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On the resonances and eigenvalues for a 1D half-crystal with localised impurity

Korotyaev, Evgeny L. and Schmidt, Karl Michael 2011. On the resonances and eigenvalues for a 1D half-crystal with localised impurity. Journal für die reine und angewandte Mathematik (Crelles Journal) 2012 (670) , pp. 217-248. 10.1515/CRELLE.2011.153

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We consider the Schrdinger operator H on the half-line with a periodic potential p plus a compactly supported potential q. For generic p, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential q and for any sequences , n {0, 1}, and , xn 0, there exists a potential p such that xn is the length of the n-th gap, n , and H has exactly n eigenvalues and 1 n antibound state in each high-energy gap. Moreover, we show that between any two eigenvalues in a gap, there is an odd number of antibound states, and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Walter de Gruyter
ISSN: 0075-4102
Last Modified: 05 Jun 2017 03:11

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