带有谱参数边界条件且权函数变号的不连续Sturm-Liouville 算子

赵迎春, 孙炯

系统科学与数学 ›› 2011, Vol. 31 ›› Issue (5) : 597-613.

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PDF(497 KB)
系统科学与数学 ›› 2011, Vol. 31 ›› Issue (5) : 597-613. DOI: 10.12341/jssms11613
论文

带有谱参数边界条件且权函数变号的不连续Sturm-Liouville 算子

    赵迎春1,2, 孙炯1
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DISCONTINUOUS STURM-LIOUVILLE OPERATOR WITH BOUNDARY CONDITION CONTAINING THE SPECTRAL PARAMETER AND INDEFINITE WEIGHT FUNCTION

    ZHAO Yingchun1,2,   SUN Jiong1
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摘要

研究了一类边界条件中含有谱参数且权函数变号的不连续Sturm-Liouville 算子L. 首先构造了一个与边值问题相关联的Krein 空间K和新算子A使得所考虑的算子L 与新算子A 的特征值相同, 证明了新算子A 在Krein 空间K 中是自共轭的. 进一步地, 通过研究算子A的谱分布, 得到了该边值问题有可数个实的
特征值、它们是上下无界的, 没有有限值的聚点, 且能够被指标化应满足下列不等式  <λn<<λ2<λ1<0<λ1<λ2<<λn<  最后给出了一个实例得到了特征值的分布情况.

Abstract

Discontinuous Sturm-Liouville operator~L with boundary condition depending on the spectral parameter and indefinite weight function is studied. Firstly, a Krein space and a new operator~A related to the boundary-value problem are constructed to make the eigenvalues of the operators~A and~L same. It is proved that the operator~A is self-adjoint in the space ~K. Then, by studying the spectral distribution of the operator~A, it is shown
that all the eignvalues of the boundary value problem are real, there exist countably infinitely many positive and negative eigenvalues, and they are unbounded from below and from above, have no finite cluster point, and can be written as   <λn<<λ2<λ1<0<λ1<λ2<<λn<   Finally, a specific example is given to get the eigenvalues distribution.

关键词

不连续~Sturm-Liouville 算子 / 特征值 /   / 权函数

Key words

Discontinuous sturm-liouville operator /   / eigenvalue /  weight function

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赵迎春, 孙炯. 带有谱参数边界条件且权函数变号的不连续Sturm-Liouville 算子. 系统科学与数学, 2011, 31(5): 597-613. https://doi.org/10.12341/jssms11613
ZHAO Yingchun, SUN Jiong. DISCONTINUOUS STURM-LIOUVILLE OPERATOR WITH BOUNDARY CONDITION CONTAINING THE SPECTRAL PARAMETER AND INDEFINITE WEIGHT FUNCTION. Journal of Systems Science and Mathematical Sciences, 2011, 31(5): 597-613 https://doi.org/10.12341/jssms11613
中图分类号: 34B24    45C05    49R50   
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