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On the Spectrum of a one-dimensional Schrödinger operator whose potential is a limit of finite-zone potentials

Published online by Cambridge University Press:  14 November 2011

B. M. Levitan
B. M. Levitan
Affiliation:
Moskovskii University, Mehmat, Moscow 117234, U.S.S.R.
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Synopsis

Denote by (αj, βj), j = 1, 2, … an infinite set of disjoint open intervals on the half-line (0, ∞). Suppose that the following conditions are fulfilled:

With the aid of the first two trace formula presented earlier by the author, we prove in this paper that there exists a function q, defined on the whole real line, such that for the Schrödinger equation −y″ + q(x)y = λy (−∞<x<∞), the intervals (αj, βj) are spectrum lacunae. As an example, we consider the case when the intervals (αj, βj) are adjacent intervals of the Cantor trinary perfect set.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 177 - 183
Copyright
Copyright © Royal Society of Edinburgh 1984

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