Abstract

We study the asymptotic behavior of the spectrum of an integral operator similar to an integral operator with a logarithmic kernel depending on the sum of arguments. By a simple change of variables, the corresponding equation is reduced to an integral equation of convolution type defined on a finite interval (as is well known, such equations in the general case cannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced to a conjugation problem for analytic functions and then to an infinite system of linear algebraic equations, the isolation of the main terms in which allows deriving a relation that determines the spectrum of the original problem.

中文翻译：

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具有特殊形式对数核的积分算子谱的渐近性

摘要

我们研究积分算子谱的渐近行为，该算子类似于具有对数核的积分算子，具体取决于参数之和。通过简单地改变变量，相应的方程就被简化为在有限区间上定义的卷积型积分方程（众所周知，一般情况下此类方程不能用求积法求解）。接下来，使用傅里叶变换，将方程简化为解析函数的共轭问题，然后简化为线性代数方程的无限系统，其中主要项的隔离允许导出确定原始问题的谱的关系。