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Smoothness of Schrödinger Operator Potential in the Case of Gevrey Type Asymptotics of the Gaps

✍ Scribed by Plamen Djakov; Boris Mityagin

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
281 KB
Volume
195
Category
Article
ISSN
0022-1236

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✦ Synopsis

Consider the Schro¨dinger equation Ày 00 þ V ðxÞy ¼ ly with a periodic real-valued L 2 -potential V of period 1; VðxÞ ¼ P 1 m¼À1 vðmÞ expð2pimxÞ: Let fl À n ; l þ n g be its zones of instability, i.e. fl À n ; l þ n g are pairs of periodic and antiperiodic eigenvalues, and b 2 ð0; 1Þ determines Gevrey classes.

then V ðxÞ is a Gevrey function, and moreover X 1 m¼À1 jvðmÞj 2 ð1 þ jmjÞ 2s e 2ajmj b o1:

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