Almost periodic potentials in higher dimensions
Vassilis G. Papanicolaou
Trans. Amer. Math. Soc. 329 (1992), 679-696
Primary 35J10; Secondary 35P05, 47F05
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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 , where is an almost periodic function in . 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 of the semigroup . For fixed and , it is known (we review the proof) that is almost periodic in with frequency module not bigger than the one of . We show that is, also, uniformly continuous on . These results imply that, if we set in the kernel of it becomes almost periodic in (for the case we must assume that ), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define , and, by integrating this times, an analog of the complex rotation number of [J-M]. We also show that, if is the kernel of the projection operator associated to , then the mean value exists. In one dimension, this (times ) is the rotation number. In higher dimensions ( included), we show that is the density of states measure of [A-S] and it is related to in a nice way. Finally, we derive a formula for the functional derivative of with respect to , which extends a result of [J-M].
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Almost periodic functions of several variables,
(complex) rotation number,
(integrated) density of states
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