Isospectral periodic Torii in dimension 2

Waters, Alden

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1173-1188 / Harvested from Numdam

Résumé

Nous considérons les potentiels analytiques à valeurs réelles en dimension deux pour l'equation de Schrödinger qui sont périodiques sur un réseau $ \mathbb{L}$. Sous certaines hypothèses sur la forme du potentiel et du réseau $ \mathbb{L}$, nous montrons qu' il y a une grande classe de potentiels analytiques Floquet rigides et denses dans l'ensemble de $ {C}^{\infty }({\mathbb{R}}^{2}/\mathbb{L})$ potentiels. Ce résultat prolonge le travail de Eskin et al., dans “Les potentiels périodiques isospectraux dans $ {\mathbb{R}}^{n}$, II.”

Abstract

We consider two dimensional real-valued analytic potentials for the Schrödinger equation which are periodic over a lattice $ \mathbb{L}$. Under certain assumptions on the form of the potential and the lattice $ \mathbb{L}$, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of $ {C}^{\infty }({\mathbb{R}}^{2}/\mathbb{L})$ potentials. The result extends the work of Eskin et al., in “On isospectral periodic potentials in $ {\mathbb{R}}^{n}$, II.”

Détail

Publié le : 2015-01-01

MR 3425258 | Zbl 1332.35239

DOI : https://doi.org/10.1016/j.anihpc.2014.06.001
Classification:  35J10,  35P05,  65M32

Bibliographie

[1] V. Matveev, A. Its, A class of solutions of the Korteweg–de Vries equation, Probl. Math. Phys. 79 no. 9 (1976) | MR 516298

[2] K. Cai, Dispersion for Schrödinger operators with one-gap periodic potentials on $ {\mathbb{R}}^{1}$ , Dyn. Partial Differ. Equ. 3 no. 1 (2006), 71 -92 | MR 2221747 | Zbl 1238.35107

[3] G. Eskin, J. Ralston, E. Trubowitz, On isospectral periodic potentials in $ {\mathbf{R}}^{n}$ , Commun. Pure Appl. Math. 37 no. 6 (1984), 715 -753 | MR 762871 | Zbl 0582.35031

[4] G. Eskin, J. Ralston, E. Trubowitz, On isospectral periodic potentials in $ {\mathbf{R}}^{n}$. ii, Commun. Pure Appl. Math. 37 no. 5 (1984), 647 -676 | MR 752594 | Zbl 0574.35021

[5] J. Garnett, E. Trubowitz, Gaps and bands of one-dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 no. 2 (1984), 258 -312 | MR 749109 | Zbl 0554.34013

[6] C. Gordon, T. Kappeler, On isospectral potentials on tori, Duke Math. J. 63 no. 2 (1991), 217 -233 | MR 1106944 | Zbl 0732.35064

[7] C. Gordon, T. Kappeler, On isospectral potentials on flat tori ii, Commun. Partial Differ. Equ. 20 no. 3–4 (1995), 709 -728 | MR 1318086 | Zbl 0849.35085

[8] Harry Hochstadt, On the determination of a Hill's equation from its spectrum, Arch. Ration. Mech. Anal. 19 (1965), 353 -362 | MR 181792 | Zbl 0128.31201

[9] T. Kappeler, M. Makarov, On Birkhoff coordinates for KdV, Ann. Henri Poincaré 2 (2001), 806 -856 | MR 1869523 | Zbl 1017.76015

[10] E. Korotyaev, Estimates for the Hill operator, J. Differ. Equ. 162 no. 1 (2000), 1 -26 | MR 1741871 | Zbl 0954.34073

[11] E. Korotyaev, Estimates for the Hill operator, ii, J. Differ. Equ. 223 (2006), 229 -260 | MR 2214934 | Zbl 1098.34070

[12] Wilhelm Magnus, Stanley Winkler, Hill's Equation, Dover Publications Inc., New York (1979) | MR 559928 | Zbl 1141.34002

[13] H.P. Mckean, E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Commun. Pure Appl. Math. 29 no. 2 (1976), 143 -226 | MR 427731 | Zbl 0339.34024

[14] J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press [Harcourt Brace Jovanovich Publishers] (1987) | Zbl 0623.34001

[15] Elias M. Stein, Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II , Princeton University Press, Princeton, NJ (2003) | MR 1976398 | Zbl 1020.30001

[16] E. Trubowitz, The inverse problem for periodic potentials, Commun. Pure Appl. Math. 30 no. 3 (1977), 321 -337 | MR 430403 | Zbl 0403.34022