On Povzner--Wienholtz-type Self-Adjointness Results for Matrix-Valued Sturm--Liouville Operators

Steve Clark, Fritz Gesztesy

备注：: LaTeX, 11 pages, an updated and corrected version to appear in Proc. Roy. Soc. Edinburgh A

英文摘要：

We derive Povzner--Wienholtz-type self-adjointness results for $m\times m$ matrix-valued Sturm--Liouville operators $T=R^{-1}\big[-\f{d}{dx}P\f{d}{dx}+Q\big]$ in $L^2((a,b);Rdx)^m$, $m\in\bbN$, for $(a,b)$ a half-line or $\bbR$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Povzner-Wienholtz-type theorems for Sturm-Liouville operators with singular coefficients

We introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the coefficients. They are assumed to satisfy conditions \[q=Q'+s;\quad \frac{1}{\sqrt{|p|}}, \frac{Q}{\sqrt{|p|}}, \frac{r}{\sqrt{|p|}} \in L^2_{loc}\left(\mathbb{R}\right), \quad s \in L^1_{loc}\left(\mathbb{R}\right), \quad\frac{1}{p}\neq 0\,\,\text{a.e.,} \] where the derivative of the function $Q$ is understood in the sense of distributions, and all function…

2021-10-22math.SP

A Glazman-Povzner-Wienholtz Theorem on graphs

The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schrödinger operator $-Δ+ q$ and suitable local regularity assumptions on $q$, guarantees its essential self-adjointness. Our aim is to extend this result to Schrödinger operators on graphs. We first obtain the corresponding theorem for Schrödinger operators on metric graphs, allowing in particular distributional potentials $q\in H^{-1}_{\rm loc}$. Moreover, we exploit recently discovered connections between Schrödinger operators on metric graphs and weighted graphs i…

2021-06-08math-phmath.FAmath.MP

Remarks on Schrödinger operators with singular matrix potentials

In this paper the asymmetric generalization of the Glazman-Povzner-Wienholtz theorem is proved for one-dimensional Schrödinger operators with strongly singular matrix potentials from the space $H_{loc}^{-1}(\mathbb{R}, \mathbb{C}^{m\times m})$. This result is new in the scalar case as well.

2013-06-03math.SP

Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds

We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner--Wienholtz--Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.

2001-07-30math.APmath.SP

Self-adjointness of Schroedinger operators with singular potentials

We study one-dimensional Schroedinger operators S with real-valued distributional potentials q in W^{-1}_{2,loc}(R) and prove an extension of the Povzner-Wienholtz theorem on self-adjointness of bounded below S thus providing additional information on its domain. The results are further specified for q in W^{-1}_{2,unif}(R).

2012-03-21math.FAmath.SP