Asymptotic expansion for nonlinear eigenvalue problems

Transcription

1 J. Math. Pures Appl. 93 1) Asymptotic expansion for nonlinear eigenvalue problems Fatima Aboud, Didier Robert Département de Mathématiques, Laboratoire Jean Leray, CNRS-UMR 669, Université de Nantes, rue de la Houssinière, 443 Nantes Cedex 3, France Received 31 March 9 Available online 11 August 9 Abstract In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is Lλ) = +P x) λ) in L ) where P is a positive elliptic polynomial in of degree m. It is nown that for d even, or d = 1, or d = 3andm 6, there exist λ C and u L ), u, such that Lλ)u =. In this paper, we give a method to prove existence of non-trivial solutions for the equation Lλ)u =, valid in every dimension d 1. This is a partial answer to a conjecture in Helffer, Robert and Xue Ping Wang 4) [13]. 9 Elsevier Masson SAS. All rights reserved. Résumé Dans cet article nous considérons un problème aux valeurs propres généralisé pour une famille d opérateurs dépendant quadratiquement d un paramètre complexe. Le modèle étudié concerne la famille Lλ) = +P x) λ) dans L ) où P un polynôme elliptique dans de degré m. Si d est paire ou si d = 1oud = 3etm 6, on sait alors qu il existe λ C et u L ), u, tels que Lλ)u =. L objet principal de cet article est de donner une méthode pour démontrer l existence de solutions non triviales pour l équation Lλ)u = pour toute dimension d 1. On répond ainsi partiellement à une conjecture formulée dans Helffer, Robert et Xue Ping Wang 4) [13]. 9 Elsevier Masson SAS. All rights reserved. Keywords: Semiclassical analysis; Nonlinear eigenvalue problems; Non-selfadjoint operators; Trace formula 1. Introduction Let us introduce the following family of differential operators, L P λ) = x + Px) λ ), 1.1) where x is the Laplace operator in x, λ is a complex parameter, P is a polynomial of degree m such that the leading homogeneous part P m of P satisfies P m x) > for every x \{} in other words we say that P is a positive-elliptic polynomial). * Corresponding author. addresses: fatima.aboud@univ-nantes.fr F. Aboud), didier.robert@univ-nantes.fr D. Robert) /$ see front matter 9 Elsevier Masson SAS. All rights reserved. doi:1.116/j.matpur.9.8.9

2 15 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Such family of operators plays an important role when studying analytic smoothness of solutions of differential operators with multiple characteristics. In 1978, B. Helffer [11], motivated by G. Métivier wor [17], has considered the following hypoelliptic differential operator, D = Dx 1 + x1 D ), x D x3 where D x = i 1 x and conjectured that it is not hypoanalytic in a neighborhood of in R 3 that means there exists u non-analytic in a neighborhood of such that Du is analytic in a neighborhood of ). He showed that if there exists λ C such that Lλ) := Dt + t λ) has a non-trivial solution in SR) then D is not hypoanalytic for other examples see [13] and references there). Quadratic families of operators with a complex parameter λ also appear in the theory of damped oscillations for dissipative problems in mechanics [9,15]. The mathematical model is a second order differential equation: Au + Bu + Cu =, 1.) where the unnown function u is defined on R with values in some Hilbert space H and u = du dt. Eq. 1.) is a model in mechanics for small oscillations of a continuum system in the presence of an impedence force [15]. Now looing for stationary solutions of 1.), that means ut) = u e λt, we have the following equation: λ A + λb + C ) u =. 1.3) So Eq. 1.3) is a nonlinear eigenvalue problem in the parameter λ C. Existence of non-null solutions for 1.3) is a non-trivial problem. For B this problem is equivalent to a true non-selfadjoint linear eigenvalue problem and non-trivial stationary solutions for 1.3) does not always exist. Nevertheless, if suitable conditions on A,B,C are satisfied, several authors [15,14,9,16] have proved that a total set of generalized eigenfunctions for 1.3) exist in the Hilbert space H. Concerning our model, we have L P λ) = +P x) λp x) + λ.in the strength of the coefficient of λ is of the same order as the operator + P) 1/, so it seems difficult to use perturbation arguments. The question we want to address here is the following: For any elliptic polynomial P of degree m, does there exist λ C and u in the Schwartz space S ), u, such that L P λ)u =? For d = 1, this was proved by Pham and Robert [18] for m even. M. Christ have generalized this result for every m [7]. In [5] the authors have proven existence of non-trivial solutions for L P when 1 d 3, assuming that m is large enough for d = 3. Later, Helffer, Robert and Wang proved in [13] the following result: Theorem 1.1. Assume that d is even and that P is a positive-elliptic polynomial of degree m. Then there exist λ C and u S ), u, such that L P λ)u =. The proof given in [13] shows that there exist an infinite number of such eigenvalues [] located in the half-plane {λ C, Rλ }. But it is not nown if the generalized eigenfunctions span all the Hilbert space L ), excepted for d = 1 [18]. For d odd, d 3, m, the problem of existence of non-zero solutions for L P is still open and it was conjectured in [13] that such solutions exist whatever the dimension d. In this paper we prove that this is true for every elliptic polynomial if d = 3 and for large classes of elliptic polynomials for d = 5, 7. We also discuss a numerical approach to prove that some coefficient in a semiclassical trace formula is not zero. For d 9 we conjecture that this coefficient is not zero hence there exists an infinite number of nonlinear eigenvalues.. Nonlinear eigenvalue problems In this section we recall some nown properties concerning nonlinear eigenvalue problems. For more details we refer to [1,16,].

3 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Let us consider the quadratic family of operators Lλ) = L + λl 1 + λ where L, L 1 are operators in an Hilbert space H. L is assumed to be self-adjoint, positive, with a domain DL ) and L 1 is L -bounded. Moreover L 1/ is in a Schatten class C p H) for some real p>. The following results are well nown. Theorem.1. Lλ) is a family of closed operators in H. λ L 1 λ) is meromorphic in the complex plane. The poles λ j of L 1 λ), with multiplicity mλ j ), coïncide with the eigenvalues with the same multiplicities, of the matrix operator A L in the Hilbert space H DL 1/ ), with domain DA L ) = DL ) DL 1/ ), where ) I A L =..4) L L 1 Let us denote Sp[L] the eigenvalues of A L which coïncide with the poles of L 1 z)). Remar.. It may happens that Sp[L] is empty. The following one-dimensional example is interesting and was discussed in [18,6,7]: L m,g λ) = d dx + x m λ ) + gx m 1..5) For every m, m even, L m, has infinity many eigenvalues but L m,m has no eigenvalue. The last statement is a consequence of the factorization, L m,m λ) = x m λ + d ) x m λ d ). dx dx So, we can compute all solutions for the equation L m,m λ)u = and see that a non-null solution u is never bounded on R. But if m is odd, L m,m λ)u =, has infinity many eigenvalues on the imaginary axis [7]. On the other side there exist sufficient general conditions to have Sp[L] [1,16]. Unfortunately these conditions are not fulfilled for our example Lλ) = x + P x) λ) when d. The following formula appears for the first time in [3] and will be very useful for our purpose. Theorem.3. For large enough N, >p) and for z C\ Sp[L], we have: TrA L z) 1 = 1 [ d! Tr Lz) 1 dz L z) ) ],.6) where each above operators are trace class. Using Lidsii Theorem [1] and.6), we get: mλ)λ z) 1 = 1 [ d! Tr Lz) 1 dz L z) ) ],.7) λ Sp[L] where mλ) is the multiplicity of the eigenvalue λ. As it was nicely remared in the paper [5], a sufficient condition for Sp[L] is that the r.h.s. in.7) is not zero. To chec this property a natural method is to introduce parameters and use semiclassical analysis. In [13] the authors also use Lidsii Theorem and semiclassical analysis on the matrix system A L.Herewe consider more directly the scalar family of operators Lz) where computations are easier even if the dependence in z is nonlinear.

4 15 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Semiclassical parametrix For simplicity we assume here that P is homogeneous of degree m and Px)>forx, x. By the scaling transformation x = τ 1/m y with h = τ m+1)/m and z = τ λ we can see that Lλ) is unitary equivalent to the semiclassical Hamiltonian τ ˆLz), where ˆLz) = h x + Px) z ). 3.8) ˆLz) is the h-weyl operator with the symbol Lz, x, ξ) = ξ + P x) z). For semiclassical analysis tools and h-weyl quantization we refer to [19]. Here we use the notation Hˆ for the h-weyl quantization of the symbol H or for convenience, H ˆ = Op w h H ). Let us recall the following definition. For a temperate symbol, possibly h-dependent, Hx,ξ, h), wehave: ) Hψx)= ˆ 1 x + y π h) n e ix y) ξ/ h H,ξ, h ψy)dydξ, 3.9) ψ SR n ). Using some semiclassical operator calculus, we can construct a good parametrix for ˆLz) 1 for z Λ, where Λ is the sector Λ = { z C, z r,π/ + δ<argz) < 3π/ δ } ; r >, δ>. Theorem 3.1. There exists a semiclassical symbol K h) z), z Λ, < h<1, such that K h) z; x,ξ) h j K j z; x,ξ), j ˆLz) 1 = Op w h K h z) ). 3.1) Moreover the asymptotic expansion has the following meaning: for every N 1 we have: ) ˆLz).Op w h h j K j z) = I + h N+ Op w h h) R N z)), j N where the symbol R h) N z) satisfies the following estimates: for every α, β Nd we have: x α β h) ξ R N z; x,ξ)) CN,α,β) μx, ξ)m + z μx, ξ) m μx, ξ) m + z μx, ξ) N α β, 3.11) where CN,α,β) is uniform in z Λ and where μx, ξ) = 1 + x m + ξ ) 1/m. Setch of proof. The method to get such a result is standard and was used many times to construct parametrix of elliptic pseudo-differential operators [1]. Usually the z-dependence is linear but here it is quadratic. Moreover here we need accurate estimates for the remainder term in the product of pseudo-differential operators depending on parameters. The necessary estimates for R h) N z; x,ξ) are established using the technics coming from the papers [8,4]. An other difficulty here is that we shall need to compute the symbols K j for j large enough. This computations are not easy, so we have to be explicit as far as possible. Using the product formula for h-pseudo-differential operators, we get at the initial step: and the induction formula, where α,β)= K j = K K z; x,ξ) = 1 Lz; x,ξ) = 1 ξ + P x) z), 3.1) l j 1 α + β =j l) 1) β j l) α!β!. Let us compute K and K 4. α,β) α ξ β x Lz) β ξ α x K l ), 3.13)

5 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) K = L z) L 3 z) + L 3z) L 4 z), L z) = Px) z ) Px)+ Px), L 3 z) = [ Px) z ) D Px)ξ ξ + Px) ξ ) + Px) z ) Px) ], where D Px)is the Hessian matrix of P in variable x. Now using 3.13) we have: { K 4 = K,β) x β Lz) β ξ K + β =4 By induction on j, we easily get that Q j K j z; x,ξ) = α = α,) ξ α Lz) α x K,β) x β Lz) β ξ K j+1 3j β = }. 3.14) Q j x, P z, ξ) Lz; x,ξ) +1, 3.15) x, P z, ξ) is a polynomial in P z), ξ), with a total degree, with coefficients depending on derivatives of Px). The following lemma will be useful later. Let us denote val[q j ], the valuation of Qj as a polynomial in P z, ξ. Let us recall the definition of valuation. Denote by I the ideal with generators ξ 1,...,ξ d,p z, inthe ring C R ξ R x ).IfQ C R ξ R x ),val[q] is the biggest integer p such that Q I p. Lemma 3.. We have: val [ Q j ] 1 j), for j + 3j, and j 1. Proof. This is easily proved by induction on j, using 3.15) and the following formula. Let Q and L be smooth functions in R n, a multiindex α N n, then we have: ) Q α L +1 = Cμ j,γ ) α γ Q γ 1L) μ 1 γ ll) μ l L μ++1, 3.16) where in the sum we have the conditions, γ j N n, μ j N, γ α, μ 1 + +μ l = μ, μ 1 γ 1 + +μ l γ l = γ. Let us recall that γ α means that γ j α j for every 1 j d. Remar 3.3. The parametrix computed above is enough to get qualitative informations. Quantitative informations are much more difficult to get except for the first orders j =, 1). When j is larger it is not so easy to compute explicitly the terms Q j x, P z, ξ). Remar 3.4. It is not difficult to extend the above results when the elliptic polynomial Px) has lower terms: P = P m + P m 1 + +P 1 + P where P j is homogeneous with degree j and P m x) > forx \{}. Then we have: P τ 1/m y ) = τp ε) y), with ε = τ 1/m = h 1/m+1) and P ε) y) = P m y) + εp m 1 y) + +ε m P y). SoP ε) is a uniform elliptic family of polynomials and we can easily see that the constructions in 3.11) are uniform in the small parameter ε. 4. A trace formula Recall that Sp[L] denote the generalized eigenvalues of the quadratic family Lz), m λ is the multiplicity of the eigenvalue λ.letf an holomorphic function in Λ such that fz) C 1 + z ) μ, z Λ. 4.17)

6 154 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) For our applications we shall choose fz)= z + λ) μ, for a suitable parameter λ C. Let be a complex contour in Λ defined as follows: = { re ±iθ } {,r r r e iθ },θ θ π θ, where r > and π <θ <π. Proposition 4.1. Assume that μ> dm+1) m. Then fa L) is a trace class operator and we have: Tr fa L ) ) = [ ] mλ)f λ) = Tr ˆLz) 1 ˆL z)f z) dz, 4.18) λ Sp[L] where Fz)dz= iπ 1 Fz)dz contour integral in the complex plane). Proof. This a direct consequence of the Cauchy integral formula and Theorem.3. Theorem 4.. For f as above, for every d 1 we have in the semiclassic regime h, modulo O h + ), mλ)f λ) C d) j f ) h j d. 4.19) λ Sp[L] j If d is odd, C d) f ) =, 4.) and for d even, C d) f ) = 1)d/ π) d ) f Px)+ η dxdη. 4.1) For the other terms j 1) we have the following qualitative information, C d) j f ) = ) A j, x)f Px) ) dx, 4.) n j where A j, x) are polynomials in γ x Px), γ j and n j depends on j. Moreover if d is odd, then we have: C d) j f ) = for d 4j ) Proof. The asymptotic expansion 4.19) is a direct consequence of 3.11) and of usual properties of trace operation for Weyl quantization. Let us compute C d) f ). We have the integral formula: C d) f ) = P x) z) ξ + P x) z) fz)dzdξ dx, where dx = π) d dx. By the residue theorem we get: C d) f ) = [ ) )] f Px)+ i ξ + f Px) i ξ dξ dx. For a>wehave: ) f Px)+ a ξ dξ dx = a d ) f Px)+ ξ dξ dx.

7 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) So by analytic extension and evaluation at a = i we get formulas 4.) and 4.1). In particular we see that for d even, there exists f satisfying 4.17) such that C d) f ). For j 1, using 3.15), we have: C d) j f ) = j+1 3j P x) z)q j x, P x) z, ξ) Lz; x,ξ) +1 fz)dzdξ dx. 4.4) Let us now prove that C d) j f ) =, for 4j + 1 d. To do that it is convenient to introduce the following integral, for u>,v>, v z) ν J,ν fu,v)= u + v z) fz)dz. 4.5) ) +1 We have easily, J,ν fu,v)= 1)! And using the residue theorem, we get: J,ν fu,v)= iν 1 u ν 1)/ J,ν f u u, v). 4.6) 1) ν+1 f v + i u ) + f v i u )). 4.7) From 4.6) and 4.7) we can compute J,ν fu,v). To prove that C d) j f ) = ford 4j +1, we shall prove that each term in the sum 4.4) vanishes, after integration in z and ξ. Suppose first that j + 1 j + 1. We have: ) x,px) z, ξ = R ν,γ x) Px) z ) ν ξ γ. Hence, is a sum of integrals lie Q j By integration by parts in z we have: ν,γ P x) z)q j x, P x) z, ξ) Lz; x,ξ) +1 fz)dzdξ, I ν f )x, ξ) = P x) z) ν fz)dz. Lz; x,ξ) +1 Iν+1 f ) = ν I 1 ν 1 f ) 1 I ν 1 f ). 4.8) So, we can assume that ν =. But we have: I l 1)l l g)x, ξ) = l! u l J,νg u, P x) ) u= ξ. So we have I lg)x, ξ) = O ξ l ) near ξ =. Now we remar that for l j + 1 and d 4j + 1wehave l< d + 1, hence ξ I l g)x, ξ) is integrable and, using the analytic dilation argument already used for j =, we get I l g)x, ξ) =, hence P x) z)q j x, P x) z, ξ) Lz; x,ξ) +1 fz)dzdξ =.

8 156 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Now, assume that j + 3j. Using Lemma 3., we have: ) x,px) z, ξ = Q j ν+ γ 1 j) R ν,γ x) Px) z ) ν ξ γ. As above, we integrate by parts in z to have the possibility to put ν at and then we use ξ γ to decrease the order of the singularity in ξ as far as possible integrability near ξ = ) of Q j dz. We conclude by the analytic dilation L argument. +1 So we have proven the last statement of Theorem 4.. We conjecture that the next following terms are not ; more precisely we claim: Conjecture. For every j N, j 1, there exists f satisfying 4.17) such that we have: C 4j 1) j f ), and C 4j 3) j f ). 4.9) In the following sections we shall chec this conjecture for d = 1, 3 and we shall compute analytic formula for C 5) 4 f ) and C7) 4 f ). Unfortunately, these analytic expressions have many terms and it is not obvious that Cd) 4 f ) for d = 5, 7, for every elliptic polynomial P. We shall see that this is true for convex polynomials for d = 7 and satisfying a technical condition if d = 5. Moreover we get, using numerical computations for particular non-convex polynomials P, that C d) 4 f ). As we shall see in the next section, the property C d) j f ) gives easily a lower bounds on the density of eigenvalues. Remar 4.3. Following Remar 3.4 we can extend our results to polyhomogeneous polynomials P = P m + P m 1 + +P 1 + P. To follow the dependence in the coefficients, we note C j f, P ) the coefficient C j f ) with polynomial P. In particular we have C d) j f, P ε) ) =, for d 4j + 1 and for every ε small enough. We have used the notations of Remar 3.4. Assume now that d = 4j 3ord = 4j 1, j 1. Then using a Taylor expansion in ε, computed for ε = h 1/m+1), we get: C j f,p h 1/m+1) ) ) γ h /m+1), 4.3) in particular if γ = C d) j f, P m ) then from Remar 3.4 we get that C d) j f, P ). So it is enough to prove the conjecture for homogeneous polynomials P. 5. Estimate the density of eigenvalues First of all let us remar that the nonlinear spectrum Sp[ ˆL] of ˆL is included in the two quarters {z C, Rz), ±Iz) > }. On one side, it is easy to see that if λ R and Lλ)u = then u =. On the other side, if Rλ) < and Lλ)u =, computing I Lλ)u, u ) we conclude that u =. Let us denote by N h R) = #{z Sp[ ˆL]; z R} and NR)= N h=1 R). Proposition 5.1. There exists C> such that N h R) Cm+1)/m h d, R 1, h ], 1]. 5.31) If C d) j f ) with d>j, then for every r>, ε> there exists c ε,r > such that N h r h ε) c ε,r h δ, h ], 1], 5.3) where δ = d j. Moreover if j = d even) then the estimates is valid with ε =. So that, in even dimension, for every R>, N h R) behaves lie h d.

9 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Proof. The proof of 5.31) is a consequence of the Weyl Ky Fan inequality as it is formulated in []. 1 Let us denote s j the eigenvalues of A ˆL A ˆL )1/, and ν h R) = #{j,s j R} its counting function. We can assume that the symbol of A ˆL is invertible in a neighborhood of. Then the Weyl Ky Fan inequality can be written [], From this inequality we get easily, R N h r) r R dr ν h r) r dr, R>. 5.33) N h R) 1 log R ν h r) r dr. 5.34) Using the semiclassical functional calculus of [1] applied to A ˆL A ˆL )1/ we get that ν h r) = Or dm+1)/m h d ) and 5.31) follows from 5.34). Let us now prove 5.3). We first remar that for every ε> there exists R ε > such that if, then we have: π/ ε arg z π/ + ε, μ R ε, t + z 1 ε) t + z ). Let us choose fλ)= λ + t) μ with large enough μ >dm+ 1)/m) and t>. We apply 4.19) to get the following inequalities: C 1 h δ t + z) μ t + z μ ) μ. C t + z z Sp[ ˆL] z Sp[ ˆL] z Sp[ ˆL] But for every μ, μ 1, large enough, such that μ μ 1 is large enough, we have: ) μ ) t + z R μ 1 μ1 μ t + z. z Sp[ ˆL] z R z Sp[ ˆL] We choose now R = r h ε to get: N h r h ε) ) 1 + μ cε,r h δ. μ R The above results concern the semiclassical regime. Now we give estimates for h = 1 and high energy regime. Corollary 5.. For R + we have: NR)= O m+1)/m). If C d) j f ) with d j >, then for every ε> there exits c ε > such that If j =, the estimate is true with ε = and c >. c ε R δm+1)/m ε NR). 1 We than J. Sjöstrand for this reference.

10 158 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) d and 3-d cases In this section we prove the following result. Theorem 6.1. For d = 1, 3, there exists f satisfying 4.17) such that for every m, we have C d) f ). More precisely, we have: C 1) f ) = 1 f 3) Px) ) P x) dx, 6.35) 16 C 3) 1 f ) = 48π We can choose fλ)= λ + t) μ with μ>dm+ 1)/m and t>. R R 3 f Px) ) Px) dx. 6.36) Proof. We compute with the explicit form we got before for K.Wehave,ford = 1, 3, C d f, x) = P z)h dz, where H = K dξ. But we have: H = P z) d 5 Pb 3 b 4,1 ) + P z) d 6 P b 3 b 4 b 4,1 ), hence and C 1) f ) = 1 8 π 3 b 4,1 4 3 b 3, + ) 3 b 4 R f 3) Px) ) P x) dx. 6.37) C 1) f ) = 1 16 f 3) Px) ) P x) dx. 6.38) C 3) f ) = π) 3 4b 4,1 b 4 b 3 ) C 3) 1 f ) = 48π R 3 R R 3 f Px) ) Px) dx 6.39) f Px) ) P x) dx. 6.4) We have seen that for d odd, d 5, C d) 7. 5-d and 7-d cases f ) =. So we have to compute Cd) 4 f ) for d = 5, 7. We have to compute in more details the term K 4 from 3.14). Recall that we have: ) C d) 4 f ) = P z) K 4 z; x,ξ)dξ fz)dz dx. 7.41) x We have to compute the following three integrals, depending on x and z C. ξ

11 I 1) β = ; β) I ) α F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) = α,) 1 ξ + P x) z) β ξ I 3) β = ; β) 1 ξ + P x) z) ) dξ; β =4, 7.4) 1 ξ + P x) z) α ξ ξ ) x α K dξ; α =, 7.43) 1 ξ + P x) z) β ξ K dξ; β =. 7.44) Using the new variable η such that ξ = P z)η plus an analytic extension), we get: ) I 1) aβ) β =, aβ)= P x) z) 8 d 16β! 1 + η β η 1 + η dη. 7.45) aβ) only when β = β 1,...,β d ) is such that β j = 4 and β = for j, orβ j = β =, j and β l = if l j, l. In the first case aβ) = a 1 and in the second case aβ) = a, where a 1 = 1 4η1 1 + η )) η ) 6 dη, 7.46) a = It is convenient to introduce the following notations: b j = dη 1 + η ) j, b j, = η1 η 1 + η dη. 7.47) ) 6 η1 dη 1 + η ) j, b j,,l = η 1 ηl dη 1 + η ) j, 7.48) where j,,l N are such that the integrals are finite. Of course these integrals can be computed with the Euler beta and gamma special functions see Appendix A for more explicit expressions). So we have a 1 = 1 6 b 6, 1 3 b 5, b 4 and a = b 6,1,1. Using integration by parts, in x or in ξ, we get the following formulas: C 4 f ) = C 4 f ; x) dx, 7.49) where and C 4 f ; x) = C 4,1 f ; x) + C 4, f ; x) + C 4,3 f ; x) 7.5) C 4,1 f ; x) = α = P z) β x P z) I 1) ) fz)dz, 7.51) β =4 C 4, f ; x) = α,) α ξ C 4,3 f ; x) = P z) x β β ξ ) α x P z Lz) ) ) K dξ fz)dz, 7.5) 3) Lz)I β fz)dz. 7.53) Now we have to compute each term. These computations are not difficult but they are very technical, so we do not give here all the details. They are performed in [1]. We use the notations:

12 16 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) j = xj, j = x j, j, = x j,x, 7.54) a 1 = 1 6 b 6, 1 3 b 5, b 4, 7.55) a = b 6,1, ) For d = 5 we get: where For d = 7 we get: where C 4 f ; x) = A x)f Px) ) + A 1 x)f Px) ) + A x)f Px) ) + A 3 x)f 3) Px) ), 7.57) A x) = π 3 j 4 4 P π 3 j 48 P, 7.58) j j< A 1 x) = 11π 3 48 j P ) π + 48 j, P ) π 3 16 P) π 4 j P ) P ), 7.59) A x) = π 3 A 3 x) = π 3 j 96 P P + π 3 j 16 j j P ) j P) j + π 48 j P ) P) + π 4 j, P ) j P ) P), 7.6) 576 P 4 π 3 96 j j j P) 4 π 3 j P) P). 7.61) 96 j j C 4 f ; x) = A x)f Px) ) + A 1 f Px) ), 7.6) A x) = π 4 1 j P ) j P) + π 3 36 j P ) P) + π 3 18 j, P ) j P ) P), 7.63) A 1 x) = π 4 We should lie to use these formulas with 4 P 4 π 4 4 j j j j P) 4 π 3 j P) P). 7.64) 4 j fλ)= λ + t) μ, with μ>dm+ 1)/m, t >, 7.65) to prove that C d) 4 f ) = C 4 f ; x) dx d = 5, 7). With f lie in 7.65) we can see easily that C d) 4 f ) for the following polynomials: 1 Px)= 1 j d α j xj m, α j >, 1 j d, d = 5, 7. Px)= 1 j, d α j,x j x, is a positive-definite quadratic form, d = 5, 7. 3 d = 7 and P is convex. 4 d = 5, P is convex and satisfies the inequalities: j P j 4 P, 7.66) 1 j< 5 1 j 5 j j, P ) 11π j P ). 7.67) j 1 j 5

13 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) For elliptic and non-convex polynomials, we can chec that C d) 4 f ), d = 5, 7, for many examples with numerical computations, supporting our conjecture that for every elliptic polynomial P, C d) 4 f ), if d = 5, 7see Appendix A). For d = 9, 11, it seems difficult to compute C d) 6 f ) by hand. We need help from symbolic and numerical computations to chec our conjecture. Acnowledgements This wor was supported by the program ANR 8-BLAN-8, NONAa, Research French Ministry. Appendix A A.1. Formulas for b j,,l We assume d 3 and j q>1. We have: where So computing in polar coordinates, + r q 1 + r ) j dr = 1 B Bx,y) = 1 t x 1 1 t) y 1 dt = b j d) = π d/ j d/). j) Now, by elementary computations we get easily A.. Numerical computations for C 4 f ) q + 1,j q + 1 ), A.68) x) y) x+ y). b j,1 d) = 1 bj 1 d) b j d) ), A.69) d b j, d) = B 5/,j d + 4 ) b j d 1), A.7) b j,1,1 d) = 1 8 B 3,j d + 4 ) b j d ). A.71) The following computations have been performed by Guy Moebs, Research Engineer, Laboratoire Jean-Leray, CNRS-University of Nantes. The method used to compute multi-dimensional integrals is Monte-Carlo, with a cut-off of the domain to reduce it in a bounded domain fitting with the behavior of the polynomial P. P is chosen non-convex, because for this case we have no mathematical proof that C 4 f ). In each example, 1 simulations are computed with at least 1 9 events. f is chosen lie in 7.65) with t = 1 and μ> large enough. Example 1. d = 5, Px)= d j=1 x j 4 + αx 1 x. α C 4 f )

14 16 F. Aboud, D. Robert / J. Math. Pures Appl. 93 1) Example. d = 7, Px)= d j=1 x j 4 + αx 1 x + βx 3 x 4. α β C 4 f ) Example 3. d = 5, Px)= d j=1 x j 6 + αx 1 x 4 + βx 3 x 4 4. References α, β) C 4 f ) 1, 1) [1] F. Aboud, Ph.D. thesis, University of Nantes, May 9. To be published in [] M.S. Agranovich, A.S. Marus, On spectral properties of elliptic pseudo-differential operators far from self-adjoint ones, Z. Anal. Anwendungen 8 3) 1989) [3] K.Kh. Boimatov, A.G. Kostyucheno, Trace formula for polynomial operators and its applications, translated from Funtsional. Anal. i Prilozhen. 5 4) 1991) [4] A. Bouzouina, D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Due Math. J. 111 ) ) 3 5. [5] S. Chanillo, B. Helffer, A. Laptev, Nonlinear eigenvalues and analytic hypoellipticity, J. Funct. Anal. 9 ) 4) [6] M. Christ, Analytic hypoellipticity, representation of nilpotent groups and non-linear eigenvalue problem, Due Math. J. 1993) [7] M. Christ, Examples of analytic non-hypoellipticity of b, Comm. PDE ) [8] M. Dauge, D. Robert, Weyl s formula for a class of pseudodifferential operators with negative order on L R n ), in: Lecture Notes in Math., vol. 156, Springer-Verlag, 1986, pp [9] A. Friedman, M. Shinbrot, Nonlinear eigenvalue problems, Acta Math ) [1] I. Gohberg, M.G. Krein, Introduction à la théorie des opérateurs non nécessairement auto-adjoints, Dunod, 197. [11] B. Helffer, Remarques sur des résultats de G. Métivier sur la nonhypoanalyticité, Séminaire de l Université de Nantes, exposé No. 9, [1] B. Helffer, D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal. 53 3) 1983) [13] B. Helffer, D. Robert, Xue Ping Wang, Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity, Algebra i Analiz 16 1) 4) [14] M.V. Keldysh, On the completeness of the eigenfunctions of some classes of non-self-adjoint linear operators, Russian Math. Surveys 6 4) 1971) [15] M.G. Krein, H. Langer, On the mathematical principles in the linear theory of damped oscillations of continua I, Integral Equations Operator Theory 1 3) 1978) [16] A.S. Marus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, [17] G. Métivier, Une classe d opérateurs non hypoelliptiques analytiques, Séminaire Goulaouic Schwartz, Exposé XII, [18] Pham The Lai, D. Robert, Sur un problème aux valeurs propres non linéaire, J. Math ) [19] D. Robert, Autour de l approximation semi-classique, Progress in Mathematics, vol. 68, Birhäuser, [] D. Robert, Nonlinear Eigenvalues Problems, Matematica Contemporanea, vol. 6, Sociedade Brasileira de Matematica, 4. [1] R. Seeley, Complex powers of an elliptic operators, in: Singular Integrals, in: Proc. Symposia Pure Math., vol. 11, Amer. Math. Soc., 1967, pp