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Regular Solutions of the Shabat Equation

✍ Scribed by Yunkang Liu

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 286 KB
Volume: 154
Category: Article
ISSN: 0022-0396
DOI: 10.1006/jdeq.1998.3541

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✦ Synopsis

The Shabat equation

where + and q are parameters, is the simplest self-similar reduction of the so-called dressing chain for constructing and analyzing exactly solvable Schro dinger equations. It is so relevant to the study of the q-oscillator algebra in quantum mechanics. The main objective of this paper is to investigate whether the Shabat equation has a nontrivial global solution f (t) # C 1 (R) that is normalizable in the sense that the corresponding potential u(t)

) is in the function space L 1 (R) and whether in the case +Â(1&q 2 )>0 it has a global solution f (t) # C 1 (R) that is regular in the sense that f (t)=-+Â(1&q 2 )+O(t &2 ) and f $(t)=O(t &3 ) as t Ä \ .

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