Avron J.E., Seiler R., Simon B.
Charge Deficiency, Charge Transport
and the Comparison of Dimensions
(75K, TeX)
ABSTRACT. 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 deficency 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.