Deformation of the spectrum for Darboux-Treibich-Verdier potential along \(\mathrm{Re}\,\tau =\frac{1}{2}\). (English) Zbl 1532.34086
The paper under review deals with the spectrum \(\sigma(L)\) of the complex Hill operator \(L\) with Darboux-Treibich-Verdier potential
\[
L=\frac{d^2}{dx^2}-6\varrho(x+z_0;\tau)-2\varrho(x+\frac{1}{2}+z_0;\tau) \text{ in } L^2 (\mathbb{R},\mathbb{C}).
\]
Here \(\varrho(z;\tau)\) is the Weierstrass elliptic function with periods 1, \(\tau\) and \(z_0 \in \mathbb{C}\) are chosen such that the differential operator \(L\) has no singularities in \(\mathbb{R}\). The author focuses on the deformation of the spectrum with \(\tau=\frac{1}{2}+ ib\) as \(b>0\) varies.
Reviewer: Erdoğan Şen (Tekirdağ)
MSC:
| 34L05 | General spectral theory of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
Keywords:
spectrum; Darboux-Treibich-Verdier potential; Hill operator; Weierstrass elliptic function; mean field equationReferences:
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