For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov-Shabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type

Grébert, B

Kappeler, T

ZORA

Zurich Open Repository andArchiveUniversity of ZurichMain LibraryStrickhofstrasse 39CH-8057 Zurichwww.zora.uzh.chYear: 2003Density of finite gap potentials for the Zakharov-Shabat systemGrébert, B; Kappeler, TAbstract: For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov-Shabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analyticpotentials of a given type.Posted at the Zurich Open Repository and Archive, University of ZurichZORA URL: http://doi.org/10.5167/uzh-21875Originally published at:Grébert, B; Kappeler, T (2003). Density of finite gap potentials for the Zakharov-Shabat system. Asymp-totic Analysis, 33(1):1-8.Asymptotic Analysis 33 (2003) 1–8 1IOS PressDensity of finite gap potentialsfor the Zakharov–Shabat systemB. Grébert a and T. Kappeler ba UMR 6629 CNRS, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes cedex 3,Franceb Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, SwitzerlandAbstract. For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov–Shabat system is dense.In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type.1. IntroductionThis paper is an addendum to our investigation of spectral properties of the Zakharov–Shabat operatorL(ψ) := i(1 00 −1) ddx +(0 ψ1ψ2 0)initiated in [6,5] (cf. also [4]). Its aim is to establish the density of finite gap potentials in weightedSobolev spaces. Here ψ = (ψ1,ψ2) and the components ψ1 and ψ2 are 1-periodic functions in theweighted Sobolev space Hw ≡ HwCof 2-periodic functionsHw :={f | f (x) :=∞∑−∞fˆ (k) eiπkx; ‖f‖w <∞}with‖f‖w :=(∑k∈Zw(k)2∣∣fˆ (k)∣∣2)1/2and w = (w(k))k∈Z is a weight, i.e., a sequence of positive numbers with w(k)  1, w(−k) = w(k)and w(k)  w(k − j)w(j) ∀j, k ∈ Z. Denote by (λ±n (ψ))n∈Z the periodic eigenvalues of L(ψ) whenconsidered on the interval [0, 2] (cf. [5] for a detailed description of the periodic spectrum). As usual,(λ±n (ψ))n∈Z is arranged in lexicographic ordering. The potential ψ is said to be a finite gap potential if{n ∈ Z | λ+n (ψ) = λ−n (ψ)} is finite. Recall from [5] that a weight w is said to be a δ-weight (δ > 0) ifw∗(k) := (1 + |k|)−δw(k) is a weight as well. The potential ψ = (ψ1,ψ2) is said to be of real type ifψ1 = ψ2. In such a case the operator L(ψ) is selfadjoint.0921-7134/03/$8.00  2003 – IOS Press. All rights reserved2 B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat systemTheorem 1.1. Let w be a δ-weight for some δ > 0. For any ε > 0 and any 1-periodic potential ψ =(ψ1,ψ2) in Hw ×Hw there is a finite gap potential ψε = (ψε1,ψε2) in Hw ×Hw at distance at most εfrom ψ, i.e., sup1j2‖ψεj − ψj‖w < ε, so thatλ+n (ψε) = λ−n (ψε), ∀|n|  Nε,where Nε ≡ Nε(ψ)  1.If ψ is of real type, i.e., ψ1 = ψ2, then ψε can be chosen to be of real type as well.Remark 1.2. The number Nε in Theorem 1.1 can be chosen uniformly on compact sets of 1-periodicpotentials in Hw.Remark 1.3. Since Hw × Hw is dense in L2([0, 1];C2) for any δ-weight w Theorem 1.1 implies thatfinite gap potentials are dense in L2([0, 1];C2).In the special case where ψ is of real type and an element of the Sobolev space Hwp with wp denotingthe Sobolev weight wp(k) := (1+ |k|)p (k ∈ Z) and p ∈ Z0, Theorem 1.1 has been proved in [10] and,independently in [3], using results established in [9] and [2] respectively.Similar results as the ones presented here for the Zakharov–Shabat operator L(ψ1,ψ2) have been ob-tained previously for the Hill operator −d2/dx2 + V in [11].To prove Theorem 1.1 (cf. [5] for a discussion of the spaces Hw) we follow the approach used in [11]:as a set-up we take the Fourier bloc decomposition introduced first for the Hill operator in [7,8] andworked out subsequently for the Zakharov–Shabat operator in [6,5]. Unlike in [11] where a contractionmapping argument was used to obtain the density results for the Hill operator, we get a short proof ofTheorem 1.1 by applying the inverse function theorem in a straightforward way. As in [11] the mainfeature of the presented proof is that it does not involve any results from inverse spectral theory (cf. [1]for related results for the Hill equation). Throughout the remainder of this paper we use the notationintroduced in [5].2. Proof of Theorem 1.1Note that for any two weights w, w′ with w  w′, Hw′ is a dense subspace of Hw. Hence withoutloss of any generality we may assume that w is a δ-weight withδ =12.As explained in [5, Section 2.6], for M  1 arbitrary, there exists N ≡ NM  1 so that for anyψ = (ψ1,ψ2) with‖ψ‖w := supj=1,2‖ψj‖w Mthe eigenvalues λ±n (ψ) for |n|  NM are of the formλ±n (ψ) = nπ + z±n , |z±n | <π4,B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat system 3where the complex numbers z±n ≡ z±n (ψ) are the two solutions in |z| < π/4 of the following systemz = α(n, z)+ ζ , (2.1)ζ2 − (ψˆ2(2n)+ β+(n, z))(ψˆ1(−2n)+ β−(n, z)) = 0 (2.2)with α(n, z) ≡ α(n, z,ψ) and β±(n, z) ≡ β±(n, z,ψ) given by (cf. [5, (2.23)–(2.26), (2.29)])α(n, z) := 〈Snψˆ2, (z −Dn)−1(Id−Qn)−1SnJψˆ1〉, (2.3)β+(n, z) := 〈Snψˆ2, (z −Dn)−1R(1)n (Id−Pn)−1Snψˆ2〉, (2.4)β−(n, z) := 〈SnJψˆ1, (z −Dn)−1R(2)n (Id−Qn)−1SnJψˆ1〉. (2.5)Here S and J denote the shift respectively involution operator given byS :  2(Z) →  2(Z), Sa(k) := a(k + 1), ∀k ∈ Z,J :  2(Z) →  2(Z), Ja(k) := a(−k), ∀k ∈ Z,and, for any n ∈ Z, 〈·, ·〉 ≡ 〈·, ·〉Z\n denotes the bilinear form (no complex conjugation)〈(ab),(cd)〉:=∑k =n(a(k)c(k) + b(k)d(k)).Finally, Dn, Pn, Qn, R(1)n and R(2)n are operators on  2(Z\n) defined byDn :=((k − n)πδkj)k,j∈Z\n, Pn := R(2)n R(1)n , Qn := R(1)n R(2)nandR(1)n a := J(ψˆ1 ∗ (z −Dn)−1)a, R(2)n a := ψˆ2 ∗ J(z −Dn)−1a.By (2.1), z+n = z−n whenever ζ+n = 0 and ζ−n = 0. Hence by (2.2) (cf. [5, Proposition 2.12]), a sufficientcondition for λ+n (ψ) = λ−n (ψ) is that there exists zn with |zn| < π/4 so thatψˆ2(2n)+ β+(n, zn,ψ) = 0, (2.6)zn − α(n, zn,ψ) = 0. (2.7)This suggests a way how to prove Theorem 1.1: given ε > 0 and a 1-periodic potential ψ ∈ Hw ×Hwwith ‖ψ‖w  M/4 we want to find Nε ≡ Nε(ψ)  NM so that there exists a 1-periodic potential ψεin Hw ×Hw with ‖ψε − ψ‖w  ε and (zn)|n|Nε with |zn| < π/4 satisfying the system of Eqs (2.6),(2.7). For the appropriate set-up of this system we review some properties of the coefficients α(n, z) andβ+(n, z) established in [5].4 B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat systemFirst let us introduce some more notation. Denote by Dr ⊆ C the open disc, Dr := {z ∈ C | |z| < r}and by BwM the open ball,BwM :={ψ = (ψ1,ψ2) | ψj ∈ Hw 1-periodic; ‖ψj‖w < M , j = 1, 2}.Concerning the coefficient α(n, z,ψ), recall from [5] that for any |n|  NM , α(n, z,ψ) is defined for(z,ψ) ∈ Dπ/4 × BwM . From the definition of α(n, z,ψ) it is straightforward to see that it is analytic onDπ/4 ×BwM .Recall the following Lemma 2.7 of [5]. (Note that we assume throughout that w is a δ-weight withδ = 1/2.)Lemma 2.1. For any (z,ψ) ∈ Dπ/4 ×BwM and any |n|  NM ,∣∣α(n, z,ψ)∣∣  4M2〈n〉 .By Cauchy’s estimate, Lemma 2.1 leads to the following estimates of the differentials dψα and dzα.Corollary 2.2. For any (z,ψ) ∈ Dπ/8 ×BwM/2 and any |n|  NM∥∥dψα(n, z,ψ)∥∥  16M〈n〉 ,∣∣dzα(n, z,ψ)∣∣  32M2π1〈n〉 .To state the next result denote byD∞π/8 ≡ D∞π/8(Z \ [−NM ,NM ]) ⊆  ∞(Z \ [−NM ,NM ];C)the open ball of radius π/8 centered at 0 in the Banach space  ∞(Z \ [−NM ,NM ];C).Proposition 2.3. The mapD∞π/8 ×BwM/2 →  2(Z \ [−NM ,NM ],C),((zn)|n|>NM ,ψ) → (α(n, zn,ψ))|n|>NMis analytic and satisfies for any N  NM∥∥(α(n, zn,ψ))|n|>N∥∥ 2M2N−1/2,∥∥dψ(α(n, zn,ψ))|n|>N∥∥ 32MN−1/2and, with Z := (zn)|n|>N ,∥∥dZ(α(n, zn,ψ))|n|>N∥∥ = ∥∥(δnk(∂zkα(n, zn,V ))|n|,|k|>N)∥∥  32M2πN−1.B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat system 5Concerning the coefficient β+(n, z,ψ) recall from [5] that for any |n|  NM , β+(n, z,ψ) is welldefined for (z,V ) ∈ Dπ/4 ×BwM .From its definition, it is straightforward to see that β+(n, z,ψ) is analytic on Dπ/4 × BwM . Recall thefollowing Proposition 2.10 in [5]. (Again note that we have chosen δ = 1/2, hence δ∗ = 1/2.)Lemma 2.4. There exists C  1 so that for any ψ ∈ BwM ,∑|n|>NM〈n〉2w(2n)2 sup|z|π/4∣∣β+(n, z,ψ)∣∣2  C.Again by Cauchy’s estimate we obtain estimates for the derivatives of β+(n, z,ψ). To state them recallthat  2w denotes the  2-Hilbert space with weight w and w1 denotes the weightw(k) := 〈k〉w(k) (∀k ∈ Z).Proposition 2.5. The mapD∞π/8 ×BwM/2 →  2w1(Z \ [−NM ,NM ];C),((zn)|n|>NM ,ψ) → (β+(n, zn,ψ))|n|>NMis analytic and satisfies for any N  NM , with C  1 given as in Lemma 2.4,(i) ‖(β+(n, zn,ψ))|n|>N∥∥w  2CN−1/2,(ii) ‖dψ(β+(n, zn,ψ))|n|>N‖w  (4C/M )N−1/2and for any Z := (zn)|n|>N ∈ D∞π/8(iii) ‖dZ (β+(n, zn,ψ))|n|>N‖w  (16C/π)N−1/2.It turns out that a convenient set-up for the system of Eqs (2.6), (2.7) is the following one: introducefor N  NME ≡ EN :={v = (vn)|n|>N | ‖v‖w <M4},where ‖v‖w = (∑|n|>Nw(2n)2|vn|2)1/2 andZ ≡ ZN :={Z = (zn)|n|>N | ‖Z‖ =( ∑|n|>N|zn|2)1/2<π8}.For any ψ = (ψ1,ψ2) ∈ BwM/4 and any v ∈ EN define ψv = (ψv1 ,ψv2 ) byψv2 (x) :=∑|n|Nψˆ2(2n) ei2πnx +∑|n|>Nvn ei2πnx (2.8)6 B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat systemandψv1 (x) :=∑|n|Nψˆ1(2n) ei2πnx +∑|n|>Nv−n ei2πnx. (2.9)The definition of ψv1 has been chosen in such a way that ψv is of real type if ψ is of real type. Noticethat for v ∈ E and ψ ∈ BwM/4, ‖ψvj ‖w  M/2 for j = 1, 2 and hence ψv ∈ BwM/2. Finally, introducefor any given ψ ∈ BwM and N  NM a map Λ ≡ Λ(N )ψ defined for any (v,Z) ∈ E × Z by Λ(v,Z) :=(Λn(v,Z))|n|>N withΛn(v,Z) :=(vn + β+(n, zn,ψv), zn − α(n, zn,ψv)).By Propositions 2.3 and 2.5, the range of Λ(N ) is contained in  2w(Z \ [−N ,N ];C)×  2(Z \ [−N ,N ];C)and is analytic. Choose on  2w ×  2 the norm defined for any (v,Z) ∈  2w ×  2 by∥∥(v,Z)∥∥ = max(‖v‖w , ‖Z‖).Lemma 2.6. For any M  1 there exists N  NM so that for any ψ ∈ BwM/4 and any (v,Z) ∈EN ×ZN ,∥∥dv,ZΛ(N )ψ − Id(N )∥∥  12 ,where Id(N ) denotes the identity map on EN ×ZN .Proof. The differential dv,ZΛ(N ) takes the formdv,ZΛ(N ) =(IdE + dv(β+n )|n|>N dZ (β+n )|n|>N−dv(αn)|n|>N IdZ − dZ (αn)|n|>N).Hence by Propositions 2.3 and 2.5 there exists N  NM so that for any ψ ∈ BwM/4 and (v,Z) ∈ E ×Z ,∥∥dv,ZΛ(N ) − Id(N )‖w∥∥dv(β+n )|n|>N∥∥w + ∥∥dZ(β+n )|n|>N∥∥w + ∥∥dv(αn)|n|>N∥∥+ ∥∥dZ (αn)|n|>N∥∥  1/2. Proof of Theorem 1.1. Let 0 < ε  1 and a 1-periodic potential ψ in Hw. Let M > 4‖ψ‖w and chooseN1  N with N as in Lemma 2.6 so that( ∑|n|>N1w(2n)2|ψˆ1(2n)|2)1/2<ε2and∥∥Z (0)(ψ)∥∥ < π/8, (2.10)whereZ (0) ≡ Z (0)(ψ) := (λ+n (ψ) − nπ)|n|>N1.B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat system 7Further definev(0) ≡ v(0)(ψ) := (ψˆ2(2n))|n|>N1.As ‖v(0)‖w  ‖ψ‖w M/4 it follows that (v(0),Z (0)) ∈ EN1×ZN1 . Hence, by Lemma 2.6, dv(0),Z (0)Λ(N1)ψis invertible. By the inverse function theorem there is η > 0 so that one can find an open neighborhood U1of (v(0),Z (0)) ≡ (v(0)(ψ),Z (0)(ψ)) contained in the open ball Bε/2(v(0),Z (0)) intersected with EN1 ×ZN1so that Λ(N1)ψ is a diffeomorphism from U1 onto the ball Bη(Λ(N1)ψ (v(0),Z (0))) in 2w(Z \ [−N1,N1]; C)×  2(Z× [−N1,N1];C).Choose N2 ≡ N2(ψ) with N2  N1 so that∥∥Λ(N2)ψ (v(0),Z (0))∥∥ < η.Then. :=((Λn(v(0),Z (0))N1<|n|N2, (0, 0)|n|>N2)is an element in Bη(Λ(N1)ψ (v(0),Z (0))). Hence there exists (v,Z) ∈ U1 so that. = Λ(N1)ψ (v,Z).In particular, for any |n| > N2vn + β+(n, zn,ψv) = 0, zn − α(n, zn,ψv) = 0,where ψv is defined as in (2.8), (2.9) with N given by N1. This means that Eqs (2.6), (2.7) are satisfiedfor ψv and (zn)|n|>N2 , hence ψv is a finite gap potential, or more precisely,λ+n(ψv)= λ−n(ψv), ∀|n| > N2.Furthermore, as (v,Z) ∈ U1 ⊂ Bε/2(v(0),Z (0)) we have∥∥ψ2 − ψv2∥∥w =( ∑|n|>N1w(2n)2|vn − v0n|2)1/2< ε/2and, by the definition (2.9) of ψv1 and the estimate (2.10),∥∥ψ1 − ψv1∥∥w ( ∑|n|>N1w(2n)2∣∣ψˆ1(2n)∣∣2)1/2+( ∑|n|>N1w(2n)2|vn|2)1/2< ε8 B. Grébert and T. Kappeler / Density of finite gap potentials for the Zakharov–Shabat systemhence∥∥ψ − ψv∥∥w = sup1j2∥∥ψj − ψvj ∥∥ < ε.This shows the claimed result with ψε := ψv and Nε := N2 for an arbitrary potential ψ ∈ Hw. If ψ is ofreal type ψε = ψv is of real type as well by the definition (2.8), (2.9) of ψv. References[1] Y. Colin de Verdière and T. Kappeler, On double eigenvalues of Hill’s operator, J. Funct. Anal. 86 (1989), 127–135.[2] J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators II, Comment. Math. Helv.62 (1987), 18–37.[3] B. Grébert and J.-C. Guillot, Gaps of one dimensional periodic AKNS systems, Forum Math. 5 (1993), 459–504.[4] B. Grébert and T. Kappeler, Gap estimates of the spectrum of the Zakharov–Shabat system, Preprint, Université PaulSabatier, 1997.[5] B. Grébert and T. Kappeler, Estimates on periodic and Dirichlet eigenvalues for the Zakharov–Shabat system, AsymptoticAnal. 25 (2001), 201–237.[6] B. Grébert, T. Kappeler and B. Mityagin, Gap estimates of the spectrum of the Zakharov–Shabat system, Appl. Math. Lett.11 (1998), 95–97.[7] T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill’s equation and action variables for KdV, Trans. Amer.Math. Soc. 351 (1999), 619–646.[8] T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math.Anal. 33 (2001), 113–152.[9] V.A. Marchenko and I.V. Ostrovsky, Approximation of periodic potentials by finite-zone potentials, Selecta Math. Soviet.6 (1987), 101–136.[10] T.V. Misyura, Finite-zone potentials for Dirac operators, Teor. Funktsii Funktsional. Anal. i Prilozhen. 33 (1980), 107–111(in Russian).[11] B. Mityagin, Manuscript, 2000.

Density of finite gap potentials for the Zakharov-Shabat system

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Density of finite gap potentials for the Zakharov-Shabat system

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