Institute for Mathematical Physics Weyl–titchmarsh Theory for Sturm–liouville Operators with Distributional Coefficients Weyl–titchmarsh Theory for Sturm–liouville Operators with Distributional Coefficients
Institute for Mathematical Physics Weyl–titchmarsh Theory for Sturm–liouville Operators with Distributional Coefficients Weyl–titchmarsh Theory for Sturm–liouville Operators with Distributional Coefficients
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通讯作者：
Jonathan Eckhardt;F. Gesztesy;Roger Nichols;G. Teschl
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作者：
Jonathan Eckhardt;F. Gesztesy;Roger Nichols;G. Teschl
我们系统地建立了任意区间（a, b）上奇异微分算子的Weyl-Titchmarsh理论，该理论与类型为τ f = 1 R−p[f ' + sf] ' + sp[f ' + sf] + qf的广义微分表达式相关联，特别是，该建立表明τ允许在W−1,1 loc （(a, b)）和H−1 loc （(a, b)）中存在分布势系数。我们研究了极大和极小Sturm-Liouville算子，极大算子Tmax的所有自伴随限制，或等价地，最小算子tmin的所有自伴随扩展，所有自伴随边界条件（分离的和耦合的），并描述了tmin的任何自伴随扩展的解。此外，我们刻画了本文的主要目标，在分离边界条件下，求出任意自伴随扩展对应的奇异Weyl-Titchmarsh-Kodaira m函数，并推导出相应的谱变换。我们还处理了主解并描述了T min的Friedrichs扩展。最后，在τ是正则的特殊情况下，我们描述了T min的Krein - von Neumann扩展，并描述了导致保正解（因此是半群）的所有边界条件。
We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type τ f = 1 r − p[f ′ + sf ] ′ + sp[f ′ + sf ] + qf , In particular, this setup implies that τ permits a distributional potential coefficient in W −1,1 loc ((a, b)) and H −1 loc ((a, b)). We study maximal and minimal Sturm–Liouville operators, all self-adjoint restrictions of the maximal operator Tmax, or equivalently, all self-adjoint extensions of the minimal operator T min , all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of T min. In addition, we characterize the principal object of this paper, the singular Weyl–Titchmarsh–Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation. We also deal with principal solutions and characterize the Friedrichs extension of T min. Finally, in the special case where τ is regular, we characterize the Krein– von Neumann extension of T min and also characterize all boundary conditions that lead to positivity preserving resolvents (and hence semigroups).