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.



Lower bound for the eigenvalue
OK, here is the sketch of the bound I mentioned. The operator in question is just $PQP$ where $P$ is the space projection to the interval $[-1,1]$ and $Q$ is the Fourier projection to the interval $[-c,c]$. So, we just need a $k$-dimensional space $V$ of functions such that $\|QPf\|\ge A(k,c)\|f\|$ for every function $f$ in that space. Then $\lambda\_k\ge A(k,c)^2$. Without thinking too much, let us take $V$ to be the space of functions that are constant on each of $k$ intervals of length $2/k$ partitioning $[-1,1]$. Assume also that $c/k\ll 1$ (otherwise the bound I mentioned is total nonsense because the norm of the operator is at most $1$). Then if the function $f$ takes value $a\_j$ on the $j$th interval, we have to estimate from below $\int\_{[-c,c]}|\widehat f|^2$ under the condition $\sum\_j|a\_j|^2=k$ (or something like that). However,  

$$
\widehat f(y)=g(y)\sum\_i a\_ie^{4\pi ij/k}
$$
where $g(y)$ is the Fourier transform of the characteristic function of an interval of length $2/k$, which, under the assumption $c/k\ll 1$ is essentially $1/k$ on the whole interval $[-c,c]$ in absolute value. Thus we are down to estimating a trigonometric polynomial of given degree on a small arc, more precisely, on $c/k$ of the full period, from below. In $L^2$, it is at least $(ac/k)^{k-1/2}$ of the full norm (choose between Berstein, Remez, and Turan for the reference name), whence $\lambda\_k\ge (ac/k)^{2k-1}$ (Here I start enumeration with $1$ unlike in your post and my comment where it starts with $0$).


That's it.


