Spectrum of the Lamé operator and application. I: Deformation along \(\operatorname{Re} \tau = \frac{ 1}{ 2} \) (Q2020416)

The authors investigate the spectrum \(\sigma(L_n)\) of the Lamé operator \[L_n:=\displaystyle\frac{d^2}{dx^2}-n(n+1)\wp(x+z_0;\tau),\,\,\,x\in\mathbb{R},\] in \(L^2(\mathbb{R},\mathbb{C})\), where \(n\in\mathbb{N}\), \(\wp(z;\tau)\) is the Weierstrass elliptic function with basic periods \(\omega_1=1\) and \(\omega_2=\tau\), and \(z_0\in\mathbb{C}\) is chosen such that \(L_n\) has no singularities on \(\mathbb{R}\). For \(n=2\), they completely determine all possible intersection points of \(\sigma(L_2)\), and then, for \(\tau=\frac{1}{2}+ib\) with \(b>0\), they show that \(\sigma(L_2)\) has exactly \(9\) different types of graphs for different \(b\)'s, and give the explicit range of \(b\) for each type of graphs. As application, for \(\tau=\frac{1}{2}+ib\) with \(b>0\), the authors prove that there are \(b\in \left(\frac{1}{2\sqrt{3}},\frac{\sqrt{3}}{2}\right)\) such that the mean field equation corresponding to \(L_n\) has no even axisymmetric solutions, but it has \(2\) even solutions which are not axisymmetric.