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

الأصل · EN

The one-dimensional Dirac operator equation* L = i pmatrix 1 & 0 0 & -1 pmatrix d/dx +pmatrix 0 & P(x) Q(x) & 0 pmatrix, P,Q ∈ L² ([0,π]), equation* 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 λₙ-,λₙ+ (periodic if n is even, or antiperiodic if n is odd) such that |λₙ± - n |<1/2. We study the asymptotics of spectral gaps γₙ =λₙ+ - λₙ- in the case P(x)=a e-2ix + A e2ix, Q(x)=b e-2ix + B e2ix, where a, A, b, B are nonzero complex numbers. We show, for large enough m, that γ± ₂ₘ=0 and align* γ₂ₘ₊₁ = ± 2 √(Ab)ᵐ (aB)ᵐ⁺¹4²ᵐ (m!)² [1 + O (² m/m²)], align* align* γ₋₍₂ₘ₊₁₎ = ± 2√(Ab)ᵐ⁺¹ (aB)ᵐ4²ᵐ (m!)² [1 + O (² m/m²)]. align*