论文标题
Darboux-treibich-Verdier潜力的必要条件，其光谱包含在$ \ mathbb {r} $中
A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$
论文作者
论文摘要
在本文中，我们研究了复杂的山丘操作员的光谱$ l = \ frac {d^2} {dx^2}+q（x;τ）$ in $ l^2（\ mathbb {r {r}，\ mathbb {c}）$ \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}{2};τ\right),\] where $ n_k \ in \ Mathbb {z} _ {\ geq 0} $带有$ \ max n_k \ geq \ geq 1 $和$ z_0 \ in \ mathbb {c} $被选择，以至于$ q（x;τ）$在$ \ \ mathbb {mathbb {r r} $上没有奇异性。对于任何固定的$τ\ in i \ mathbb {r} _ {> 0} $，我们在$（n_0，n_1，n_2，n_3）$上给出了必要且充分的条件，以保证频谱$σ（l）$是 \ [σ（l）=（ - \ infty，e_ {2g}] \ cup [e_ {2g-1}，e_ {2g-2}] \ cup \ cdots \ cup \ cup \ cup [e_ {1}，e_ {0}，e_ {0}] 因此，对于Darboux-Treibich-Verdier的潜力而言，这是INCE在1940年的显着结果。我们还确定了每个有界间隔$（e_ {2j-1} $，$ e_ {2j-2}）$中（反）周期性特征值的数量，该数字在\ cite {hhv}中概括了最新的结果{hhv}其中lamécase $ n_1 = n_2 = n_2 = n_2 = n_3 = 0 $。
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 \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}{2};τ\right),\] where $n_k\in\mathbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;τ)$ has no singularities on $\mathbb{R}$. For any fixed $τ\in i\mathbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $σ(L)$ is \[σ(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1}, E_{0}],\quad E_j\in \mathbb{R},\] 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 $(E_{2j-1}$, $E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the Lamé case $n_1=n_2=n_3=0$ was studied.
