Deformation of the spectrum for Darboux–Treibich–Verdier potential along $\operatorname{Re}\tau=\frac{1}{2}$

Erjuan Fu

doi:10.4171/jst/453

JournalsjstVol. 13, No. 1pp. 347–381

Deformation of the spectrum for Darboux–Treibich–Verdier potential along $Re τ = \frac{1}{2}$

Erjuan Fu
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, China

$Deformation of the spectrum for Darboux–Treibich–Verdier potential along $\operatorname{Re}\tau=\frac{1}{2}$ cover$

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Abstract

In this paper, we study the spectrum $σ (L)$ of the complex Hill operator with Darboux–Treibich–Verdier potential

L = \frac{d ^{2}}{d x ^{2}} - 6 ℘ (x + z_{0}; τ) - 2 ℘ (x + \frac{1}{2} + z_{0}; τ) in L^{2} (R, C),

where $℘ (z; τ)$ is the Weierstraß elliptic function with periods $1$ and $τ$ , and $z_{0} \in C$ is chosen such that $L$ has no singularities on $R$ . We give a complete picture of the deformation of the spectrum with $τ = \frac{1}{2} + ib$ as $b > 0$ varies. A new idea of the proof is to apply the result of the mean field equation and its connection with this operator.

Cite this article

Erjuan Fu, Deformation of the spectrum for Darboux–Treibich–Verdier potential along $Re τ = \frac{1}{2}$ . J. Spectr. Theory 13 (2023), no. 1, pp. 347–381

DOI 10.4171/JST/453