Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. II. Internal gaps

Suslina, T.

doi:10.1090/S1061-0022-04-00810-6

Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. II. Internal gaps
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by T. A. Suslina
Translated by: the author

St. Petersburg Math. J. 15 (2004), 249-287

DOI: https://doi.org/10.1090/S1061-0022-04-00810-6

Published electronically: January 29, 2004

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Abstract:

The discrete spectrum in the spectral gaps is studied in the case of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. The main goal is to find asymptotics (for the large coupling constant) of the number of eigenvalues that have been “born” (or have “died”) at the edges of the gap. The high-energy (Weyl) asymptotics and the threshold asymptotics are distinguished. At the right edge of the gap, a competition between the Weyl contribution and the threshold contribution may occur. The case of a semiinfinite gap was studied in part I of the paper.

References

H. Davenport, On Waring’s problem for cubes, Acta Math. 71 (1939), 123–143. MR 26, DOI 10.1007/BF02547752
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
I. M. Kamenetzky, Sur l’interpolation au moyen des dérivées et sur les procédés d’interpolation correspondants. I, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 356–358 (French). MR 0002003
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
M. Gontcharoff, Sur quelques séries d’interpolation généralisant celles de Newton et de Stirling, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 17–48 (Russian, with French summary). MR 0002002
K. A. Hirsch, On skew-groups, Proc. London Math. Soc. 45 (1939), 357–368. MR 0000036, DOI 10.1112/plms/s2-45.1.357
B. Helffer, Around Floquet eigenvalues, Preprint, Mittag-Leffler Inst., 2002.
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
T. A. Suslina, On discrete spectrum in the gaps of a two-dimensional periodic elliptic operator perturbed by a decaying potential, Preprint, Mittag-Leffler Inst., 2002; Contemp. Math. (to appear).