Inverse Indefinite Spectral Problem for Second Order Differential Operator with Complex Periodoc Coefficients

The inverse spectral problem for indefinite strings

Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form \[-u"=z\,u\,ω+z^2u\,\upsilon\] on an interval $[0,L)$, where $ω$ is a real-valued distribution in $H^{-1}_{\mathrm{loc}}[0,L)$, $\upsilon$ is a non-negative Borel measure on $[0,L)$ and $z$ is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of …

Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil

The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem

广义不定弦的迹公式与逆谱理论

Trace formulas and inverse spectral theory for generalized indefinite strings

本文研究了广义不定弦的逆谱理论，证明了相关的 Szegő 型定理，完整刻画了特定系数类的谱测度集合，并将其应用于保守 Camassa-Holm 流，证明了低正则性阶梯型初始条件下全局弱解的存在性。

Herglotz-Nevanlinna函数的连分数展开与Stieltjes型广义不定弦

Continued fraction expansions of Herglotz-Nevanlinna functions and generalized indefinite strings of Stieltjes type

本研究借助Herglotz-Nevanlinna函数的连分数展开结果，刻画Stieltjes型广义不定弦的谱数据，并通过显式公式求解其逆谱问题。

层状超材料的均匀化与内谱

Homogenisation of Laminated Metamaterials and the Inner Spectrum

本文研究快速变号系数散度型方程在层状超材料中的均匀化，引入电导率内谱刻画极限退化或四阶非局部化，并建立不定权Sturm–Liouville算子理论。