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]**Joseph Avron and Barry Simon,*Almost periodic Schrödinger operators. II. The integrated density of states*, Duke Math. J.**50**(1983), no. 1, 369–391. MR**700145**, https://doi.org/10.1215/S0012-7094-83-05016-0**[C-L]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[F]**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****[J-M]**R. Johnson and J. Moser,*The rotation number for almost periodic potentials*, Comm. Math. Phys.**84**(1982), no. 3, 403–438. MR**667409****[L]**L. Loomis,*Abstract harmonic analysis*, Van Nostrand, Princeton, N.J., 1953.**[S.B]**Barry Simon,*Schrödinger semigroups*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 3, 447–526. MR**670130**, https://doi.org/10.1090/S0273-0979-1982-15041-8**[S.G]**Günter Scharf,*Fastperiodische Potentiale*, Inauguraldissertation zur Erlangung der Würde eines Doktors der Philosophie vorgelegt der Philosophischen Fakultät II der Universität Zürich, Buchdruckerei Birkhäuser AG, Basel, 1965 (German). MR**0201721****[S.M]**M. A. Šubin,*Almost periodic functions and partial differential operators*, Uspehi Mat. Nauk**33**(1978), no. 2(200), 3–47, 247 (Russian). MR**0636410****[W]**David Vernon Widder,*The Laplace Transform*, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR**0005923**

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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