We define and analyze the rotation number for the almost periodic -d 2 Schrόdinger operator L = -^ + q(x). We use the rotation number to discuss (i) the spectrum of L\ (ii) its relation to the Korteweg-de Vries equation.

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It has been pointed out to us by B. Simon that the proof of Proposition 3.3, p. 413, is incorrect. The statement, however, is valid as it stands and we want to present a proof of that statement: According to Proposition 2.1, we have to show that m+(x+t,) converges uniformly for any sequence t, for which q(x+t,) does. We show first the convergence for x =0 and subsequently verify the uniform convergence for all x~ll~ We change the notation and write for fixed z in Imz~0, m[ ql = m + (0, z; q) .

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