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The spectrum of operators elliptic along the orbits of \({\mathbb{R}}^ n\) actions. (English) Zbl 0643.58033

A method of studying differential operators on an open manifold, developed by A. Connes, is to consider the open manifold as a leaf in a foliation of a compact manifold. The authors apply this method in the more general setting of a group action of \(R^ n\) on a compact Hausdorff space. They extend the ideas and methods of Bellisard and Testard to prove a “spectral duality theorem” in this general setting.
The main application is a proof that a periodic elliptic operator on \(R^ n\) has no eigenvalues, off of the discontinuities of the spectral density function, which have \(L^ 2\)-eigenfunctions. The authors hope to use the general setting of their results for future applications to almost-periodic Schrödinger operators.
Reviewer: M.K.Murray

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J22 Exotic index theories on manifolds
35P99 Spectral theory and eigenvalue problems for partial differential equations
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[1] Atiyah, M. F.:Elliptic operators and compact groups. Lecture Notes in Mathematics Vol.41, Berlin, Heidelberg, New York: Springer 1970 · Zbl 0207.22601
[2] Bellisard, J., Testard, D.: Almost periodic hamiltonians: An algebraic approach, pre-print, 1981, Centre de Physique Theorique, CNRS Marseille
[3] Connes, A.: Sur la theorie non commutative de l’interation. Lecture Notes in Mathematics, Vol.725. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0412.46053
[4] ??, A survey of foliations and operator algebras. Proc. Symposia Pure Math.38, Part I, 521-628 (1982)
[5] Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403-438 (1982) · Zbl 0497.35026
[6] Kozlov, S. M., Shubin, M. A.: On the coincidence of the spectra of random elliptic operators. Math. USSR Sbornik51, 455-471 (1985) · Zbl 0568.35091
[7] Lazarov, C.: Spectral invariants of foliations. Mich. Math. J.33, 231-243 (1986). · Zbl 0609.57016
[8] Moore, C., Schochet, C.: Global analysis on foliated spaces. Berlin, Heidelberg, New York: Springer (to appear) · Zbl 1091.58015
[9] Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math.3, 463-490 (1982) · Zbl 0545.34023
[10] Shubin, M. A.: The spectral theory and the index of elliptic operators with almost periodic coefficients. Russ. Math. Surveys34, 109-157 (1979) · Zbl 0448.47032
[11] Shubin, M. A.: Spectrum and its distribution function for a transversally elliptic operator. Funct. Anal. Appl.15, 74-76 (1981) · Zbl 0502.35068
[12] Wiener, N.: The fourier integral and certain of its applications. Cambridge: The University Press 1933 · Zbl 0006.05401
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