The band-structure of a one-dimensional, periodic system in a scaling limit. (English) Zbl 0412.34013


34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L99 Ordinary differential operators
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
Full Text: DOI


[1] Mathieu, É., J. Math. Pures Appl. Sér. 2, 13, 137-203 (1868)
[2] Hill, G. W., Acta Math., 8, 1-36 (1886) · JFM 18.1106.01
[3] Eastham, M. S.P., The Spectral Theory of Periodic Differential Equations (1974), Hafner: Hafner New York · Zbl 0285.34015
[4] Reed, M.; Simon, B., (Methods of Modern Mathematical Physics, Vols. I-IV (1972-1979), Academic Press: Academic Press New York), (especially Vol. IV) · Zbl 0242.46001
[5] Goldstein, S., (Proc. Roy. Soc. Edinburgh, 49 (1929)), 210-223 · JFM 55.0260.02
[6] Harrell, E., Commun. Math. Phys., 60, 73-95 (1978) · Zbl 0395.34023
[7] S. Coleman; S. Coleman
[8] Kato, T., J. Phys. Soc. Japan, 4, 334-339 (1949)
[9] Harrell, E., (Proc. Amer. Math. Soc., 69 (1978)), 271-276 · Zbl 0345.47007
[10] Simon, B., Ann. Phys. (N.Y.), 58, 76-136 (1970)
[11] Fröman, N.; Fröman, P. O., JWKB Approximation, Contributions to the Theory (1965), North-Holland: North-Holland Amsterdam · Zbl 0129.41907
[12] Berry, M. V.; Mount, K. E., Rep. Prog. Phys., 35, 315-397 (1972)
[13] E. Harrell and B. Simon; E. Harrell and B. Simon
[14] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1969), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0108.26903
[15] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions. Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, Vol. 55 (1964), National Bureau of Standards: National Bureau of Standards Washington) · Zbl 0171.38503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.