磁拉普拉斯算子在非光滑域和规范变换中的所有自伴随扩展

C. Oliveira, W. Monteiro
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引用次数: 1

摘要

我们使用边界三元组找到磁性薛定谔算子的所有自伴随扩展的参数化,在具有紧边界的拟凸域$\Omega$中,以及在$\textrm{W}^{1}_{\infty}(\overline{\Omega})$中具有分量的磁势。这也给出了拉普拉斯算子在非正则域上的所有自伴随扩展的一个新的表征。然后讨论了这类自伴随扩展的规范变换,并推广了Dirichlet拉普拉斯算子的Dirichlet磁算子的规范等价的一个表征;还讨论了包括不规则螺线管在内的与Aharonov-Bohm效应的关系。特别地,在(有界)拟凸域的情况下,证明了如果某些扩展(通过与光滑单位函数的乘法)与具有零磁势的实现是一致等价的,那么对于所有自伴随实现都是相同的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations
We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schrodinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.
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