Remarks on linear differential systems with measurable coefficients
HTML articles powered by AMS MathViewer
- by Russell A. Johnson PDF
- Proc. Amer. Math. Soc. 100 (1987), 491-504 Request permission
Abstract:
We discuss several topics having to do with linear differential equations with locally integrable coefficients. These include gap labelling and Cantor spectrum for the random Schrodinger operator, and Coppel’s perturbation theorem.References
- Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249
- 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, DOI 10.1215/S0012-7094-83-05016-0
- J. Béllissard and E. Scoppola, The density of states for almost periodic Schrödinger operators and the frequency module: a counterexample, Comm. Math. Phys. 85 (1982), no. 2, 301–308. MR 676003
- J. Béllissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Functional Analysis 48 (1982), no. 3, 408–419. MR 678179, DOI 10.1016/0022-1236(82)90094-5
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196
- Corrado De Concini and Russell A. Johnson, The algebraic-geometric AKNS potentials, Ergodic Theory Dynam. Systems 7 (1987), no. 1, 1–24. MR 886368, DOI 10.1017/S0143385700003783 I. Dinaburg and Ya. Sinai, On the one-dimensional Schrodinger equation with quasi-periodic potential, Functional Anal. Appl 9 (1975), 8-21.
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
- Robert Ellis and Russell A. Johnson, Topological dynamics and linear differential systems, J. Differential Equations 44 (1982), no. 1, 21–39. MR 651685, DOI 10.1016/0022-0396(82)90023-7
- A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
- R. Giachetti and R. Johnson, Spectral theory of second-order almost periodic differential operators and its relation to classes of nonlinear evolution equations, Nuovo Cimento B (11) 82 (1984), no. 2, 125–168 (English, with Italian and Russian summaries). MR 770735, DOI 10.1007/BF02732870
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- Russell A. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), no. 1, 54–78. MR 818861, DOI 10.1016/0022-0396(86)90125-7 —, $m$-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl. (to appear).
- Russell A. Johnson, The recurrent Hill’s equation, J. Differential Equations 46 (1982), no. 2, 165–193. MR 675906, DOI 10.1016/0022-0396(82)90114-0
- R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409
- Russell A. Johnson, Kenneth J. Palmer, and George R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal. 18 (1987), no. 1, 1–33. MR 871817, DOI 10.1137/0518001
- Shinichi Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 225–247. MR 780760, DOI 10.1016/S0924-6509(08)70395-7
- Shinichi Kotani, On an inverse problem for random Schrödinger operators, Particle systems, random media and large deviations (Brunswick, Maine, 1984) Contemp. Math., vol. 41, Amer. Math. Soc., Providence, RI, 1985, pp. 267–280. MR 814717, DOI 10.1090/conm/041/814717
- Shinichi Kotani, Support theorems for random Schrödinger operators, Comm. Math. Phys. 97 (1985), no. 3, 443–452. MR 778625
- H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
- Richard K. Miller and George R. Sell, Volterra integral equations and topological dynamics, Memoirs of the American Mathematical Society, No. 102, American Mathematical Society, Providence, R.I., 1970. MR 0288381
- V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Differencial′nye Uravnenija 4 (1968), 391–396 (Russian). MR 0229912
- Jürgen Moser and Jürgen Pöschel, An extension of a result by Dinaburg and Sinaĭ on quasiperiodic potentials, Comment. Math. Helv. 59 (1984), no. 1, 39–85. MR 743943, DOI 10.1007/BF02566337
- S. P. Novikov, A periodic problem for the Korteweg-de Vries equation. I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54–66 (Russian). MR 0382878
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280 K. Palmer, On Cherry’s example of a differential equation with a homoclinic point, preprint, 1985.
- Friedrich Riesz and Béla Sz.-Nagy, Vorlesungen über Funktionalanalysis, Hochschulbücher für Mathematik, Band 27, VEB Deutscher Verlag der Wissenschaften, Berlin, 1973 (German). Dritte Auflage; Übersetzung aus dem Französischen nach der 2. und 3. Auflage: Siegfried Brehmer und Brigitte Mai. MR 0344831
- Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429–458. MR 341458, DOI 10.1016/0022-0396(74)90067-9
- Robert J. Sacker and George R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), no. 3, 320–358. MR 501182, DOI 10.1016/0022-0396(78)90057-8
- Sol Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66 (1957), 270–284. MR 88720, DOI 10.2307/1969999
- James F. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359–390. MR 368080, DOI 10.1090/S0002-9947-1975-0368080-X
- George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
- R. È. Vinograd, On a problem of N. P. Erugin, Differencial′nye Uravnenija 11 (1975), no. 4, 632–638, 763 (Russian). MR 0473360
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 491-504
- MSC: Primary 34A30; Secondary 34C35, 47B99, 54H20
- DOI: https://doi.org/10.1090/S0002-9939-1987-0891153-3
- MathSciNet review: 891153