On subordinacy and spectral multiplicity for a class of singular differential operators
On subordinacy and spectral multiplicity for a class of singular differential operators
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关于一类奇异微分算子的从属性和谱重数
DOI：
10.1017/s0308210500021648
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
D. Gilbert
中科院分区：
文献类型：
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作者：
D. Gilbert
研究了与奇异微分表达式相关的自伴算子 H 的谱重数。基于 I. S. Kac 的早期工作和最近关于从属性的研究结果，根据以下方面建立了谱多重性为 1 或 2 的完整必要和充分条件：(i) Titchmarsh-Weyl m 函数的边界行为，以及 (ii) Lu = λu, λεℝ 在端点 a 和 b 处的解的渐近性质。特别地，它表明，当且仅当 L 在 a 和 b 处都处于极限点情况，并且 Lu = λu 的解不从属于 a 或 b 处的所有 λ 的集合具有正勒贝格测度时，H 才具有重数 2。结果是完全通用的，仅受到对系数 p(r)、q(r) 和 w(r) 的最小限制，以及当 L 在两个端点处于极限圆情况时的分离边界条件的假设。
The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the form is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.