Erjuan Fu
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View article: Deformation of the spectrum for Darboux–Treibich–Verdier potential along $\operatorname{Re}\tau=\frac{1}{2}$
Deformation of the spectrum for Darboux–Treibich–Verdier potential along $\operatorname{Re}\tau=\frac{1}{2}$ Open
In this paper, we study the spectrum \sigma(L) of the complex Hill operator with Darboux–Treibich–Verdier potential L=\frac{d^2}{dx^2}-6\wp(x+z_0;\tau)-2\wp\Bigl(x+\frac{1}{2}+z_0;\tau\Bigr)\quad \text{in }L^2(\mathbb{R}, \mathbb{C}), wher…
View article: Tube classes over elementary vanishing cycles
Tube classes over elementary vanishing cycles Open
Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mappi…
View article: A new approach towards Lefschetz $(1, 1)$-Theorem
A new approach towards Lefschetz $(1, 1)$-Theorem Open
Let $S$ be a complex projective surface. Lefschetz originally proved Lefschetz $(1, 1)$--Theorem by studying a Lefschetz pencil of hyperplane sections of $S$ and the Abel--Jacobi mapping. In this paper, we attack Lefschetz $(1, 1)$--Theore…
View article: Spectrum of the Lamé operator along $\mathrm{Re}τ={1}/{2}:$ The genus $3$ case
Spectrum of the Lamé operator along $\mathrm{Re}τ={1}/{2}:$ The genus $3$ case Open
In this paper, we study the spectrum $σ(L)$ of the Lamé operator \begin{equation*}L=\frac{d^2}{dx^2}-12\wp(x+z_0;τ)\quad \text{in}\;\;L^2(\mathbb{R}, \mathbb{C}), \end{equation*} where $\wp(z;τ)$ is the Weierstrass elliptic function with p…
View article: A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$
A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$ Open
In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;τ)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}…