arXiv 1202.24442012-02-11math.SPmath.CA

关于不定Sturm-Liouville问题Riesz基性质的一个必要方面

On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems

Aleksey Kostenko

备注：: 26 pages
发表信息：: Math. Nachr. 287, no. 14-15, 1710-1732 (2014)
DOI：: 10.1002/mana.201300104

摘要：

1996年，H. Alummer观察到不等式\[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \]对于$[-1,1]$上的某类函数$f$满足某个正常数$K>0$，如果问题\[ -y"=λ\, r(x)y,\quad y(-1)=y(1)=0 \]的特征函数形成希尔伯特空间$L^2_{|r|}(-1,1)$的Riesz基。这里，假设权重$r\in L^1(-1,1)$满足$xr(x)>0$ a.e.在$[-1,1]$上。我们提出了两个标准的Weyl-Titchmarsh $m$ -函数的Weyl-Titchmarsh不等式是有效的。利用这些结果，我们证明了这个不等式是有效的，如果与谱问题相关联的操作符满足线性预解式增长条件。特别地，我们证明了当$r$为奇数时，特征函数的Riesz基性质等价于线性预解式的增长.

英文摘要：

In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \[ -y"=λ\, r(x)y,\quad y(-1)=y(1)=0 \] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$. We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. Using these results we show that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if $r$ is odd.

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