论文标题
通过边界和扰动理论，单数Sturm-Liouville操作员的光谱特性
Spectral Properties of Singular Sturm-Liouville Operators via Boundary Triples and Perturbation Theory
论文作者
论文摘要
我们将边界三元组和扰动理论的理论应用于具有两个极限圈端点的半结合的Sturm-Liouville操作员的设置。对于一般的边界条件，我们获得了有关其特征值和本征函数的完善和新结果。 在边界三重设置中，我们获得了简单的标准，用于识别当参数是矩阵时具有双重特征值的哪些自相关扩展。我们还确定了Friedrichs扩展的进一步光谱特性，并且（当操作员为正时）Von Neumann-Krein扩展。 在最近的一些标量Aronszajn-Donoghue类型结果中，我们发现，当边界条件仅限于与扰动参数的空间中的仿射线相对应时，实数只能是Sturm-Liouville操作员的两个扩展的特征值。此外，我们确定了可以通过扰动理论达到的那些Sturm-liouville操作员的大部分光谱表示。
We apply both the theory of boundary triples and perturbation theory to the setting of semi-bounded Sturm-Liouville operators with two limit-circle endpoints. For general boundary conditions we obtain refined and new results about their eigenvalues and eigenfunctions. In the boundary triple setup, we obtain simple criteria for identifying which self-adjoint extensions possess double eigenvalues when the parameter is a matrix. We also identify further spectral properties of the Friedrichs extension and (when the operator is positive) the von Neumann-Krein extension. Motivated by some recent scalar Aronszajn-Donoghue type results, we find that real numbers can only be eigenvalues for two extensions of Sturm-Liouville operator when the boundary conditions are restricted to corresponding to affine lines in the space from which the perturbation parameter is taken. Furthermore, we determine much of the spectral representation of those Sturm-Liouville operators that can be reached by perturbation theory.
