Renormalized Oscillation Theory for Singular Linear Hamiltonian Systems
Renormalized Oscillation Theory for Singular Linear Hamiltonian Systems
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DOI：
10.1016/j.jfa.2022.109525
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发表时间：
2020-09
影响因子：
1.7
通讯作者：
P. Howard;A. Sukhtayev
中科院分区：
文献类型：
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作者：
P. Howard;A. Sukhtayev
在一类一般的线性哈密顿系统的区间上，至少有一个奇异端点可以是极限点、极限环或极限中间点，我们证明了通过考虑与适当选择的C2n的拉格朗日子空间的路径相关的Maslov指数，可以自然地得到重正化的振动结果。在第一部分的分析中，我们将我们的线性哈密顿系统与一族定义的自伴算子族联系在一起，在第二部分中，我们使用重整化振荡的方法来计算这些算子在固定区间(λ1，λ2)上的本征值的个数，这些闭包不与算子的本质谱相交。我们用两个说明性的例子来结束分析，说明该理论如何在实践中得以实施。这推广了作者以前关于正则线性哈密顿系统的工作。
Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C 2 n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ 1, λ 2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.