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The first instability interval for Hill equations
with symmetric single well potentials


Author: Min-Jei Huang
Journal: Proc. Amer. Math. Soc. 125 (1997), 775-778
MSC (1991): Primary 34B30, 34L15
DOI: https://doi.org/10.1090/S0002-9939-97-03705-2
MathSciNet review: 1363425
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Abstract: For Hill equations with symmetric single well (or symmetric single barrier) potentials, the first instability interval is absent when and only when the potential is constant.


References [Enhancements On Off] (What's this?)

  • 1. M. Ashbaugh and R. Benguria, Optimal lower bounds for eigenvalue gaps for Schrödinger operators with symmetric single-well potentials and related results, Maximum Principles and Eigenvalue Problems in Partial Differential Equations (P. W. Schaefer, ed.), Pitman Res. Notes Math. Ser., vol. 175, Longman Sci. Tech., Harlow, 1988, pp. 134-145. MR 90c:35157
  • 2. -, Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc. 105 (1989), 419-424. MR 89f:81028
  • 3. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1-96. MR 7:382d
  • 4. M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh and London, 1973.
  • 5. H. Hochstadt, Functiontheoretic properties of the discriminant of Hill's equation, Math. Zeit. 82 (1963), 237-242. MR 27:5955
  • 6. -, On the determination of a Hill's equation from its spectrum , Arch. Rational Mech. Anal. 19 (1965), 353-362; 23 (1966), 237-238. MR 31:6019; MR 34:411
  • 7. -, On a Hill's equation with double eigenvalues, Proc. Amer. Math. Soc. 65 (1977), 373-374. MR 56:3404
  • 8. R. Lavine, The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc. 121 (1994), 815-821. MR 94i:35144
  • 9. W. Magnus and S. Winkler, Hill's Equation, John Wiley and Sons, New York, 1966. MR 33:5991
  • 10. P. Ungar, Stable Hill equations, Comm. Pure Appl. Math. 14 (1961), 707-710. MR 31:423

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Additional Information

Min-Jei Huang
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043
Email: mjhuang@math.nthu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-97-03705-2
Keywords: Hill equation, eigenvalue, instability interval, symmetric single well potential.
Received by editor(s): July 10, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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