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APPEARED IN BULLETIN OF THE

arXiv:math/9204235v1 [math.SP] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 299-303SPECTRAL THEORY AND REPRESENTATIONSOF NILPOTENT GROUPSP. Levy-Bruhl, A. Mohamed, and J. NourrigatAbstract.

We give an estimate of the number N(λ) of eigenvalues < λ for the imageunder an irreducible representation of the “sublaplacian” on a stratified nilpotent Liealgebra. We also give an estimate for the trace of the heat-kernel associated with thisoperator.

The estimates are formulated in term of geometrical objects related to therepresentation under consideration. An important particular case is the Schr¨odingerequation with polynomial electrical and magnetical fields.1.

IntroductionWe first consider the Schr¨odinger operator with polynomial electrical and mag-netical fieldsP =nXj=1(Dj −Aj(x))2 + V (x),where Aj and V are real polynomials on Rn, of degree ≤r, and with V ≥0.We define Bjk = ∂Aj/∂xk −∂Ak/∂xj and assume that there is no rotation of thecoordinates axis making all the Bjk and V independent of one of the xi’s. Thiscase has been investigated in [9]: defineM(x) =X|α|≤r|∂αV (x)|1/(|α|+2) +Xα,j,k|∂αBjk(x)|1/(|α|+2),M(x, ξ) = |ξ| + M(x) .Let N(λ) be the number of eigenvalues of P < λ and N0(λ) the volume in R2n ofthe set of points (x, ξ) such that M(x, ξ)2 ≤λ.

Then there is a C > 1, independentof λ such that C−1N0(C−1λ) ≤N(λ) ≤CN0(Cλ) for all λ > 0. A more preciseequivalent in particular cases is given in [13].

This result was proved before, in adifferent formulation, by Fefferman, when the Aj’s are zero [1]. If V is the squareof a polynomial, P is of the form π(−∆) where ∆is a sublaplacian on a stratifiedr-step nilpotent Lie group and π an irreducible representation [2, 9].

In the generalcase, (V ≥0), our following results needs a very small change to be applied. Thisexample is one of the motivations for the present generalization.1991 Mathematics Subject Classification.

Primary 35P20, 22E27.Key words and phrases. Representations of nilpotent Lie groups, spectral theory for partialdifferential equations.Received by the editors May 15, 1991 and, in revised form, September 24, 1991The results have been presented at the 1991 meeting on “Harmonic analysis and representationtheory” (R. S. Howe.

org.) (held January 27 to February 2), in Oberwolfach, Germanyc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2P. LEVY-BRUHL, A. MOHAMED, AND J. NOURRIGAT2.

Statement of the resultsLet G be a stratified r-steps nilpotent Lie algebra. In other words, we assumethat G can be written as a direct sum of subspaces Gj (1 ≤j ≤r) such that[Gj, Gk] ⊂Gj+kif j + k ≤r,[G, Gk] = 0if j + k > r,and such that G is generated as Lie algebra by the subspace G1.

We choose a basisX1, . .

. Xp of G1, and we shall be interested in spectral properties of the imageunder irreducible representations of the sublaplacian ∆= −Ppj=1 X2j .Let π be an irreducible, unitary, nontrivial representation of the connected simplyconnected group exp G associated to G. It is well known that the operator π(−∆),defined on the space Sπ of C∞vectors of the representation π, has a unique selfad-joint extension, still denoted π(−∆), whose spectrum is a sequence (λj) (j ∈N) ofpositive eigenvalues, such that λj ≤λj+1 and λj →+∞.

Let us denote by N(λ)the number of eigenvalues λj ≤λ. Our first goal is to give an estimate for N(λ) interms of geometrical objects associated to the representation π.We denote by δt (t > 0) the natural dilations of G; that is, the linear maps definedby δt(X) = tjX if X ∈Gj.

Let also G∗be the dual of G, δ∗t the transpose of δt,and ||| ||| a homogeneous norm on G∗, i.e., a nonnegative continuous, subadditivefunction on G∗, only vanishing at the origin and satisfying |||δ∗t l||| = t|||l||| if t > 0and l ∈G∗.By the Kirillov theory, the representation π is associated with an orbit Oπ of thecoadjoint representation of the group G = exp G in G∗. The G invariant measureson Oπ are proportional, and we denote by µ the canonical one; that is, with thecorrect normalization for the character formula [4, 12].

For each λ > 0, we setN0(λ) = µ(l ∈Oπ, |||l|||2 ≤λ) .We can now state the estimate for N(λ).Theorem 1. There exists a constant C > 1, independent of λ, and of the repre-sentation π such thatC−1N0(C−1λ) ≤N(λ) ≤CN0(Cλ),for all λ > 0 .Manchon gives in [7] the proof of a conjecture of Karasev-Maslov [3].

His resultdoes not apply to our operator, but rather to −Pj Y 2j , where (Yj) is a basis ofG as a vector space. Such an operator is also selfadjoint with compact resolvent,and N(λ) is given by a Weyl type formula.

Such a formula does not make sense ingeneral in our case, the integral involved in it being nonconvergent.It is also well known, since π is irreducible, that for t > 0, the operator exp(−tπ(∆))is trace-class. We can estimate its traceZ(t) = Tr(exp(−tπ(∆)))by the following result, using the functionZ0(t) =ZOπexp(−t|||l|||2) dµ(l) .

SPECTRAL THEORY3Theorem 2. There exists a constant C > 1, independent of t > 0, and of therepresentation π such thatC−1Z0(Ct) ≤Z(t) ≤CZ0(C−1t) .As noticed in the introduction, our results can be used for the Schr¨odinger op-erator with polynomial electrical and magnetical fields.3.

Sketch of the proofRemark. We give here a sketch of the proof of Theorem 1.

The reader may consult[6] for the details.We construct Hilbert spaces of sequences controlling the norm of the Sobolevspaces associated with π(−∆) and use the minimax formula.For this construction, we need a suitable set of functions in L2(Rn). The prop-erties of these functions have been suggested both by the ideas of Perelomov [11](coherent states) and of Meyer [8] (wavelets).We realize the representation π in the following form: for X ∈G, π(X) is adifferential operator on Rn,(∗)π(X) = A1(X)∂/∂x1 + A2(x1, X)∂/∂x2+ · · · + An(x1, .

. .

, xn−1, X)∂/∂xn + iB(x, X),where the Aj and B are real linear forms in X and polynomials in x ∈Rn, A1is independent of x, and Aj only depends on (x1, . .

. , xj−1).

For every finite se-quence I = (i1, . .

. , im) of positive integers smaller than p, π(XI) is the iteratedcommutatorπ(XI) = (ad π(Xi1)) · · · (ad π(Xim−1))π(Xim) .Let π(X)(x, ξ) be the complete symbol of the operator π(X).

The orbit Oπ is theset of linear forms X →−iπ(X)(x, ξ) for (x, ξ) in R2n, and the canonical measureµ is then dxdξ. We can also choose the homogeneous norm|||l||| =X|l(Xi)|1/|I| .The functions Mπ(x, ξ) and Mπ(x) are defined byMπ(x, ξ) =X|I|≤r|π(XI)(x, ξ)|1/|I|andMπ(x) = infξ Mπ(x, ξ) .For the representation π under consideration, Mπ(x) > 0.

We prove the theoremsusing this “concrete” realization of π. We are not able to construct an orthonormalbasis of L2 so that the Sobolev spaces associated to our operator are l2 weightedspaces, but the following result is a substitute and allows the use of minimax formu-las.

Let p(x, ξ) = x, (x, ξ) ∈Rn. For every finite sequence (α1, .

. .

, αm) of positiveintegers smaller than p, we define π(Xα) = π(Xα1) · · · π(Xαm), and foru ∈C∞0 (Rn): ∥u∥m,π =X|α|=m∥π(Xα)u∥21/2.

4P. LEVY-BRUHL, A. MOHAMED, AND J. NOURRIGATTheorem 3.

“Coherent states.” Let π be an induced representation of G realizedin the form (∗) with Mπ(x) > 0 for all x ∈Rn. (π is not necessarily irreducible.

)For every a > 0 small enough, there exists (ψj) and (Ψj) (j ∈N) in C∞0 (Rn), asequence Ωj of compact sets in R2n, a point (xj, ξj) ∈Ωj, and C > 0 with:1. u = Pj(u, Ψj)ψj for all u in L2(Rn).2. (supp Ψj) ⊂(supp)ψj ⊂p(Ωj) for all j ∈N.3.

Every finite subset A of N contains a part B such that #B ≥(#A)/C andsuch that for every sequence (λj) of complex numbersXj∈B|λj|2 ≤CXj∈Bλjψj2.4. For every integer m ≥0, there exists Cm > 0 with the properties : If u ∈C∞0 (Rn) is of the form u = Pj λjψj (λj ∈C not necessarily given by (u, Ψj))then ∥u∥2m,π ≤CmPj |λj|2Mπ(xj, ξj)2m.

If in addition λj = (u, Ψj) then we havePj |(u, Ψj)|2Mπ(xj, ξj)2m ≤Cm∥u∥2m,π.5. R2n ⊂Sj(Ωj) and every (x, ξ) is contained in at most C of the Ωj’s.

Further,1/C ≤Vol(Ωj) ≤C.6. For (x, ξ) in Ωj, we have the estimate (1/C)Mπ(xj, ξj) ≤Mπ(x, ξ) ≤CMπ(xj, ξj).7.

C and the constants Cm only depend on a and G not on π.The construction of the coherent states is first performed locally: we associateto each x ∈Rn a representation σx, equivalent to π such that σx has the form(∗) with Mσx(0) = Mπ(x) and the coefficients of the polynomials in the expressionof σx(δ−1t(X)) where t = Mπ(x) are bounded, independently of x and π. UsingFourier series and induction on the dimension n, “local coherent states” for σx areconstructed. Certain properties in Theorem 3 are only satisfied if u is supported inthe ball of radius a, centered at 0.

The coherent states are then obtained by meansof a partition of unity, modeled on the symplectic diffeomorphisms related to theintertwining operators between π and σx.In the proof, the following theorem is used (the notation is as before).Proposition 4. There exists C > 0 such that for every representation π of theform (∗) and all u ∈S(Rn)X|α|≤m∥Mπ(x)m−|α|π(Xα)u∥2 ≤C∥u∥2m,π .In this form, this theorem requires G to be stratified, however, the proof ofLemma 4.5 in [10] works without this hypothesis.

The function M is then morecomplicated, reflecting the structure of G. It is possible to formulate our presentresults in this set-up. The same method also applies to more general operatorsthan the model presented here; we need only a control of (π(P)u, u) in terms of the∥∥m,π norm.References1.

C. L. Fefferman, The uncertainty principle, Bull. Amer.

Math. Soc.

9 (1983), 129–206.2. B. Helffer and J. Nourrigat, Hypoellipticit´e maximale pour des op´erateurs polynˆomes dechamps de vecteurs, Progr.

Math., vol. 58, Birkhauser, Boston, MA, 1985.

SPECTRAL THEORY53. Karasev and V. Maslov, Algebras with general commutation relations and their applica-tions II, J. Soviet Math.

15 (3) (1981), 273–368.4. A. Kirillov, Unitary representations of nilpotent groups, Russian Math.

Survey 14 (1962),53–104.5. P. G. Lemari´e, Base d’ondelettes sur les groupes de Lie stratifi´es, Bull.

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France117 (1989), 211–232.6. P. L´evy-Bruhl, A. Mohamed and J. Nourrigat, Etude spectale d’op´erateurs sur des groupesnilpotents, S´eminaire “Equations aux D´eriv´ees Partielles”, ´Ecole Polytechnique (Palaiseau),Expos´e 18, 1989–90; preprint, 1991.7.

D. Manchon, Formule de Weyl pour les groupes de Lie nilpotents, Th`ese, Paris, 1989.8. Y. Meyer, Ondelettes et op´erateurs, Hermann, Paris, 1990.9.

A. Mohamed and J. Nourrigat, Encadrement du N(λ) pour des op´erateurs de Schr¨odingeravec champ magn´etique, J. Math.

Pures Appl. (9) 70 (1991), 87–99.10.

J. Nourrigat, In´egalit´es L2 et repr´esentations de groupes nilpotents, J. Funct. Anal.

74(1987), 300–327.11. A. Perelomov, Generalized coherent states and their applications, Texts Monographs Phys.,Springer, 1981.12.

L. Pukanszki, Le¸cons sur les repr´esentations des groupes, Dunod, Paris, 1967.13. B. Simon, Non classical eigenvalue asymptotics, J. Funct.

Anal. 53 (1983), 84–98.(P.

Levy-Bruhl and J. Nourrigat) D´epartement de Mathematiques, Universit´e de Reims,B. P. 347, 51062 Reims CEDEX, France(A. Mohamed) D´epartment de Mathematiques, Universit´e de Nantes, 2 rue de laHoussini`ere, 44072 Nantes CEDEX, France

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