2020/01/30 by Zhijie Chen, Erjuan Fu, Chen, Zhijie +3
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2001.11244
openalex publication_date 2020/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the spectrum of the complex Hill operator L=(d2)/(dx2)+q(x;τ) in L2(ℝ,ℂ) with the Darboux-Treibich-Verdier potential q(x;τ):=-∑k=03nk(nk+1)\wp ( x+z0+\tfracωk2;τ), where nk∈ℤ≥ 0 with max nk≥ 1 and z0∈ℂ is chosen such that q(x;τ) has no singularities on ℝ. For any fixed τ∈ iℝ>0, we give a necessary and sufficient condition on (n0,n1,n2,n3) to guarantee that the spectrum σ(L) is σ(L)=(-∞, E2g]∪[E2g-1, E2g-2]∪ ⋯ ∪[E1, E0], Ej∈ ℝ, and hence generalizes Ince's remarkable result in 1940 for the Lamé potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval (E2j-1, E2j-2), which generalizes the recent result in \citeHHV where the Lamé case n1=n2=n3=0 was studied.