# Spectrum of the Lam\'{e} operator along $\mathrm{Re}\tau={1}/{2}:$ The genus $3$ case

@inproceedings{Fu2021SpectrumOT, title={Spectrum of the Lam\'\{e\} operator along \$\mathrm\{Re\}\tau=\{1\}/\{2\}:\$ The genus \$3\$ case}, author={Erjuan Fu}, year={2021} }

In this paper, we study the spectrum σ(L) of the Lamé operator L = d2 dx2 − 12℘(x + z0; τ) in L(R, C), where ℘(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 λ + 8 5 η1λ 2 − 3g2λ + 9g3 − 6η1g2 = 0. We also study the deformation…

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