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On the band gap structure of Hill’s equation. (English) Zbl 1095.34014

For Hill’s equation it is known that solutions are linear combinations of functions having the form \(y(x,\lambda)=\exp(i 2\pi \theta(\lambda) x)p(x,\lambda)\), in which \(p\) is periodic in \(x\) and \(\exp(i 2\pi \theta(\lambda))\) is a Floquet multiplier. The parameter \(\theta(\lambda)\) is real-valued when \(\lambda\) lies in a stability interval and complex-valued when \(\lambda\) lies in an instability interval. Another way to formulate the stability problem is to consider \(\theta\) as a parameter in \((-1/2,1/2]\) and examine the family, parametrized by \(\theta\), of periodic eigenvalue problems for the function \(p\); the classical work of Bloch shows that the union of the ranges of the eigenvalues \(\lambda_m(\theta)\), \(m\in {\mathbb N}\), \(\theta\in (-1/2,1/2]\), is the same as the union of the stability intervals of the original equation.
In this work, the authors show that formally replacing \(\theta\) by \(i\vartheta\), and allowing \(\vartheta\) to range over \({\mathbb R}\), one obtains nonselfadjoint eigenvalue problems for the functions \(p\) such that the union of the ranges of the real eigenvalues as \(\vartheta\) ranges over \({\mathbb R}\) yields precisely the instability intervals of the original problem.

MSC:

34B24 Sturm-Liouville theory
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] Allaire, G.; Capdeboscq, Y., Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Mat., 181, 247-282 (2002) · Zbl 1072.35026
[2] Ammari, H.; Santosa, F., Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math., 64, 2018-2033 (2004) · Zbl 1060.35140
[3] Avellaneda, M.; Bardos, C.; Rauch, J., Contrôlabilité exacte, homogénéisation et localisation d’ondes dans un milieu non-homogène, Asymptot. Anal., 5, 481-494 (1992) · Zbl 0763.93006
[4] Bakhvalov, N.; Panasenko, G., Homogenization: Averaging Processes in Periodic Media, Mathematics and Its Applications, vol. 36 (1990), Kluwer Academic: Kluwer Academic Dordrecht
[5] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, vol. 5 (1978), North-Holland: North-Holland Amsterdam · Zbl 0411.60078
[6] Bloch, F., Über die Quantenmechanik der Electronen in Kristallgittern, Z. Phys., 52, 555-600 (1928) · JFM 54.0990.01
[7] Capdeboscq, Y., Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A, 132, 567-594 (2002) · Zbl 1066.82530
[8] Conca, C.; Planchard, J.; Vanninathan, M., Fluids and Periodic Structures, Research in Applied Mathematics, vol. 38 (1995), Wiley/Masson: Wiley/Masson New York/Paris · Zbl 0910.76002
[9] Eastham, M. S.P., The Spectral Theory of Periodic Differential Equations (1973), Scottish Academic Press: Scottish Academic Press Edinburgh · Zbl 0287.34016
[10] Figotin, A.; Klein, A., Midgap defect modes in dielectric and acoustic media, SIAM J. Appl. Math., 58, 1748-1773 (1998) · Zbl 0963.78005
[11] Floquet, G., Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. 2, 12, 47-89 (1883) · JFM 15.0279.01
[12] Gohberg, I. C.; Krejn, M. G., Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien (1971), Dunod: Dunod Paris, Traduit du russe par Guy Roos, Monographies Universitaires de Mathématiques, No. 39
[13] Hochstadt, H., A special Hill’s equation with discontinuous coefficients, Amer. Math. Monthly, 70, 1826 (1963) · Zbl 0117.05103
[14] Hochstadt, H., On the determination of a Hill’s equation from its spectrum, Arch. Rational Mech. Anal., 19, 353-362 (1965) · Zbl 0128.31201
[15] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0148.12601
[16] Krönig, R. L.; Penney, W. G., Quantum mechanics in crystals lattices, Proc. Roy. Soc. London, 130, 499-531 (1931) · Zbl 0001.10601
[17] Lions, J. L., Some Methods in the Mathematical Analysis of Systems and Their Controls (1981), Science Press/Gordon and Breach: Science Press/Gordon and Breach Beijing/New York · Zbl 0542.93034
[18] Magnus, W.; Winkler, S., Hill’s Equation, Interscience Tracts in Pure and Applied Mathematics, vol. 20 (1966), Interscience Wiley: Interscience Wiley New York · Zbl 0158.09604
[19] McKean, H. P.; van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30, 217-274 (1975) · Zbl 0319.34024
[20] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978), Academic Press (Harcourt Brace Jovanovich Publishers): Academic Press (Harcourt Brace Jovanovich Publishers) New York · Zbl 0401.47001
[21] Rellich, F., Perturbation Theory of Eigenvalue Problems, Notes on Mathematics and Its Applications (1969), Gordon and Breach: Gordon and Breach New York · Zbl 0181.42002
[22] Trubowitz, E., The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30, 321-337 (1977) · Zbl 0403.34022
[23] Wilcox, C. H., Theory of Bloch waves, J. Anal. Math., 33, 146-167 (1978) · Zbl 0408.35067
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