Compactness of Floquet isospectral sets for the matrix Hill’s equation

Carlson, Robert

doi:10.1090/S0002-9939-00-05634-3

Compactness of Floquet isospectral sets for the matrix Hill’s equation
HTML articles powered by AMS MathViewer

by Robert Carlson

Proc. Amer. Math. Soc. 128 (2000), 2933-2941

DOI: https://doi.org/10.1090/S0002-9939-00-05634-3

Published electronically: April 7, 2000

PDF | Request permission

Abstract:

Let $\mathcal {M}(Q)$ denote the set of self adjoint $K \times K$ potentials for the matrix Hill’s equation having the same Floquet multipliers as $-D^2 + Q$. Elementary methods are used to show that $\mathcal {M}(Q)$ has compact closure in the space of continuous matrix valued functions.

References

Lars V. Ahlfors, Complex analysis: An introduction of the theory of analytic functions of one complex variable, 2nd ed., McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0188405
Bruno Després, The Borg theorem for the vectorial Hill’s equation, Inverse Problems 11 (1995), no. 1, 97–121. MR 1313602
Allan Finkel, Eli Isaacson, and Eugene Trubowitz, An explicit solution of the inverse periodic problem for Hill’s equation, SIAM J. Math. Anal. 18 (1987), no. 1, 46–53. MR 871819, DOI 10.1137/0518003
Charles T. Fulton and Steven A. Pruess, Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems, J. Math. Anal. Appl. 188 (1994), no. 1, 297–340. MR 1301734, DOI 10.1006/jmaa.1994.1429
John Garnett and Eugene Trubowitz, Gaps and bands of one-dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 (1984), no. 2, 258–312. MR 749109, DOI 10.1007/BF02566350
Katsunori Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. (4) 149 (1987), 185–206. MR 932784, DOI 10.1007/BF01773933
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). MR 1136595
Peter D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28 (1975), 141–188. MR 369963, DOI 10.1002/cpa.3160280105
Wilhelm Magnus and Stanley Winkler, Hill’s equation, Dover Publications, Inc., New York, 1979. Corrected reprint of the 1966 edition. MR 559928
H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
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. MR 427731, DOI 10.1002/cpa.3160290203
James Ralston and Eugene Trubowitz, Isospectral sets for boundary value problems on the unit interval, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 301–358. MR 967643, DOI 10.1017/S0143385700009470
E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. MR 430403, DOI 10.1002/cpa.3160300305