Almost periodic potentials in higher dimensions

Author:
Vassilis G. Papanicolaou

Journal:
Trans. Amer. Math. Soc. **329** (1992), 679-696

MSC:
Primary 35J10; Secondary 35P05, 47F05

DOI:
https://doi.org/10.1090/S0002-9947-1992-1042290-0

MathSciNet review:
1042290

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Abstract | References | Similar Articles | Additional Information

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].

**[A-S]**J. Avron and B. Simon,*Almost periodic Schrödinger operators*. II.*The integrated density of states*, Duke Math. J.**50**(1983), 369-391. MR**700145 (85i:34009a)****[C-L]**E. Coddington and N. Levinson,*Theory of ordinary differential equations*, McGraw-Hill, New York, 1955. MR**0069338 (16:1022b)****[F]**W. Feller,*An introduction to probability theory and its applications*, vol. II, 2nd ed., Wiley, New York, 1971. MR**0270403 (42:5292)****[J-M]**R. Johnson and J. Moser,*The rotation number for almost periodic potentials*, Comm. Math. Phys.**84**(1982), 403-438; erratum: Comm. Math. Phys.**90**(1983), 317-318. MR**667409 (83h:34018)****[L]**L. Loomis,*Abstract harmonic analysis*, Van Nostrand, Princeton, N.J., 1953.**[S.B]**B. Simon,*Schrödinger semigroups*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), 447-526. MR**670130 (86b:81001a)****[S.G]**G. Scharf,*Fastperiodische Potentiale*, Helv. Phys. Acta**24**(1965), 573-605. MR**0201721 (34:1603)****[S.M]**M. A. Shubin,*Almost periodic functions and partial differential operators*, Russian Math. Surveys**33:2**(1978), 1-52. MR**0636410 (58:30522)****[W]**D. V. Widder,*The Laplace transform*, Princeton Univ. Press, Princeton, N.J., 1941. MR**0005923 (3:232d)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1042290-0

Keywords:
Almost periodic functions of several variables,
Schrödinger semigroup,
resolvent,
(complex) rotation number,
(integrated) density of states

Article copyright:
© Copyright 1992
American Mathematical Society