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.