For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d\psi}{dx}-c^2x^2\psi $$ ( in fact, $\psi_{k,c}$ is called the prolate spheroidal wave function).

Consider a compact integral operator $$ F_c(\psi)(x)=\int_{-1}^1\frac{\sin(c(x-y))}{\pi(x-y)}\psi(y)dy. $$

It is known that $\psi_{k,c}$ are the eigenfunctions of the operator $F_c$. So, $$ F_c=\lambda_k\psi_{k,c}, $$ where $\lambda_k$ are eigenvalues of $F_c$. (Note, it is known that $\lambda_k\leq \frac{c}{2}\left(\frac{ec}{4k}\right)^{2k}$.)

I would like to find a lower bound for $\lambda_k$.

Any references and ideas will be very helpful.

Thank you.