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Estimates on Periodic and Dirichlet Eigenvalues for the Zakharov-Shabat System

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Estimates on Periodic and Dirichlet Eigenvalues for the Zakharov-Shabat System Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system B. Gr´ebert1, T. Kappeler2 December 19, 2002 1. UMR 6629 CNRS, Universite de Nantes, 2 rue de la Houssi`ere, BP 92208, 44322 Nantes cedex 3, France. 2. Institut fur¨ Mathematik, Universit¨at Zuric¨ h, Winterthurerstrasse 190, CH-8057 Zuric¨ h, Switzerland. Abstract Consider the 2 2 first order system due to Zakharov-Shabat, × 1 0 0 LY := i Y 0 + 1 Y = λY 0 1 0 − 2 with 1; 2 being complex valued functions of period one in the weighted w w Sobolev space H HC : Denote by spec( ; ) the set of peri- ≡ 1 2 odic eigenvalues of L( 1; 2) with respect to the interval [0; 2] and by specDir( 1; 2) the set of Dirichlet eigenvalues of L( 1; 2) when con- sidered on the interval [0; 1]. It is well known that spec( 1; 2) and specDir( 1; 2) are discrete. Theorem Assume that w is a weight such that, for some δ > 0, w (k) = (1 + k )−δw(k) is a weight as well. Then for any bounded −δ j j subset B of 1-periodic elements in H w Hw there exist N 1 and M × ≥ ≥ 1 so that for any k N, and ( ; ) B, the set spec( ; ) λ j j ≥ 1 2 2 1 2 \f 2 1 C λ kπ < π=2 contains exactly one isolated pair of eigenvalues j j − j g λ+; λ− and spec ( ; ) λ C λ kπ < π contains a f k k g Dir 1 2 \ f 2 j j − j 2 g single Dirichlet eigenvalue µk. These eigenvalues satisfy the following estimates (i) w(2k)2 λ+ λ− 2 M; jk|≥N j k − k j ≤ (λ++λ−) (ii) P w(2k)2 k k µ 2 M: jk|≥N j 2 − kj ≤ P Furthermore spec( ; ) λ; k N and spec ( ; ) µ 1 2 nf k j j ≥ g Dir 1 2 nf k j k N are contained in λ C λ < Nπ π=2 and its cardinality j j ≥ g f 2 j j j − g is 4N 2, respectively 2N 1. − − When = (respectively = ); L( ; ) is one of the op- 2 1 2 − 1 1 2 erators in the Lax pair for the defocusing (resp. focusing) nonlinear Schr¨odinger equation. Contents 1 Introduction 3 1.1 Results . 3 1.2 Comments . 5 1.3 Method of proof . 7 1.4 Related work . 8 2 Periodic eigenvalues 8 2.1 Lyapunov-Schmidt decomposition . 8 2.2 P -equation . 11 2.3 Q-equation . 17 2.4 Estimates for α(n; z) . 19 2.5 Estimates for β(n; z) . 20 2.6 z-equation . 24 2.7 ζ-equation . 25 2.8 Proof of Theorem 1.1 . 26 2.9 Improvement of Theorem 1.1 for L selfadjoint . 30 2 3 Riesz spaces and normal form of L 32 3.1 Riesz spaces . 32 3.2 Normal form of L . 34 4 Dirichlet eigenvalues 36 4.1 Dirichlet boundary value problem . 36 4.2 Decomposition . 38 4.3 Proof of Theorem 1.2 . 40 A Appendix A: Spectral properties of L( 1; 2) 41 B Appendix B: Proof of Lemma 2.8 44 1 Introduction 1.1 Results Consider the Zakharov-Shabat operator (see [ZS]) 1 0 d 0 1 L( 1; 2) := i + 0 1 dx 2 0 − w w where ; are 1-periodic elements in the weighted Sobolev space H HC 1 2 ≡ of 2-periodic functions 1 Hw := f(x) = f^(k)eiπkx f < f j k kw 1g −∞ X with 2 2 1=2 f w := (2 w(k) f^(k) ) k k Z j j Xk2 and w = (w(k))k2Z a weight, i.e. a sequence of positive numbers with w(k) 1; w( k) = w(k) ( k Z) and the following submultiplicative property ≥ − 8 2 w(k) w(k j)w(j) k; j Z: ≤ − 8 2 3 As an example of such a weight we mention the Sobolev weights sN N ≡ (s (k)) Z; s (k) := k , where, for convenience, n k2 N h i k := 1 + k ; h i j j or more generally, the Abel-Sobolev weight w (w (k)) Z a;b ≡ a;b k2 w (k) := k aebjkj (a 0; b 0): a;b h i ≥ ≥ wa;b ^ iπkx An element H is a complex valued function f(x) = k2Z f(k)e , 2 b which admits an analytic extension f(x + iy) to the strip y < π such that b b a j Pj a 1 f(x + i ) and f(x i ) are both in the Sobolev space HC H ( ; C). De- π − π ≡ S note by spec( 1; 2) the periodic spectrum of L( 1; 2) when considered on the interval [0; 2] and by specDir( 1; 2) the Dirichlet spectrum of L( 1; 2) when considered on [0; 1]. It is well known that both, spec( 1; 2) and specDir( 1; 2) are discrete. The main purpose of this paper is to study the asymptotics of the large (in absolute value) eigenvalues in spec( 1; 2) and specDir( 1; 2) for 1-periodic w functions 1; 2 in H . To formulate our first result we need to introduce some more notation: we say that w is a δ-weight for δ > 0 if w (k) := k −δw(k) ∗ h i is a weight as well. Notice that the Abel-Sobolev weight wa;b is a δ-weight iff 0 < δ a. Let ≤ 1 1 δ := δ = inf(δ; ) : ∗ ^ 2 2 Further let 1=2 ρ := ( ^ (2n) + β+(n))( ^ ( 2n) + β−(n)) n 2 0 1 − 0 with an arbitrary, but fixed choice of the square root and ^ (k + n) ^ ( k j) β+(n) := 2 1 − − ^ (j + n) 0 (k n)π (j n)π 2 kX;j6=n − − ^ ( k n) ^ (k + j) β−(n) := 1 − − 2 ^ ( j n) 0 (k n)π (j n)π 1 − − kX;j6=n − − The first result concerns the periodic eigenvalues (cf. section 2). 4 Theorem 1.1 Let M 1; δ > 0 and w a δ-weight. Then there exist con- stants 1 C < and ≥1 N < so that the following statements hold: ≤ 1 ≤ 1 w For any n N and any 1-periodic functions 1; 2 H with j w M, j j ≥ C π 2 k k ≤ the set spec( 1; 2) λ λ nπ < 2 contains exactly one isolated pair of eigenvalues \λ+f; λ2− . Thesej j − eigenvaluesj g satisfy f k k g (i) w(2n)2 λ+ λ− 2 C; jn|≥N j n − n j ≤ 3δ∗ 2 + − 2 (ii)P n w(2n) min (λ λ ) 2ρn C; jn|≥N h i j n − n j ≤ C π (iii)Pspec( 1; 2) λn n N is contained in λ λ < Nπ 2 and its cardinalitynfis 4Nj j j 2≥. g f 2 j j j − g − Theorem 1.2 Let M 1; δ > 0 and w be a δ-weight. Then there exist ≥ constants 1 C < and N N 0 < (with N given by Theorem 1.1) so that the following≤ statements1 hold:≤ 1 0 w For any n N and any 1-periodic functions 1; 2 H with j w M, the setj jspec≥ ( ; ) λ C λ nπ < π 2contains exactlyk k one≤ Dir 1 2 \ f 2 j j − j 2 g eigenvalue denoted by µn. These eigenvalues satisfy: 2 + 2 (i) 0 w(2n) µ λ C; jn|≥N j n − n j ≤ (ii) spec ( ; ) µ n N 0 is contained in λ C λ < N 0π π P Dir 1 2 nf n j j j ≥ g f 2 j j j − 2 g and its cardinality is 2N 0 1. − Statement (iii) in Theorem 1.1 and (ii) in Theorem 1.2 are obtained in a standard way. For the convenience of the reader we prove it in Appendix A. In section 3, we consider the Riesz spaces En, i.e. the images of the Riesz projectors associated to L( 1; 2) for a small circle around nπ with n suf- + j j ficiently large. We analyze the restriction of L λn to En and study the asymptotic properties of eigenfunctions in E for−n . n j j ! 1 1.2 Comments Operator L( 1; 2): The Zakharov-Shabat operator occurs in the Lax pair dM representation dt = [M; A] of the focusing (NLS−) and defocusing (NLS+) nonlinear Schr¨odinger equation i@ ' = @2 2 ' 2'; t − x j j 5 M := L('; ') ; M := L('; ') + − − (whereas the operators A are rather complicated third order operators, given in [FT]). One can show that spec L('; ') respectively spec L('; ') − is a complete set of conserved quantities for NLS+ respectively NLS−. We mention that L( 1; 2) is unitarily equivalent to the AKNS operator (see [AKNS], [MA]) 0 1 d q p LAKNS := − + − 1 0 dx p q where := q + ip ; = q ip: 1 − 2 − − Hence the selfadjoint operator M+ corresponds to an operator LAKNS with the functions q; p being real valued. Selfadjoint case: We emphasize that Theorem 1.1 and Theorem 1.2 do not require L( 1; 2) be selfadjoint. However, in the selfadjoint case, the decay rate of the asymptotics in Theorem 1.1 (ii) can be improved from 3δ∗ to 4δ∗, 4δ∗ 2 + − 2 n w(2n) min (λn λn ) 2ρn M: h i j − j ≤ jnX|≥N (This is proved in section 2.9). L2-case: Theorem 1.1 (i) and Theorem 1.2 (i) no longer hold for H w = L2 (i.e.
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