$ satisfies, a.e.: $$= - {Index}\, (P_\omega U P_\omega)\tag8.9$$ \endproclaim \demo{Proof} The proof of this statement is an adaptation of the one given in section 4, theorem 4.2; integrating the basic equality 3.7 for the index over probability space brings us into the situation we had encountered in the proof of theorem 4.2 since the average of the triple product (8.8) is invariant under translations. \qed\enddemo \vskip 0.3in \heading {\bf Appendix A} \endheading \vskip 0.3in The purpose of this appendix is to show that hypothesis 3.1 on the regularity and decay of the integral kernel of spectral projections is guaranteed whenever the Fermi energy is placed in a gap. Although we have not attempted to give optimal conditions on the vector potentials, the conditions are mild enough to cover the physically interesting models. \proclaim {Theorem (A.1)} Let $H(A,V)$ be a one particle Schr\"odinger operator in $n=2,3$ dimensions with differentiable vector potential $A$ and scalar potential $V$ which is in the Kato class $K_{n=2,3}$ (which includes Coulombic singularities).\newline a) The integral kernel for spectral projections for $H(A,V)$, $p(x,y)$ is jointly continuous in $x$ and $y$.\newline b) Suppose, in addition, that $H(A,V)$ has a gap in the spectrum. Then the spectral projection below the gap has integral kernel which decays exponentially with $|x-y|$. \endproclaim \proclaim{ Remark } The two parts of the theorem have rather different proofs. The $K_n$ condition is natural for (a). Part (b) only requires form boundedness of $V$ which is slightly weaker than the $K_n$ condition. \endproclaim\demo{Proof} {(a)} $\exp (-tH)(x,y)$ has a jointly continuous integral kernel by the path integral (Ito) way of writing the kernel -- see, e.g.\ \cite{\semigroup }. Because $H$ is bounded below and has a gap, $P = g(H)$ where $g$ is a smooth function of compact support. Since $f(y) \equiv \exp (2y) g(y)$ can be approximated by polynomials $\exp (-y)$ uniformly, we can write $$ g(H) = lim\, g_j (H),\quad g_j(H) \equiv \exp(-H) f_j (H) \exp(-H),\eqno(A.1)$$ where the operators $f_j$ converge to $f$ in norm as $L^2\rightarrow L^2$ operators and each $f_j(H)$ is a polynomial in $\exp (-H)$. On general principles (see, e.g.\, \cite{\semigroup }), $\exp (-H)$ is a bounded operator from $L^1$ to $L^2$ and from $L^2$ to $L^\infty$. Thus the limit in (A.1) gives a bounded operator from $L^1$ to $L^\infty$ and so in infinity norm for the integral kernel (see e.g. \cite{\semigroup }). Since $g_j$ has a continuous integral kernel the result follows. \newline {(b) } Let $B_{\vec a} \equiv e^{i\,\vec a \cdot \vec x}$, $a \in {\Bbb C}$, be a complex boost. Then: $$B_a\, H(A,V)\, B_{-a} = H(A,V) + \vec a \cdot \vec a + \vec a \cdot (-i\,\vec \nabla -\vec A).\tag A.2$$ This gives an analytic family of type $B$ in the sense of Kato \cite{\kato } if the form domain is independent of $\vec a$. In particular, this is the case if $V$ is form bounded relative to the kinetic energy. By the diamagnetic inequality it is enough to check that $V$ is bounded relative to the Laplacian. $K_n$ implies form boundedness (see \cite{\semigroup } ). In particular, if $P$ is a spectral projection associated with a gap, then the gap is stable and: $$p_a (x,y) = e^{i\, a \cdot x} p (x,y) e^{-i\, a \cdot y}\tag A.3$$ is real analytic in $\vec a$ uniformly in $x$ and $y$. In particular, (A.2) says that $p(x,y)$ is exponentially decaying in $|x-y|$. This is a version of the Combes--Thomas argument \cite{\ct }. \qed\enddemo \proclaim{Remarks } 1. For potentials $V$ which are perturbations of Landau Hamiltonian, an adaptation of the above method gives decay which is faster than any exponential.\newline 2. It is easy to construct families of Schr\"odinger operators, with ergodic $A$ and $V$ so that $H(A,V)$ has gaps in the spectrum. \newline 3. 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