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) \(\sum_{k \geq n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg | ^2 \leq M\).


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
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[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
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