Abstract
We give a simple proof of Guillemin’s theorem on the determination of the magnetic field on the torus by the spectrum of the corresponding Schrödinger operator.
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References
G. Eskin, Inverse spectral problem for the Schrödinger equation with periodic vector potential. Comm. Math. Phys. 125 (1989), 263–300.
G. Eskin, J. Ralston, Inverse spectral problems in rectangular domains. Commun. PDE 32 (2007), 971–1000.
G. Eskin, J. Ralston, E. Trubowitz, On isospectral periodic potentials in ℝ n, I and II. Commun. Pure and Appl. Math. 37 (1984), 647–676, 715–753.
C. Gordon, Survey of isospectral manifolds. Handbook of Differential Geometry 1 (2000), North Holland, 747–778.
V. Guillemin, Inverse spectral results on two-dimensional tori. Journal of the AMS 3 (1990), 375–387.
V. Guillemin, D. Kazdhan, Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301–312.
J. Hadamard, Le Problème de Cauchy et les Equations aux Dérivées Partielles Linéaires Hyperboliques. Hermann, Paris, 1932.
L. Hörmander, The Analysis of Linear Partial Differential Operators, III. Springer-Verlag, Vienna, 1985.
M. Krein, Solution of the inverse Sturm-Liouville problem (Russian). Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 21–24.
M. Krein, Determination of the density of a nonhomogeneous cord by its frequency spectrum. Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348.
V.A. Marchenko, Operatory Shturma Liuvilliâ i ikh prilozhen’iâ, Naukova Dumka, Kiev, 1977.
T. Sunada, Riemannian coverings and isospectral manifolds. Ann. of Math. 121 (1985), 169–186.
J. Zak, Dynamics of electrons in solids in external fields. Phys. Rev. 168 (1968), 686–695.
S. Zelditch, Spectral determination of analytic, bi-axisymmetric plane domains. Geometric Funct. Anal. 10 (2000), 628–677.
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© 2009 Birkhäuser Verlag Basel/Switzerland
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Eskin, G., Ralston, J. (2009). Remark on Spectral Rigidity for Magnetic Schrödinger Operators. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 191. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9921-4_19
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DOI: https://doi.org/10.1007/978-3-7643-9921-4_19
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-9920-7
Online ISBN: 978-3-7643-9921-4
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