Abstract
The singularities of the stratification of the space of the Hermitian matrices according to the multiplicities of the eigenvalues are described as an informal complexification of the previous study of the space of the real symmetric matrices. The degeneration of the spectral sequence associated to this stratification provides some strange combinatorial identities. The eigenvector bundles over the manifold of the Hermitian matrices with simple spectra are equiped with the natural connections, describing also the adiabatic approximation to the oscillations of the linear systems defined by the slowly varying skew Hermitian matrices. The curvature of this connection is singular at the codimension 3 variety of the Hermitian matrices having multiple eigenvalues. The resulting jumps of the integrals of the curvature form at the crossings of this variety by the moving surface of integration are responsible for the quantum Hall effect.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V.I. Arnold.Modes and quasimodes. Funkt. Anal. and its Appl.,6:2 (1972), 94–101.
M. Shapiro, A. Vainstein.Stratification of Hermitian matrices, the Alexander mapping and the bundle of eigenvspaces. Preprint, Weizmann Inst., 1994, 6pp.
V.I. Arnold.On some problems in singularity theory. Trudy Seminare S.L. Sobolev, Novosibirsk, vol. 1 (1976), pp. 5–15, (in Russian; English translation in: Singularities, Proc. Symp. Pure Math., vol. 401, AMS, 1983, pp. 57–69).
M.V. Berry.Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. London, vol. A-392 (1984), pp. 45–57.
Y. Aharonov, J. Anadan. Phys. Rev. Lett. Vol. 58 (1987), pp. 1593.
S.M. Rytov.On the transition from wave to geometric optics. Doklady Ac. of Sc. USSR,18:2 (1938), 263–266.
V.V. Vladimirskii.On the rotation of the polarization plane along a curved ray. Doklady Ac. of Sc. USSR,31:3 (1941), 222–225.
A. Yu. Ishlinskii.Mechanics of Special Gyroscopical Systems. Ukrainian Acad. of Sc., Kiev, 1952 (see also: A. Yu. Ishlinskii. Orientation, gyroscopes, and inertial naviagation. Nauka, Moscow, 1976, page 195).
Y. Collin de Verdière.Effect quantique de Hall. Seminaire de Theorie Spectrale et Geometrie, no. 11 (1993), Institut Fourier, Grenoble.
A.B. Givental.Quaternionic Sturn theory and symplectic geometry. Lectures at Strasbourg (June 1991) and at Bologna (September 1991).
F. Klein.Einleitung in die höhere Geometrie. 3d Aufl., Berlin, 1926.
Author information
Authors and Affiliations
Additional information
This article is an expanded version of a lecture read 25 March, 1994, at the conference on Real Algebraic Geometry at the Institut Henry Poincaré, Paris
Rights and permissions
About this article
Cite this article
Arnold, V.I. Remarks on eigenvalues and eigenvectors of Hermitian matrices, berry phase, adiabatic connections and quantum Hall effect. Selecta Mathematica, New Series 1, 1–19 (1995). https://doi.org/10.1007/BF01614072
Issue Date:
DOI: https://doi.org/10.1007/BF01614072