n l n for some l n = l xn,ξn } (4.5) l xn,ξ n (Y ) = 1 i σ(λ(y ))(x n, ξ n ), where σ(λ(y )) denotes the symbol of λ(y ). Moreover, we impose that ξ n + and x n x 0. so Note that 1 i σ(λ(y ))(x n, ξ n ) = λ x n ξ n (Y ), l xn,ξ n = λ x n ξ n. It can be shown that Γ x0 is a closed G-invariance homogeneous set in G. The necessity of the conjecture was proved by J. Nourrigat (cf Chapter 3 in [HelNo3]) and a microlocal version for systems is given in [No3]. The sufficiency is proved in many cases, containing in particular the case when [G 2, G 2 ] = 0 (this condition will be always satisfied in our examples). There is a general proof for systems given by J. Nourrigat in the 80 s [No6, No7]. A particular case is the case when P = π 0,H (P ). In this case Γ is the so called spectrum of the induced representation : Γ = Sp (π 0,H ) = G H Remark There is a microlocal version of the conjecture. The set Γ x0 is then replaced by ξ the set Γ x0,ξ 0 which is defined by adding the property that lim n n + ξ 0. ξ n ξ 0 This set is also closed, G-stable and dilation invariant. Of course we also have : Γ x0,λξ 0 = Γ x0,ξ Folland s result Let V be a real vector space admitting a decomposition as a direct sum of spaces V i (1 i r). For every t > 0, we define the dilation h t on V by h t ( x j ) = t j x j, for x j V j. (4.6) 23

24 We say that a differential operator P on V is homogeneous of degree m if P (f h t ) = t m (P f) h t, f C (V ). (4.7) Let X 1,, X p a system of C real vector fields homogeneous of degree 1 and satisfying the rank r Hörmander condition. Let G the free nilpotent Lie Algebra of rank r with p generators. Let L(V ) the Lie algebra of C vector fields on V. Then there exists a unique linear map from G into L(V ) such that : λ(y j ) = X j λ([a j, a k ]) = [λ(a j ), λ(a k )], for a j G j, a k G k. (4.8) On can verify that λ is an homomorphism. We observe indeed that an homogeneous vector of degree > r is necessarily identicall 0. We now define H as the subspace of G generated by the a s such that λ(a) vanishes at 0. It is immediate to see that H is a subalgebra of G and stable by the dilations of G : λ δ t = h t λ. (4.9) Moreover the codimension of H is equal to dim V. Folland has proved the Proposition There exists a global diffeomorphism θ from V onto R dim V with Jacobian 1, such that for all a G and f C 0 (R dim V ), we have : Π 0,H (a)f = (λ(a)(f θ)) θ 1. (4.10) Moreover, the map f f θ sends S(R dim V ) onto S(V ). The Witten situation. Let us consider the Witten case. We take an homogeneous polynomial Φ of degree r on R n. We take V 1 = R n x, V 2 = = V r 1, V r = R t. We define : X j = xj, X j+n = ( xj Φ) t. We assume that r 2. Then it is clear that H in G, is generated by G 1 G 2 G r 1 where G 1 is generated by the X j+n (j = 1,, n). 24

25 What is the spectrum of Π 0,H? Coming back to the definition, we obtain that l in G is in the spectrum of the representation, if there exists a sequence (x n, τ n ) such that : l 1 (X j+n ) = lim n + ( xj Φ)(x n )τ n, for j = 1,, n ; l 2 ([X k, X j+n ]) = lim n + ( xk xj Φ)(x n )τ n for j = 1,, n, k = 1,, n, l 2 ([X k+n, X j+n ]) = 0 for j = 1,, n, k = 1,, n, l r ( ad X α X l+n ) = lim n + ( x α xl Φ)(0)τ n for α + 1 = q r. Note that the Hörmander condition is satisfied if the polynomial Φ is not identically zero. Note also that one can also say when the representation π τ is irreducible for τ 0. Remark It is better to work immediately with a smaller algebra taking account of the bracket properties of the vector fields X j. If for the hypoellipticity of π 0,H (P ), it is the injectivity of π(p ) for any π in the spectrum of π 0,H. The property that π τ (P ) for τ 0 can be obtained on the basis that π(p ) is injective for all π in the spectrum of π 0,H which are degenerate on G r and non trivial. The investigation of the proof in Helffer-Nourrigat permits to separate the case τ > 0 and the case τ < 0. Let us treat the easy case. This is the case when Φ is elliptic outside 0 or more generally when Φ does not vanish on the unit sphere S n 1. In this case, we get easily that the only representations which belong to this degenerate spectrum are the representations which are 0 on G 2. One get immediately that for these π s (π not trivial), π(p ) is injective. Actually we get π(p ) = π( j X2 j ). Of course this result can be proved quite easily by the criterion given in Proposition When Φ is not elliptic, there are another cases where one can give an answer but we postpone this to the next section. The simplest example is Φ(x 1, x 2 ) = ±x 2 1x 2 2. But this case can again be treated by the proposition

26 4.8 Discussion on Rothschild-Stein and Helffer-Métivier- Nourrigat We would like the properties of Σ p 1 j=1 Y 2 j + i 2 b jk [Y j, Y k ], j,k where b jk is a real antisymmetric matrix. Where the Y j are a basis of G 1 and G 1 generates a stratified Lie Algebra : G = r j=1g j. Let us introduce some definitions. If ρ is a real antisymmetric matrix, we can define its trace norm by ρ 1 = j ρ j, (4.11) where the ρ j are the eigenvalues of ρ. We denote by S the subspace of the real antisymmetric matrices such that s jk [Y j, Y k ] = 0. (4.12) j,k Theorem Suppose that G is not an Heisenberg algebra. Then is hypoelliptic if and only if : sup Tr (bρ) < 1. (4.13) ρ 1 1, ρ S (Here S is the set of ρ s such that Tr (sρ) = 0, s S.) The sufficiency part is proved in [RoSt] and we do not need the assumption that G is not an Heisenberg algebra. The case of an Heisenberg algebra can be treated separately and discrete phenomena appear. For the comparison with Witten Laplacians. It is enough to concentrate on the following case. The space G 1 admits a decomposition : G 1 = G 1 G 1, (4.14) 26

27 with Moreover and Our operator is dim G 1 = dim G 1. [G 2, G 2 ] = 0, dim G r = 1. P := (X j) 2 + (X j ) 2 + i j [X j, X j ]. (4.15) Actually, we are not exactly interested in this operator but more on the properties of Π(P ) is some representation of the envelopping algebra. This representation is an induced representation, with H = G 1 G 2 G r 1. 27

28 5 Maximal hypoellipticity for systems and spectral theory for Witten Laplacians We develop in more detail a variant of the previous point of view. The language of the nilpotent groups has been in some sense eliminated from the presentation. 5.1 Introduction Here we are mainly inspired by the presentation given by J. Nourrigat in [No1] of results of Helffer-Nourrigat and Nourrigat which appear in a less explicit form in the book [HelNo3]. This is in some sense a particular case of the big programm developed by J. Nourrigat at the end of the 80 s for understanding the subelliptic systems [No1]-[No7]. Our aim is to analyze the maximal hypoellipticity of the system L j = (X j iy j ), where X j = xj and Y j = ( xj Φ(x)) t in a neighborhood of 0 R n+1 and to see at the same time how the technics used for this analysis will lead to some information on the question concerning the Witten Laplacian. We assume that the real function Φ is such that the rank r Hörmander condition is satisfied at 0. This is an immediate consequence of the condition : α with 1 α r s. t α x Φ(0) 0. (5.1) Let us start with rather formal considerations. By maximal hypoellipticity for the system we mean the analysis of the inequality : X j u 2 + ( ) Y j u 2 C L j u 2 + u 2. (5.2) j j This system is microlocally elliptic outside of ξ = 0, Φ(x) = 0. So we are more precisely interested in the microlocal hypoellipticity in a conic neighborhood V ± of (x, t; ξ, τ) = (0; 0, ±1) that is with the microlocalized version of the inequality : j j χ ±(x, t, D x, D t )X j u 2 + ( j χ ±(x, t, D x, D t )Y j u 2 ) C j L ju 2 + u 2, (5.3) where χ ± is a pseudodifferential of order 0 which localizes in V ±. 28

29 It is interesting to observe that in the right hand side, we can write : L j u 2 = L jl j u u, j j and that ( L jl j = Xj 2 + j j j Y 2 j i j [X j, Y j ] ). 5.2 Microlocal hypoellipticity and semi-classical analysis Observing the translation invariance with respect to t, it is natural to ask for the existence of the inequality : π τ (X j )v 2 + ( ) π τ (Y j )v 2 C π τ (L j )v 2 + v 2, (5.4) j j for all v C 0 (V(0)), where V(0) is a neighborhood of 0 in R n, j π τ (L j ) = π τ (X j ) iπ τ (Y j ) = xj + τ( j Φ)(x). (5.5) The constant should be independent of τ. Using the partial Fourier transform with respect to t, one can indeed show that the proof of (5.4) uniformly with respect to τ is the main point for getting the maximal estimate. We now give two remarks : 1. The first one is that the estimate is trivial to obtain for τ in a bounded set. 2. The second one is that according to the point where we are interested for the microhypoellipticity, we have to consider the inequality for ±τ 0 (τ large). Take τ > 0 for simplicity. If we put τ = 1, the inequality we want to analyze h becomes, after division by τ 2 : (h xj )v 2 + ( ) ( xj Φ) v 2 C (0) h,φ v v + h2 v 2, (5.6) j j 29

30 for all v C 0 (V(0)), where (0) h,φ = h2 + Φ 2 h Φ. (5.7) The Hörmander s condition gives as a consequence of the microlocal subelliptic estimate the existence of V(0), h 0 > 0 and C > 0 such that : h 2 2 r v 2 C( j (h xj )v 2 + j ( xj Φ) v 2 (5.8) for h ]0, h 0 ] and v C 0 (V(0). So we finally obtain the existence of V(0), h 0 > 0 and C > 0 such that : h 2 2 r v 2 C (0) h,φv v (5.9) for h ]0, h 0 ] small enough. So the maximal microellipticity (actually the subelliticity would have been enough) implies some semi-classical localized lower bound for the semi-classical Witten Laplacian of order 0. Remark We refer to [HelNo4] for an analysis of a semi-classical subelliptic uncertainty principle. Of course, many semi-classical results can be obtained by other technics, particularly in the case when Φ is a Morse function with a critical point at 0, so it is more in degenerate cases that these old results can be relooked for giving new results. In the case when r = 2, we will see that, when Φ is a Morse function, the condition for the maximal microlocal hypoellipticity at (0; 0, 1) is that Φ is not a local minimum. Let us now express what it means for the semi-classical Witten Laplacian. Definition We denote by L 0 the set of all polynomials P of degree r vanishing at 0 (P E r ) such that there exists a sequence x n 0, τ n + and d n 0 such that : d α n τ n ( x α Φ)(x n ) x α P (0). (5.10) Remark If the Hörmander condition of rank r can not be improved, we have : lim n + dr nτ n 0. 30

31 In the case, when Φ is a Morse function, the set L 0 is simply the quadratic approximation of Φ at 0 up to a multiplicative positive constant. Helffer-Nourrigat s theorem gives in semiclassics : Theorem We assume that (5.1) is satisfied at rank r. Then, if no polynomial in L 0 except 0 has a local minimum at the origin, the inequality (5.9 ) is satisfied for h small enough. Moreover, the condition is necessary for getting the maximal estimate (5.6). Remark There is an equivalent way to express the condition. There exists a neighborhood V of 0 and two constants d 0 and c 0, such that : inf x x 1 d (Φ(x) Φ(x 1 )) c 0 sup x x 1 d Φ(x) Φ(x 1 )) for all x 1 V and for all d [0, d 0 [. We refer also to F. Trèves [Tr]. Remark At least when Φ is a Morse function, the trial function χ exp Φ show that h there are no hope to have an estimate when Φ has a local minimum. Remark This could be interesting to apply when Φ has isolated critical points. Remark Maire s results [Mai] should permit to get the semi-classical subelliptic estimate unnder the assumption that Φ is analytic and thhat Φ has no local minimum at the origin. 5.3 Around the proof and spectral byproducts for the Witten Laplacians Proposition The set L 0 has the following properties : 1. If P L 0 and y R n, then the polynomial defined by is also in L 0. Q(t) = P (t + y) P (y) 31

32 2. If P L 0 and λ > 0, then Q(t) = P (λt) is also in L L 0 is a closed subset of E r. Definition A set in E r satisfying the three conditions of Proposition 5.9 will be called canonical. To each polynomial P E r, we can associate a system of differential operators in R n by π P (X j ) = D xj, π P (Y j ) = xj P, π P (L j ) = π P (X j ) iπ P (Y j ). (5.11) In order to prove maximal estimates like : D xj u 2 + λ 2 ( xj P ) u 2 C (D xj iλ( xj P )) u 2, (5.12) j j we should also consider operators obtained from above by reduction of the number of variables. After a suitable linear change of variables x t, we get an integer k = k(p ) and a family depending on τ R n k of operators on R k π P,τ (X j ) = D tj for j = 1,, k ; π P,τ (X j ) = τ j for j = k + 1 n, π P,τ (Y j ) = tj ˆP, for j = 1,, k ; πp,τ (Y j ) = 0 for j = k + 1, n, (5.13) with ˆP (t) = P (x). ( ˆP is independent of the variables t k+1,, t n ). It is easy to show : Proposition If a polynomial P E r, P 0 has no local minimum in R n, then the system π P,τ (L j ) is injective in S(R k(p ) ). One observes indeed that for τ = 0 (which is the only non trivial case), a solution of π P,τ (L j )u = 0 is necessarily (up to a multiplicative constant) u = exp ˆP (t). The function being in S(R k(p ) ) and positive should have a maximum in contradiction with the property of ˆP. The next proof appears in a different language in Helffer-Nourrigat [HelNo3] and is the core of the proof : 32

33 Proposition Let L be a canonical subset of E r. We assume that for any P L \ {0} and for any τ R n k(p ) the system π P,τ (L j ) is injective. Then there exists a constant c 0 > 0 such that : π P (X j )u 2 + π P (Y j )u 2 c 0 π P (L j )u 2. (5.14) j j for all P L and for all u S(R n ). We shall not give the complete proof of the proposition but we would like to emphasize on various points of the proof which will actually have also consequences for the analysis of the Witten Laplacian. Continuity. An intermediate step is to show that π P,τ (X j )u 2 + π P,τ (Y j )u 2 c 0 (P, τ) j j j j π P,τ (L j )u 2, (5.15) for all v S(R k(p ) ), and to control the uniformity of the constant c 0 (P, τ) with respect to τ. This leads to (5.14) for some constant c 0 (P ) depending on P L \ {0}. Then the second difficult point is to control the uniformity of the constant c 0 (P ) with respect to P. Control at. We would first like to mention : Lemma With R P (x) = 1 α r α x P (x) 1 α, we have the existence of a constant c 1 > 0 such that ( ) R P u 2 c 1 π P (X j )u 2 + π P ((Y j )u 2 j (5.16) for all P E r and for all u S(R n ). Remark This is much more precise that what is obtained by the approach of Helffer- Mohamed. 33

34 Inequality with remainder. The proof of this proposition being by induction, let us look at the step based on the recursion assumption that the proof has been proved for the rank (r 1). An intermediate result is the following : Lemma If L is canonical and if (5.14) is valid for all P L E r 1, then there exists c 1 > 0 such that : π P (X j )u 2 + ( ) π P (Y j )u 2 c 1 π P (L j )u 2 + [P ] 2 r u 2, j j j (5.17) for all P L and for all u S(R n ), with [P ] r = P (α) (0) 1/r. α =r We shall use this in the following way. Theorem Let us assume that Φ is a polynomial in E r depending effectively on all the variables R n (in other words π Φ is irreducible or equivalently k(φ) = n or there are no translational invariance). Let L Φ the smallest canonical closed set containing Φ. Suppose that L Φ E r 1 does not contain any non zero polynomial having a local minimum, then the Witten Laplacian (0) Φ is with compact resolvent. Remarks The condition of irreducibility is necessary for having compact resolvent (see [HelNo3] and the discussion in the last section). 2. Let us observe that the condition is stable if one replace Φ by λφ, with λ > 0. It is the same for the results of Helffer-Nier [HelNi] 3. It seems reasonable that by further work, one can treat the case of more general functions Φ, which are no more polynomials. We shall present a more pedestrian approach in the last section in the case when the function Φ is a sum of homogeneous functions at. 34

35 4. Another interesting problem would be to determine in the spirit of Helffer-Mohamed the essential spectrum of the Witten Laplacian, when the condition of compactness is not satisfied. We are interested indeed in determining the existence of a gap. 5. Another idea which could be efficient would be to analyze in the same way, the corresponding Witten Laplacian on the one forms. The existence of a gap for (0) Φ should be a consequence of the microhypoellipticity of the operator on one forms : ( j L jl j ) I ihess Φ t 6. Other examples of this type are considered in [GHH]. Applications. Let us analyze how one can apply our Theorem Let us assume that Φ is homogeneous of degree r without translational invariance (here the point of view with the induced representation is also working). In order to apply the theorem we have to determine the set L Φ E r 1. One has to determine the polynomials of order r 1 appearing as limits of λ r nφ(x + h n ) λ r nφ(h n ). The coefficients of this limit polynomial P should satisfy : lim n + λr nφ α (h n ) = P α (0). with the additional condition that λ n 0. Elliptic case. Let us treat the elliptic case corrresponding to Φ 0 on the unit sphere. Let us show that the limit polynomial is necessarily of degree 1. If it was not the case, there will be some α with α 2, such that lim n + λr nφ α (h n ) > 0. This implies : 1 C 1 h n r α λ r n, 35

36 and we get also that h n +. We now use the ellipicity. We have : C 3 Φ(h n ) λ r n 1 C 2 h n r 1 λ r n 1 C 1 C 2 h n α 1. This leads to a contradiction. So P is a polynomial of degree one which can not have any local minimum. Second example Φ = x 2 1x 2 2. We have r = 4. Using the ellipticity of Φ, we first see that P should be of order less or equal to 2. But is P is effectively of order 2, if we observe that Φ(x) = 2(x x 2 2), we will get the same property for P (one can actually show that P (x) = γx 2 1 or P (x) = γx 2 2). This should forbid that P has a local minimum. 36

37 6 Semi-classical analysis for the Schrödinger operator 6.1 Harmonic approximation Introduction The harmonic oscillator plays a crucial role in Quantum Mechanics. This is not only the fact that its spectrum can be computed explicitely. As we shall see here, it gives the more natural approximation in order to analyze the spectrum of the Schrödinger near the bottom in the generic situation where there is a unique non degenerate minimum. We shall analyze in this section this approximation, some large dimension aspects and also high order approximations. We refer for this sections to the books of Cycon-Froese- Kirsch-Simon [CFKS] and to [Hel2] The case of dimension 1 We start with the simplest one-well problem: S h v := h 2 d 2 /dx 2 + v(x), (6.1) where v is a C - function tending to and admitting a unique minimum at say 0 with v(0) = 0. Let us assume that v (0) > 0. (6.2) In this very simple case, the harmonic approximation is an elementary exercise. We first consider the harmonic oscillator attached to 0: h 2 d 2 /dx v (0)x 2. (6.3) Using the dilation x = h 1 2 y, we observe that this operator is uniquely equivalent to h [ d 2 /dy ] v (0)y 2. (6.4) Consequently, the eigenvalues are given as λ n (h) = h λ n (1) = (2n + 1)h v (0) 2, (6.5) 37

38 and the corresponding eigenfunctions are u h n(x) = h 1 4 u 1 n ( x ) (6.6) h 1 2 with 1 v u 1 (0) y 2 n(y) = P n (y) exp (6.7) 2 2 We are just looking for simplification at the first eigenvalue. We consider the function u h,app. 1 x χ(x)u h 1(x) = c χ(x)h 1 4 exp v (0) 2 x 2 2h, where χ is compactly supported in a small neighborhood of 0 and equal to 1 in a smaller neighborhood of 0. We now get v (Sv h (0) h 2 )uh,app. 1 = O(h 3 2 ). and χ give expo- The coefficients corresponding to the commutation of Sv h nentially small terms and the main contribution is (v(x) 1 2 v (0)x 2 )χ(x)u h 1(x) L 2 which is easily seen as O(h 3 2 ). Then the spectral theorem gives the existence for Sv h of an eigenvalue λ(h) such that v (0) λ(h) h 2 C h 3 2 In particular, we get the inequality v (0) λ 1 (h) h + Ch 3 2. (6.8) 2 1 We normalize by assuming that the L 2 -norm is one. For the first eigenvalue, we have seen that, by assuming in addition that the function is strictly positive, we determine completely u h 1(x). 38