IJPAM: Volume 52, No. 1 (2009)

EIGENVALUES OF A SELF-DIFFERENTIAL OPERATOR

Yuri Dimitrov$^1$, G.A. Edgar$^2$
$^{1,2}$Department of Mathematics
Ohio State University
Columbus, OH 43210, USA
$^1$e-mail: yuri@math.ohio-state.edu
$^2$e-mail: edgar@math.ohio-state.edu


Abstract.In this paper we study the asymptotic behavior of the eigenvalues of the compact operator $Tf(x)={\int_0^1}\varphi(x,s)f(s)ds$, where $f\in C[0,1]$ with $f(0)=0$, and

()

We show that the eigenvalues $\{A_n\}_{n=-\infty}^\infty$ of $T$ satisfy

()

and that $A_n^{-1}$ are zeros of the power series \begin{equation*}
{\sum_{m=0}^\infty} \frac{B_mx^m}{2^{(m^2-m)/2}m!}\,,
\end{equation*} where $B_m$ are the Bernoulli numbers. An asymptotic estimate of a partial theta function is involved in the proof.

Received: March 12, 2009

AMS Subject Classification: 39B22, 45C05

Key Words and Phrases: functional differential equation, self-differential, theta function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 1