Almost periodic potentials in higher dimensions

Abstract:

This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator $L = - \Delta /2 - q$, where $q$ is an almost periodic function in ${R^d}$. It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well). We start with the kernel $k(t,x,y)$ of the semigroup ${e^{ - tL}}$. For fixed $t > 0$ and $u \in {R^d}$, it is known (we review the proof) that $k(t,x,x + u)$ is almost periodic in $x$ with frequency module not bigger than the one of $q$. We show that $k(t,x,y)$ is, also, uniformly continuous on $[a,b] \times {R^d} \times {R^d}$. These results imply that, if we set $y = x + u$ in the kernel ${G^m}(x,y;z)$ of ${(L - z)^{ - m}}$ it becomes almost periodic in $x$ (for the case $u = 0$ we must assume that $m > d/2$), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define ${w_m}(z) = {M_x}[{G^m}(x,x;z)]$, and, by integrating this $m$ times, an analog of the complex rotation number $w(z)$ of [J-M]. We also show that, if $e(x,y;\lambda )$ is the kernel of the projection operator ${E_\lambda }$ associated to $L$, then the mean value $\alpha (\lambda ) = {M_x}[e(x,x;\lambda )]$ exists. In one dimension, this (times $\pi$) is the rotation number. In higher dimensions ($d = 1$ included), we show that $d\alpha (\lambda )$ is the density of states measure of [A-S] and it is related to ${w_m}(z)$ in a nice way. Finally, we derive a formula for the functional derivative of ${w_m}(z;q)$ with respect to $q$, which extends a result of [J-M].