We have the following integral operator
$$
Ku(t)=\int_0^1 G(t,s)\ u(s)\ ds,\quad u\in L^2[0,1],
$$
where
$$G(t,s)=
\begin{cases}
s(1-t)& 0\leq s\leq t\leq 1\\
t(1-s)& 0\leq t\leq s\leq 1
\end{cases}$$
The eingenvalues of $K$ are
$$\lambda_n=\frac{1}{\pi^2 n^2}, \quad n=1,2,3,\ldots$$
My question is as follows. Can we find an operator, say $K'$, related to this (with a sum?) such that its eigenvalues are $n+\frac{1}{\pi^2 n^2}$?
Thank you.