Isospectral periodic Torii in dimension 2
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, p. 1173-1188
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We consider two dimensional real-valued analytic potentials for the Schrödinger equation which are periodic over a lattice 𝕃. Under certain assumptions on the form of the potential and the lattice 𝕃, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of C ( 2 /𝕃) potentials. The result extends the work of Eskin et al., in “On isospectral periodic potentials in n , II.”

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

DOI : https://doi.org/10.1016/j.anihpc.2014.06.001
Classification:  35J10,  35P05,  65M32
Keywords: Inverse problems, Spectral theory, Schrödinger equations
@article{AIHPC_2015__32_6_1173_0,
     author = {Waters, Alden},
     title = {Isospectral periodic Torii in dimension 2},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {6},
     year = {2015},
     pages = {1173-1188},
     doi = {10.1016/j.anihpc.2014.06.001},
     zbl = {1332.35239},
     mrnumber = {3425258},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_6_1173_0}
}
Waters, Alden. Isospectral periodic Torii in dimension 2. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1173-1188. doi : 10.1016/j.anihpc.2014.06.001. http://www.numdam.org/item/AIHPC_2015__32_6_1173_0/

[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 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 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 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