The purpose of this Note is to highlight the spectral instability of some
non-selfadjoint differential operators, by studying the growth rate of the
norms of the spectral projections $\Pi_n$ associated with their eigenvalues.
More precisely, we are concerned with the complex Airy operator and even
anharmonic oscillator. We get asymptotic expansions for the norm of the
spectral projections associated with the large eigenvalues, extending the
results of Davies and Davies-Kuijlaars

Henry, Raphaël

Comptes Rendus Mathematique

English

arXiv

Raphaël Henry

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Spectral instability of some non-selfadjoint anharmonic oscillators

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 1043–1046Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential EquationsSpectral instability of some non-selfadjoint anharmonic oscillatorsInstabilité spectrale de certains oscillateurs anharmoniques non-autoadjointsRaphaël HenryUniversité Paris-Sud, 91405 Orsay cedex, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 5 November 2012Accepted after revision 19 November 2012Presented by the Editorial BoardThe purpose of this Note is to highlight the spectral instability of some non-selfadjointdifferential operators, by studying the growth rate of the norms of the spectral projectionsΠn associated with their eigenvalues. More precisely, we are concerned with someanharmonic oscillators A(m, θ) = − d2dx2+ eiθ |x|m with |θ | < min{ (m+2)π4 , (m+2)π2m }, definedon L2(R). We get asymptotic expansions for the norm of the spectral projections associatedwith the large eigenvalues of A(1, θ) and A(2k, θ), k  1, extending the results of Davies(2000) [4] and Davies and Kuijlaars (2004) [5].© 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.r é s u m éNotre objectif est de mettre en évidence l’instabilité spectrale de certains opérateursdifférentiels non-autoadjoints, via l’étude de la croissance des normes des projecteursspectraux Πn associés à leurs valeurs propres. Nous nous intéressons à certains oscillateursanharmoniques A(m, θ) = − d2dx2+eiθ |x|m avec |θ | < min{ (m+2)π4 , (m+2)π2m }, définis sur L2(R).Nous étendons les résultats de Davies (2000) [4] et Davies et Kuijlaars (2004) [5] endonnant un développement asymptotique de la norme des projecteurs spectraux associésaux grandes valeurs propres pour les opérateurs A(1, θ) et A(2k, θ), k  1.© 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.1. Spectral instability and pseudospectraIt is well known that the spectral theorem implies some control of stability for the spectrum of selfadjoint operators: ifA is a selfadjoint operator acting on the Hilbert space H, the spectrum of its perturbations A + εB, with ε > 0 and anyB ∈L(H), ‖B‖ 1, lies entirely inside an ε-neighborhood of the spectrum σ(A). In other words, the norm of the resolventof A near the spectrum blows up like the inverse distance to the spectrum. It has also been known for several years (seefor instance [13]) that such a behavior could not be expected in general in the case of non-selfadjoint operators. One canunderstand it thanks to the notion of ε-pseudospectra of an operator A, defined as the family of sets σε(A), indexed byε > 0,σε(A) ={ξ ∈ ρ(A): ∥∥(A− ξ)−1∥∥ > 1ε}∪ σ(A).The link between spectral instability and pseudospectra appears more clearly in the following equivalent formulation, whichis a weak version of the Roch and Silbermann theorem [11]:E-mail address: raphael.henry@math.u-psud.fr.1631-073X/$ – see front matter © 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.http://dx.doi.org/10.1016/j.crma.2012.11.0111044 R. Henry / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 1043–1046σε(A) =⋃ω∈L(H),‖B‖1σ(A+ εB)(see also [12] and references therein).In the following, we deal with the instability indices associated with an isolated eigenvalue λ ∈ σ(A). The instabilityindex associated with λ is defined as κ(λ) = ‖Π(λ)‖, where Π(λ) denotes the spectral projection associated with λ. Ofcourse κ(λ)  1 in any case, and κ(λ) = 1 when A is selfadjoint. These numbers κ(λ) are closely related to the size ofε-pseudospectra around λ. Indeed, if σλε denotes the connected component of σε(A) containing λ, and if we assume forsimplicity that σλε ∩ σ(A) = {λ} and σλε is bounded, then the perimeter |∂σ λε | of σλε satisfies (see [3])∣∣∂σ λε ∣∣  2πεκ(λ). (1)In the finite dimensional setting at least, instability indices give a better description of pseudospectra: if A ∈ Mn(C) is adiagonalizable matrix with distinct eigenvalues λ1, . . . , λn , Embree and Trefethen show [13] that there exists ε0 > 0 suchthat, for all ε < ε0,⋃λk∈σ (A)D(λk, εκ(λk) +O(ε2)) ⊂ σε(A) ⊂ ⋃λk∈σ (A)D(λk, εκ(λk) +O(ε2)). (2)In the case of an infinite dimensional space, the validity of this statement should be investigated, as well as the dependanceon λk of the O(ε2) terms.In the following, we study the instability indices of simple non-selfadjoint differential operators introduced by Daviesin [4], for which the instability phenomenon described above will appear clearly. Let us define the anharmonic oscillatorsA(m, θ) = − d2dx2+ eiθ |x|m (3)with|θ | < min{(m + 2)π4,(m + 2)π2m}, (4)defined on L2(R) in [4] by taking the closure of the associated quadratic form defined on C∞0 (R), which is sectorial if θsatisfies (4). According to [4], its spectrum consists of a sequence of discrete simple eigenvalues, denoted in nondecreasingmodulus order by λn = λn(m, θ), |λn| → +∞. The associated spectral projections are of rank 1, and E.-B. Davies showedin [4] that for all m ∈ ]0,+∞[ and θ 	= 0 satisfying (4), for all α > 0, there exists N = N(m, θ,α) 0 such that the instabilityindices κn(m, θ) of A(m, θ) satisfy κn(m, θ)  nα for n  N . This statement has been improved in the case m = 2 of theharmonic oscillator (sometimes referred as the Davies operator), since E.-B. Davies and A. Kuijlaars showed [5] that κn(2, θ)grows exponentially fast as n → +∞, with an explicit rate c(θ): there exists an explicit c(θ) such thatlimn→+∞1nlogκn(2, θ) = c(θ). (5)The purpose of this Note is to prove that this statement actually holds for the so-called complex Airy operator A(1, θ) andfor the even anharmonic oscillators A(2k, θ), k  1.2. Non-selfadjoint anharmonic oscillatorsWe first deal with the complex Airy operator A(1, θ) defined in (3). We show that the corresponding instability indicesκn(1, θ) grow like in (5) as n → +∞. More precisely, we get asymptotic expansions in powers of n−1 as n → +∞.Theorem 2.1. Let 0 < |θ | < 3π/4. There exists a real sequence (α j(θ)) j1 such that the instability indices κn(1, θ) of A(1, θ) satisfy,as n → +∞,exp(−C(θ)(n − 1/2))κn(1, θ) = K (θ)√n(1 ++∞∑j=1α j(θ)n− j)+O(n−∞), (6)whereC(θ) = πm3/2θ |sin θ | and K (θ) =12√3|sin θ |m1/4θ,withmθ =√1 + sin2(2θ/3)sin2 θ− 2 cos(θ/3) sin(2θ/3)sin θ> 0.R. Henry / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 1043–1046 1045Sketch of the proof. Let us first recall that all the eigenvalues of A(m, θ), m ∈ N, have associated spectral projections ofrank 1, see Lemma 5 in [4]. Hence, one can easily check that [3]κn(m, θ) = ‖un‖2〈un, ūn〉 , (7)where un denotes an eigenfunction associated with the n-th eigenvalue of A(m, θ).We get rid of the singularity of the potential at x = 0 by decomposing A(1, θ) into its Dirichlet and Neumann realizationsAD(1, θ) and AN (1, θ) on R+ . We then compute their instability indicesκD/Nn (1, θ) =∫R+ |Ai(μD/Nn + eiθ/3x)|2 dx| ∫R+ Ai(μD/Nn + eiθ/3x)2 dx|, (8)given by formula (7), where x → Ai(μD/Nn + eiθ/3x) is the n-th eigenfunction of AD/N(1, θ), μDn (resp. μNn ) being then-th (negative) zero (resp. critical point) of the Airy function Ai (see [2,8]). We estimate the numerator in (8) by usingthe well-known asymptotic expansion of Ai at infinity in the complex plane (see [1]), and the Laplace method brings anexp(cθ |μD/Nn |3/2) term in κ D/Nn (1, θ), cθ > 0. The integral in the denominator of (8), after deformation of the path of inte-gration by homotopy, is equal to+∞∫μD/NnAi2(x)dx = Ai′2(μD/Nn ) (9)(it is indeed straightforward, using Airy equation, to check that x → xAi2(x) − Ai′ 2(x) is a primitive for Ai2). Hence theexpansion of Ai′(−z) as z → +∞, given in [1], provides an asymptotic expansion for (9) in powers of |μD/Nn |−3/2. Thestatement follows from the behavior of μD/Nn as n → +∞, since we have asymptotic expansions for (n − 1/4)−2/3|μDn |(resp. (n − 3/4)−2/3|μNn |) in powers of (n − 1/4)−2 (resp. (n − 3/4)−2). Notice that the exponential instability appears as soon as θ 	= 0.We have a similar statement for even anharmonic oscillators:Theorem 2.2. Let k ∈ N∗ and θ be such that 0 < |θ | < (k+1)π2k . If κn(2k, θ) denotes the n-th instability index of A(2k, θ) = − d2dx2+eiθ x2k, then there exist K (2k, θ) > 0 and a real sequence (C j(2k, θ)) j1 such thate−ck(θ)nκn(2k, θ) = K (2k, θ)√n(1 ++∞∑j=1C j(2k, θ)n− j)+O(n−∞) (10)as n → +∞, withck(θ) =2(k + 1)√πΓ ( k+12k )ϕk(ξk)Γ ( 12k )> 0, (11)whereξk =(tan(|θ |/(k + 1))sin(k|θ |/(k + 1)) + cos(k|θ |/(k + 1)) tan(|θ |/(k + 1))) 12k, (12)ϕk(ξ) = Imξei θ2(k+1)∫0(1 − t2k)1/2 dt. (13)Sketch of the proof. We first perform an analytic dilation and a scale change to recover the semiclassical selfadjoint an-harmonic oscillator Ph(2k) = −h2 d2dx2 + x2k − 1, with h = hn = |λn(2k, θ)|−k+12k . The n-th instability index of A(2k, θ) thenwritesκn(2k, θ) =∫R|ψh(eiθ2(k+1) x)|2 dx∫Rψ2h (x)dx(14)where ψh solves Ph(2k)ψh = 0, ψh ∈ L2(R) (see (7), after deformation of the integration path in the denominator). Thecomplex WKB method (see [10,14,6]) and the analysis of the Stokes lines of Ph(2k) provide an asymptotic expansion of1046 R. Henry / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 1043–1046ψh(ei θ2(k+1) x) as h → 0, which enables us to determine the asymptotic behaviour of the numerator in (14), using again theLaplace method.On the real axis, ψh is treated separately in its oscillatory region [−1 + δ,1 − δ], δ > 0, and in the neighbourhood of theturning points ±1. Hence, the stationary phase method leads to an asymptotic expansion in powers of h of the denominatorin (14). Finally, the statement follows from the Bohr–Sommerfeld quantization rule for hn (see [7, Exercise 12.3]) or Weylformula [9]. In the harmonic case k = 1 (Davies operator), the first term in (10) yields the Davies–Kuijlaars theorem [5]:limn→+∞1nlog‖Πn‖ = c1(θ) = 4ϕ1(1√2 cos(θ/2))= 2 Re f(eiθ/4√2 cos(θ/2))where f (z) = log(z + √z2 − 1) − z√z2 − 1.3. Eigenfunctions and semigroupsThe following theorem has been proved in [2] in the case of complex Airy operator A(1, θ), and in [4] in the harmoniccase (k = 1), as well as for A(2k, θ), k  2, |θ | < π2 . The proof actually extends to any operator A(2k, θ) with |θ | < (k+1)π2k :Theorem 3.1. For any m = 1,2k, k  1, and any θ satisfying (4), the eigenfunctions of A(m, θ) form a complete set of the space L2(R).Notice however that the eigenfunctions of A(1, θ) and A(2k, θ), k  1, cannot form a Riesz basis because of the growthof the instability indices as n → +∞.Theorem 3.1 and the previous estimates enable us to study the convergence of the operator series defining the semigroupe−tA(m,θ) associated with A(m, θ) when decomposed along the projections Πn .The following statement extends the result of [5] in the harmonic case.Corollary 3.2. Let |θ |  π/2 and e−tA(m,θ) be the semigroup generated by A(m, θ), λn = λn(m, θ) the eigenvalues of A(m, θ), andΠn = Πn(m, θ) the associated spectral projections.Let T (θ) = c1(θ)/ cos(θ/2), where c1(θ) is the constant in (11). The series Σm,θ (t) = ∑+∞n=1 e−tλn(m,θ)Πn(m, θ) is not normallyconvergent in cases m = 1 for any t > 0, and m = 2 for t < T (θ); in cases m = 2 for t > T (θ), and m = 2k for any t > 0, k  2, theseries converges normally towards e−tA(m,θ) and, for N sufficiently large and for some constants C1 = C1(k, θ) and C2 = C2(θ), thefollowing estimate holds∥∥e−tA(m,θ)(I − Π<N)∥∥ ⎧⎨⎩C1√Neck(θ)n exp(−t Re λN), k  2,C2√Nexp(−2 cos(θ/2)(t − T (θ))N), k = 1, t > T (15)where Π<N = Π1 + · · · + ΠN−1 denote the projection on the first N − 1 eigenspaces.AcknowledgementsI am greatly indebted to Professor Bernard Helffer for his help, advice and comments; I am also grateful to Thierry Ra-mond, Christian Gérard and André Martinez for their valuable discussions. I acknowledge the support of the ANR NOSEVOL.References[1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.[2] Y. Almog, The stability of the normal state of superconductors in the presence of electric currents, SIAM J. Math. Anal. 40 (2) (2008) 824–850.[3] A. Aslanyan, E.-B. Davies, Spectral instability for some Schrödinger operators, Numer. Math. 85 (2000) 525–552.[4] E.-B. Davies, Wild spectral behaviour of anharmonic oscillators, Bull. London Math. Soc. 32 (2000) 432–438.[5] E.-B. Davies, A. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2) 70 (2004) 420–426.[6] C. Gérard, A. Grigis, Precise estimates of tunneling and eigenvalues near a potential barrier, J. Differential Equations 72 (1988) 149–177.[7] A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction, London Math. Soc. Lecture Note Ser., vol. 196, 1994.[8] B. Helffer, On pseudo-spectral problems related to a time dependent model in superconductivity with electric current, Confluentes Math. 3 (2) (2011)237–251.[9] B. Helffer, D. Robert, Asymptotique des niveaux d’énergie pour des hamiltoniens à un degré de liberté, Duke Math. J. 49 (4) (1982) 853–868.[10] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.[11] S. Roch, B. Silbermann, C∗-algebras techniques in numerical analysis, J. Operator Theory 35 (1996) 241–280.[12] J. Sjöstrand, Lecture notes: Spectral properties of non-self-adjoint operators, Journ. Equ. Dériv. Partielles (2009), Exp. No. I, 111 pp.[13] L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ,2005.[14] A. Voros, Spectre de l’équation de Schrödinger et méthode BKW, Publications Mathématiques d’Orsay, 81.09, 1981.

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Spectral instability of some non-selfadjoint anharmonic oscillators

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