×

Eigenvalues of a self-differential operator. (English) Zbl 1181.45002

The authors are presenting a detailed study of the following simple but interesting problem: Describe the asymptotic behavior of the eigenvalues \(A_n\), \(n\in\mathbb{Z}\), of the compact Fredholm operator \[ (Tf)(x)=\int^1_0 \varphi(x,s) f(s)\,ds, \] on the space \(C_0(0,1)\) of real-valued continuous functions on \([0,1]\), vanishing at \(x= 0\), where \[ \varphi(x,s)= \begin{cases} 0,\quad &\text{if }1> s> 2x,\\ 1/2,\quad &\text{if }2x> s> 2x- 1,\\ 1,\quad &\text{if }2x- 1> s> 0.\end{cases} \] It is shown that the following formulas are valid: \[ \lim_{n\to\pm\infty} A_n= 0,\quad \lim_{n\to\infty} {A_n\over A_{-n}}= -1,\quad \lim_{n\to\infty} {A_n\over A_{n-1}}= {1\over 4}. \] Also it is proven that the inverses of the eigenvalues are the zeros of the function \[ \sum^\infty_{m=0} {B_m x^m\over 2^{(m^2- m)/2} m!}, \] with \(B_m\) the Bernoulli numbers. The asymptotics of the eigenvalues is also discussed, and estimates are provided.

MSC:

45C05 Eigenvalue problems for integral equations
45M05 Asymptotics of solutions to integral equations
47G10 Integral operators
PDFBibTeX XMLCite