Spectral theory of some non-selfadjoint linear differential operators
Spectral theory of some non-selfadjoint linear differential operators
复制标题
一些非自伴线性微分算子的谱论
DOI：
10.1098/rspa.2013.0019
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
B. Pelloni
中科院分区：
文献类型：
--
作者：
D. A. Smith;B. Pelloni
我们给出了常系数线性微分算子的谱性质的刻画，这些算子作用于有界区间上定义的函数，并由一般线性边界条件决定。边界条件可能使得所得到的算子不是自伴随的。我们将这样一个算子S的谱性质与偏微分方程∂tq±iSq=0的相应边值问题解的性质联系起来。也就是说，我们能够在算子的特征函数族的性质，特别是这个族是否为基，与相关边值问题的唯一解的存在性和性质之间建立显式的对应关系。当这种唯一解存在时，我们将其表示为复轮廓积分，该积分是用Fokas等人最近提出的一种变换方法得到的。这种表示中被积函数的解析性对于研究关联算子的谱理论是至关重要的。
We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂tq±iSq=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.