transactions of the american mathematical society Volume 329, Number 2, February 1992 ALMOST PERIODIC POTENTIALS IN HIGHER DIMENSIONS VASSILISG. PAPANICOLAOU 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 = -A/2 - q, where q is an almost periodic function in Rd . 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~'L . For fixed / > 0 and u e Rd , 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]x Rd x Rd . These results imply that, if we set y=x + u in the kernel Gm(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 wm(z) = MxlGm(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 ; X) is the kernel of the projection operator Ex associated to L , then the mean value a(X) = Mx[e(x, x ; X)] exists. In one dimension, this (times n) is the rotation number. In higher dimensions (d = 1 included), we show that da(X) is the density of states measure of [A-S] and it is related to wm(z) in a nice way. Finally, we derive a formula for the functional derivative of wm(z; q) with respect to q , which extends a result of [J-M], 1. Introduction. Schrödinger operators Let Cb(Rd) be the class of bounded continuous functions in Rd , d > 1. This class is complete with respect to the supnorm, which we will always denote by || • ||. For a real-valued q in Cb(Rd) we define (1.1) L = L(q) = -A/2-q, where A is the Laplacian operator. L is a Schrödinger operator with potential -q . It is well known (see [S.B, Theorem B.12.1], etc.) that L, as an operator of L2(Rd), is essentially self adjoint with a unique self adjoint extension (with zero boundary conditions at infinity) and L2-spectrum o(L) bounded below. In fact, if we set X0= info(L), then A0 > -|kll • Furthermore, for each t > 0, e~'L is a well-defined semigroup. It turns out that this semigroup possesses a kernel k(t, x, y) which is nonnegative, continuous in (t, x, y) if t > 0, Received by the editors November 20, 1989. 1980 Mathematics Subject Classification(1985 Revision). Primary 35J10. Key words and phrases. Almost periodic functions of several variables, Schrödinger semigroup, resolvent, (complex) rotation number, (integrated) density of states. © 1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page 679 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use