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
MathSciNet review: 1042290
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
- 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)
- E. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
- W. Feller, An introduction to probability theory and its applications, vol. II, 2nd ed., Wiley, New York, 1971. MR 0270403 (42:5292)
- 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. Loomis, Abstract harmonic analysis, Van Nostrand, Princeton, N.J., 1953.
- B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447-526. MR 670130 (86b:81001a)
- G. Scharf, Fastperiodische Potentiale, Helv. Phys. Acta 24 (1965), 573-605. MR 0201721 (34:1603)
- M. A. Shubin, Almost periodic functions and partial differential operators, Russian Math. Surveys 33:2 (1978), 1-52. MR 0636410 (58:30522)
- D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, N.J., 1941. MR 0005923 (3:232d)
Keywords: Almost periodic functions of several variables, Schrödinger semigroup, resolvent, (complex) rotation number, (integrated) density of states
Article copyright: © Copyright 1992 American Mathematical Society