TY - JOUR

T1 - On the resonances and eigenvalues for a 1D half-crystal with localised impurity

AU - Korotyaev, Evgeny L.

AU - Schmidt, Karl Michael

N1 - Funding Information: The first author was supported by the 14.740.11.0581 Federal Program ‘‘Development of Scientific Potential of Higher Education for the period 2009–2013’’.

PY - 2012/9

Y1 - 2012/9

N2 - We consider the Schrödinger 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 (σ n)∞ 1, σ n ε {0,1}, and (x n) 1 ∞ ε l 2, x n ≧ 0, there exists a potential p such that x n is the length of the n-th gap, n εℕ, and H has exactly σ n eigenvalues and 1 - σ 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.

AB - We consider the Schrödinger 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 (σ n)∞ 1, σ n ε {0,1}, and (x n) 1 ∞ ε l 2, x n ≧ 0, there exists a potential p such that x n is the length of the n-th gap, n εℕ, and H has exactly σ n eigenvalues and 1 - σ 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.

UR - http://www.scopus.com/inward/record.url?scp=84870180548&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2011.153

DO - 10.1515/CRELLE.2011.153

M3 - Article

AN - SCOPUS:84870180548

SP - 217

EP - 248

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 670

ER -