Geodesic
The continuous spectrum and the effect of parametric resonance. The case of bounded operators
Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 684-702

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The paper is concerned with the Mathieu-type differential equation $u''=-A^2 u+\varepsilon B(t)u$ in a Hilbert space $H$. It is assumed that $A$ is a bounded self-adjoint operator which only has an absolutely continuous spectrum and $B(t)$ is almost periodic operator-valued function. Sufficient conditions are obtained under which the Cauchy problem for this equation is stable for small $\varepsilon$ and hence free of parametric resonance. Bibliography: 10 titles.
Keywords: parametric resonance, continuous spectrum, stability. 
V. V. Skazka. The continuous spectrum and the effect of parametric resonance. The case of bounded operators. Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 684-702. http://geodesic.mathdoc.fr/item/SM_2014_205_5_a4/
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[1] L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Ergebnisse der Mathematik und ihrer Grenzgebiete. N.F., 16, Springer-Velag, Berlin–Göttingen–Heidelberg, 1959, vii+271 pp. | DOI | MR | MR | Zbl | Zbl

[2] V. A. Yakubovich, V. M. Starzhinskii, Parametricheskii rezonans v lineinykh sistemakh, Nauka, M., 1987, 328 pp. | MR | Zbl

[3] V. S. Belonosov, V. N. Dorovskii, A. S. Belonosov, S. V. Dorovskii, “Gidrodinamika gazosoderzhaschikh sloistykh sistem”, Uspekhi mekhaniki, 3:2 (2005), 37–70

[4] V. S. Belonosov, “The spectral properties of distributions and asymptotic methods in perturbation theory”, Sb. Math., 203:3 (2012), 307–325 | DOI | DOI | MR | Zbl

[5] V. I. Derguzov, “Ob ustoichivosti reshenii uravnenii Gamiltona s neogranichennymi periodicheskimi operatornymi koeffitsientami”, Matem. sb., 63(105):4 (1964), 591–619 | MR | Zbl

[6] V. I. Derguzov, “Dostatochnye usloviya ustoichivosti gamiltonovykh uravnenii s neogranichennymi periodicheskimi operatornymi koeffitsientami”, Matem. sb., 64(106):3 (1964), 419–435 | MR | Zbl

[7] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York, 1966, xix+592 pp. | MR | MR | Zbl | Zbl

[8] E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., 31, rev. ed., Amer. Math. Soc., Providence, R. I., 1957, xii+808 pp. | MR | MR | Zbl

[9] L. Schwartz, Analyse. Deuxième partie: Topologie générale et analyse fonctionnelle, Collect. Enseign. Sci., 11, Hermann, Paris, 1970, 433 pp. | MR | Zbl | Zbl

[10] R. E. A. C. Paley, N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., 19, Amer. Math. Soc., New York, viii+184 pp. | MR | Zbl | Zbl