论具有紧凑逆的非负对称算子理论中的比尔曼问题

Pub Date : 2023-12-29 DOI:10.1134/S0016266323020090

M. M. Malamud

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引用次数: 0

摘要

Abstract 描述了具有以下性质的关于 \(\Bbb R^2\) 和 \(\Bbb R^3\) 的大类非负薛定谔算子： 1.这些算子中的每个算子对 \(\Bbb R^2\) （在 \(\Bbb R^3\) 中）度量为零的适当无界集的限制是一个具有紧凑前溶剂的非负对称算子（迪里希特问题算子）；2.在关于势的某些附加假设下，这种限制的弗里德里希斯扩展具有连续（有时是绝对连续）的谱，填充正半轴。所获得的结果给出了 M. S. Birman 问题的一个解决方案。

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On the Birman Problem in the Theory of Nonnegative Symmetric Operators with Compact Inverse

Large classes of nonnegative Schrödinger operators on \(\Bbb R^2\) and \(\Bbb R^3\) with the following properties are described:

1. The restriction of each of these operators to an appropriate unbounded set of measure zero in \(\Bbb R^2\) (in \(\Bbb R^3\)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;

2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.

The obtained results give a solution of a problem by M. S. Birman.

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