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Published January 1994 | public
Journal Article Open

Charge deficiency, charge transport and comparison of dimensions


We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.

Additional Information

© 1994 Springer-Verlag. Received: January 19, 1993. Part of this work was written while the authors enjoyed the hospitality of the Landau Center at the Hebrew University. The work is supported by BSF, DFG, GIF, NSF, the fund for the Promotion of Research at the Technion and the Technion VPR-Steigman research fund. One of the authors (RS) should like to acknowledge the hospitality of the Mittag-Leffler Institute. We are grateful to S. Agmon, E. Akkermans, J. Bellissard, S, Borac, J. Fröhlich, I. Kaplansky, M. Klein, A. Pnueli and U. Sivan for useful discussions and comments.

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August 20, 2023
August 20, 2023