limit-point criteria for the matrix sturm-liouville operator and its powers

معايير نقطة النهاية لمؤثر شتورم-ليوفيل المصفوفي وقواه

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ID: 142965
2017
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Abstract
We consider matrix Sturm-Liouville operators generated by the formal expression \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] in the space \(L^2_n(I)\), \(I:=[0, \infty)\). Let the matrix functions \(P:=P(x)\), \(Q:=Q(x)\) and \(R:=R(x)\) of order \(n\) (\(n \in \mathbb{N}\)) be defined on \(I\), \(P\) is a nondegenerate matrix, \(P\) and \(Q\) are Hermitian matrices for \(x \in I\) and the entries of the matrix functions \(P^{-1}\), \(Q\) and \(R\) are measurable on \(I\) and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices \(P\), \(Q\) and \(R\) that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by \(l^k[y]\) (\(k \in \mathbb{N}\)). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.
ندرس مؤثرات شتورم-ليوفيل المصفوفية المتولدة عن التعبير الشكلي \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] في الفضاء \(L^2_n(I)\)، حيث \(I:=[0, \infty)\). لتكن الدوال المصفوفية \(P:=P(x)\) و \(Q:=Q(x)\) و \(R:=R(x)\) من الرتبة \(n\) (\(n \in \mathbb{N}\)) معرّفة على \(I\)، بحيث تكون \(P\) مصفوفة غير متدهورة، و\(P\) و\(Q\) مصفوفتين هيرميتيتين لكل \(x \in I\)، ومدخلات الدوال المصفوفية \(P^{-1}\) و\(Q\) و\(R\) قابلة للقياس على \(I\) وقابلة للتكامل على كل فترة جزئية مغلقة ومحدودة منه. يتمثل الهدف الرئيسي من هذه الورقة في إيجاد شروط على المصفوفات \(P\) و\(Q\) و\(R\) تضمن تحقق حالة النقطة الحدية للمؤثر المتماثل المغلق الأصغري المتولد عن \(l^k[y]\) (\(k \in \mathbb{N}\)). وبوجه خاص، نحصل على شروط النقطة الحدية لمؤثرات شتورم-ليوفيل ذات المعاملات التوزيعية ذات القيم المصفوفية.
Reference Key
braeutigam2017opusculalimit-point Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors ;Irina N. Braeutigam
Journal zhonghua yi xue za zhi
Year 2017
DOI
http://dx.doi.org/10.7494/OpMath.2017.37.1.5
URL
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3701.pdf
Keywords
quantitative methods

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