Asymptotics of spectral gaps of 1D Dirac operator with two exponential terms potential

2013/12/08 by Anahtarci, Berkay, Djakov, Plamen
#34L10 #34L40 #47E05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1312.2219

Abstract

The one-dimensional Dirac operator L = i \beginpmatrix 1 amp; 0
0 amp; -1 \endpmatrix (d)/(dx) +\beginpmatrix 0 amp; P(x)
Q(x) amp; 0 \endpmatrix, P,Q ∈ L² ([0,π]), considered on [0,π] with periodic and antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n ∈ ℤ, there are two (counted with multiplicity) eigenvalues λ_n^-,λ_n⁺ (periodic if n is even, or antiperiodic if n is odd) such that |λ_n^± - n |<1/2. We study the asymptotics of spectral gaps γ_n =λ_n⁺ - λ_n^- in the case P(x)=a e^-2ix + A e^2ix, Q(x)=b e^-2ix + B e^2ix, where a, A, b, B are nonzero complex numbers. We show, for large enough m, that γ_{± 2m}=0 and γ_2m+1 = ± 2 \frac√(Ab)^m (aB)^m+14^2m (m!)² [ 1 + O ( (log² m)/(m²)) ], γ_-(2m+1) = ± 2\frac√(Ab)^m+1 (aB)^m4^2m (m!)² [ 1 + O ( (log² m)/(m²)) ].

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