×

Gap estimates of the spectrum of Hill’s equation and action variables for KdV. (English) Zbl 0924.58074

Summary: Consider the Schrödinger equation \(-y'' + Vy = \lambda y\) for a potential \(V\) of period 1 in the weighted Sobolev space \((N \in \mathbb{Z}_{\geq 0}, \;\omega \in \mathbb{R}_{\geq 0})\) \[ H^{N, \omega}(S^1; \mathbb{C}) :=\left \{ f(x) = \sum^{\infty}_{k= - \infty} \;\Hat{\Hat f}(k) e^{i 2 \pi kx} \;\bigg | \| f \|_{N, \omega} < \infty\right \} \] where \(\Hat{\Hat f}(k) \;(k \in \mathbb{Z})\) denote the Fourier coefficients of \(f\) when considered as a function of period 1, \[ \| f \|_{N, \omega} \;:= \;\bigg( \sum_k (1+| k|)^{2N} e^{2 \omega | k | } | \;\;\Hat{\Hat{f}} (k) | ^2 \bigg)^{1/2} < \infty , \] and where \(S^1\) is the circle of length 1. Denote by \(\lambda_k \equiv \lambda_k (V) \;(k \geq 0)\) the periodic eigenvalues of \( - \frac{d^2}{dx^2} + V\) when considered on the interval \([0,2],\) with multiplicities and ordered so that Re\( \lambda_j \leq\text{Re} \lambda_{j+1} \;(j \geq 0).\) We prove the following result.
Theorem. For any bounded set \({\mathcal B} \subseteq H^{N, \omega} (S^1; \mathbb{C}),\) there exist \(n_0 \geq 1\) and \(M \geq 1\) so that for \(k \geq n_0\) and \(V \in{\mathcal B}\), the eigenvalues \(\lambda_{2k}, \lambda_{2k -1}\) are isolated pairs, satisfying (with \(\{ \lambda_{2k}, \lambda_{2k-1} \} = \{ \lambda^+_k , \lambda^-_k \})\)
(i) \(\sum_{k \geq n_0} (1+k)^{2N} e^{2 \omega k} | \lambda_k^+ - \lambda^-_k | ^2 \leq M\),
(ii) \(\sum_{k \geq n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg | ^2 \leq M\).

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
35Q35 PDEs in connection with fluid mechanics
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37N99 Applications of dynamical systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] D. Bättig, A. M. Bloch, J.-C. Guillot, and T. Kappeler, On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS, Duke Math. J. 79 (1995), no. 3, 549 – 604. · Zbl 0855.58035
[2] D. Bättig, T. Kappeler, and B. Mityagin, On the Korteweg-de Vries equation: convergent Birkhoff normal form, J. Funct. Anal. 140 (1996), no. 2, 335 – 358. · Zbl 0868.35099
[3] D. Bättig, T. Kappeler, B. Mityagin, On the Korteweg-de Vries equation: frequencies and initial value problem, Pacific J. Math. 181 (1997), 1-55. CMP 98:06 · Zbl 0899.35096
[4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: KdV-equation, Geom. Funct. Anal. 3 (1993), 209-262. · Zbl 0787.35098
[5] B. A. Dubrovin, Igor Moiseevich Krichever, and S. P. Novikov, Integrable systems. I, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 179 – 284, 291 (Russian).
[6] H. Flaschka and D. W. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (1976), no. 2, 438 – 456. · Zbl 1109.35374
[7] Thomas Kappeler, Fibration of the phase space for the Korteweg-de Vries equation, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 3, 539 – 575 (English, with French summary). · Zbl 0731.58033
[8] Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. · Zbl 0592.34011
[9] H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143 – 226. · Zbl 0339.34024
[10] H. P. McKean and E. Trubowitz, Hill’s surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1042 – 1085. · Zbl 0428.34026
[11] Jürgen Pöschel and Eugene Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, vol. 130, Academic Press, Inc., Boston, MA, 1987. · Zbl 0623.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.