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Constructing resonance calabashes of Hill's equations using step potentials

Published online by Cambridge University Press:  01 July 2000

MEIRONG ZHANG  and
SHAOBO GAN
MEIRONG ZHANG
Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing 100084, People's Republic of China; e-mail: mzhang@math.tsinghua.edu.cn The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy
SHAOBO GAN
Affiliation:
The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy School of Mathematical Science, Peking University, Beijing 100871, People's Republic of China
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Abstract

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger operators with certain step potentials. This gives counter-examples to the Alikakos–Fusco conjecture on the second periodic eigenvalues. Using this simple model, we can also construct infinitely many resonance pockets, which are much like calabashes emanating from a cane, of one-parameter Hill's equations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 1 , July 2000 , pp. 153 - 164
Copyright
2000 Cambridge Philosophical Society

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