2021/11/30 by Erjuan Fu, Fu, Erjuan · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Combinatorics #FOS: Mathematics #Genus #Lambda #Mathematical analysis #Mathematical functions and polynomials #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Operator (biology) #Physics #Polynomial #Quantum mechanics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Spectrum (functional analysis) #Zero (linguistics)
paper · pdf · doi:10.48550/arxiv.2111.15059
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2021/11/30 · openalex created_date 2021/12/06 · openalex updated_date 2026/08/06
In this paper, we study the spectrum σ(L) of the Lamé operator L=(d2)/(dx2)-12\wp(x+z0;τ) in L2(ℝ, ℂ), where \wp(z;τ) is the Weierstrass elliptic function with periods 1 and τ, and z0∈ℂ is chosen such that L has no singularities on ℝ. We prove that a point λ∈ σ(L) is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if λ is a zero of the following cubic polynomial: (4)/(15) λ3+(8)/(5)η1 λ2-3g2 λ+9g3-6η1 g2=0. We also study the deformation of the spectrum as τ=(1)/(2)+ib with b>0 varying. We discover 7 different types of graphs for the spectrum as b varies around the double zeros of the spectral polynomial.