The semi-classical spectrum and the Birkhoff normal form
aa r X i v : . [ m a t h - ph ] F e b The semi-classical spectrumandthe Birkhoff normal form
Yves Colin de Verdi`ere ∗ November 23, 2018
Introduction
The purposes of this note are • To propose a direct and “elementary” proof of the main result of [3], namelythat the semi-classical spectrum near a global minimum of the classicalHamiltonian determines the whole semi-classical Birkhoff normal form (de-noted the BNF) in the non-resonant case. I believe however that the methodused in [3] (trace formulas) are more general and can be applied to any nondegenerate non resonant critical point provided that the corresponding crit-ical value is “simple”. • To present in the completely resonant case a similar problem which is NOTwhat is done in [3]: there, only the non-resonant part of the BNF is provedto be determined from the semi-classical spectrum!
Let us give a semi-classical Hamiltonian ˆ H on R d (or even on a smooth connectedmanifold of dimension d ) which is the Weyl quantization of the symbol H ≡ H + ~ H + ~ H + · · · .Let us assume that H has a global non degenerate non resonant minimum E at the point z : it means that after some affine symplectic change of variables ∗ Institut Fourier, Unit´e mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martind’H`eres Cedex (France); [email protected] = E + P dj =1 ω j ( x j + ξ j ) + · · · where the ω j ’s are > < ω < ω < · · · < ω d . We will denote E = H ( z ).We assume also that lim inf ( x,ξ ) →∞ H ( x, ξ ) > E . Let us denote by λ ( ~ ) < λ ( ~ ) ≤ · · · ≤ λ N ( ~ ) ≤ · · · the discrete spectrum ofˆ H . This set can be finite for a fixed value of ~ , but, if N is given, λ N ( ~ ) existsfor ~ small enough. Definition 1.1
The semi-classical spectrum of ˆ H is the set of all λ N ( ~ ) ( N =1 , · · · ) modulo O ( ~ ∞ ) . NO uniformity with respect to N in the O ( ~ ∞ ) isrequired.Definition 1.2 The semi-classical Birkhoff normal form is the following formalseries expansion in
Ω = (Ω , · · · , Ω d ) and ~ : ˆ B ≡ E + ~ E + d X j =1 ω j Ω j + X l + | α |≥ c l,α ~ l Ω α with Ω j = (cid:0) − ~ ∂ j + x j (cid:1) . The series ˆ B is uniquely defined as being the Weylquantization of some symbol B equivalent to the Taylor expansion at z of H bysome automorphism of the semi-classical Weyl algebra (see [2]). The main result is the
Theorem 1.1 ([3])
Assume as before that the ω j ’s are linearly independent overthe rationals. Then the semi-classical spectrum and the semi-classical Birkhoffnormal form determine each other. The main difficulty is that the spectrum of ˆ B is naturally labelled by d − uples k ∈ Z d + while the semi-classical spectrum is labelled by N ∈ N . We will denoteby ψ the bijection ψ : N → k of N onto Z d + := { k = ( k , · · · , k d ) |∀ j, k j ∈ Z , k j ≥ } given by ordering the numbers h ω | k i in increasing order: they are pair-wisedistincts because of the non-resonant assumption. We have the following result 2 heorem 2.1
The semi-classical spectrum is given by the following power seriesin ~ : λ N ( ~ ) ≡ E + ~ (cid:18) E + 12 h ω | ψ ( N ) + 12 i (cid:19) + ∞ X j =2 ~ j P j ( ψ ( N )) (1) where the P j ’s are polynomials of degree j given by P j ( Z ) = X l + | α | = j c l,α (cid:18) Z + 12 (cid:19) α . This result is an immediate consequence of results proved by B. Simon [5] andB. Helffer-J. Sj¨ostrand [4] concerning the first terms, and by J. Sj¨ostrand in [6](Theorem 0.1) where he proved a much stronger result. ω j ’s ω j ’s Because E = lim ~ → λ ( ~ ), we can substract E and assume E = 0.By looking at the limits, as ~ → µ N := lim λ N ( ~ ) / ~ ( N fixed), we knowthe set of all E + P dj =1 ω j ( k j + ) , ( k , · · · , k d ) ∈ Z d + . Let us give 2 proofs that the µ N ’s determine the ω j ’s. Using the partition function: from the µ N ’s, we know the meromorphicfunction Z ( z ) := X e − zµ N .Z ( z ) := e − z ( E + P dj =1 ω j ) X k ∈ Z d + e − z h ω | k i , We have Z ( z ) = e − z ( E + P dj =1 ω j ) Π dj =1 (1 − e − zω j ) − , The poles of Z are P := ∪ j =1 , ··· ,d { πi Z ω j } . The set of ω j is hence determinedup to a permutation. We fix now ω = ( ω , · · · , ω d ) with ω < ω < · · · .From the knowledge of the ω j ’s, we get the bijection ψ .2. A more elementary proof: substract µ = E + P ω j from the wholesequence and denote ν N = µ N − µ . Then ω = ν . Then remove themultiples of ω . The first remaining term is ω . Remove all integer linearcombinations of ω and ω , the first remaining term is ω , · · · .2 Determining the c l,α ’s Let us first fix N : from Equation (1) and the knowledge of λ N mod O ( ~ ∞ ) weknow the P j ( ψ ( N ))’s for all j ’s.Doing that for all N ’s and using ψ determine the restriction of the P j ’s to Z d + and hence the P j ’s. For simplicity, we will consider the completely resonant case ω = · · · = ω d = 1and work with the Weyl symbols. Let us denote by Σ = P ( x j + ξ j ).The (Weyl symbol of the) QBNF is then of the form B ≡ Σ + ~ P , + ∞ X n =2 X j + l = n ~ j P l,j where P l,j is an homogeneous polynomial of degree 2 l in ( x, ξ ), Poisson commut-ing with Σ: { Σ , P l,j } = 0 .For example, the first non trivial terms are: • for n = 2: P , + ~ P , + ~ P , • for n = 3: P , + ~ P , + ~ P , + ~ P , .The semi-classical spectrum splits into clusters C N of N + 1 eigenvalues in aninterval of size O ( ~ ) around each ~ ( N + d + P , ) with N = 0 , , · · · .The whole series B is however NOT unique, contrary to the non-resonant case,but defined up to automorphism of the semi-classical Weyl algebra commutingwith Σ.Let G be the group of such automorphisms (see [2]). The natural question isroughly: Is the QBNF determined modulo G from the semi-classical spec-trum, i.e. from all the clusters? G The linear part of G is the group M of all A ’s in the symplectic group whichcommute with ˆ H , i.e. the unitary group U ( d ).We have an exact sequence of groups:0 → K → G → M → . The Moyal bracket of any A with H reduces to the Poisson bracket K (the “pseudo-differential” part):Let S = S + · · · in the Weyl algebra (the formal power series in ( ~ , x, ξ ) withthe Moyal product and the usual grading degree( ~ j x α ξ β ) = 2 j + | α | + | β | ) g S ( H ) = e iS/ ~ ⋆ H ⋆ e − iS/ ~ preserves Σ iff { S n , Σ } = 0. This implies that n is even and S n is a polynomialin z j z k ( z j = x j + iξ j ). Then K is the group of all g S ’s with { S, Σ } = 0. References [1] Laurent Charles & San V˜u Ngo. c. Spectral asymptotics via the Birkhoff nor-mal form.
ArXiv:math-sp/0605096 , Duke Math. Journal :463–511 (2008).[2] Yves Colin de Verdi`ere. An extension of the Duistermaat-Singer Theorem tothe semi-classical Weyl algebra.
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Math. Res.Lett. 14:711–719 (2007). [4] Bernard Helffer & Johannes Sj¨ostrand. Puis multiples en semi-classique I.
Commun. PDE. :337–408 (1984).[5] Barry Simon. Semi-classical analysis of low lying eigenvalues I: Non degen-erate minima. Ann. IHP (phys. th´eo.) : 295–307 (1983).[6] Johannes Sj¨ostrand. Semi-excited states in nondegenerate potential wells. Asymptotic Analysis6