  For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$
form a basis with remarkable properties for the space of band-limited functions
with bandwidth $c$. They have been largely studied and used after the seminal
work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications
rely heavily of the behavior and the decay rate of the eigenvalues
$(\lambda_n(c))_{n\geq 0}$ of the time and frequency limiting operator, which
we denote by $\mathcal Q_c.$ Hence, the issue of the accurate estimation of the
spectrum of this operator has attracted a considerable interest, both in
numerical and theoretical studies. In this work, we give an explicit integral
approximation formula for these eigenvalues. This approximation holds true
starting from the plunge region where the spectrum of $\mathcal Q_c$ starts to
have a fast decay. As a consequence of our explicit approximation formula, we
give a precise description of the super-exponential decay rate of the
$\lambda_n(c).$ Also, we mention that the described approximation scheme
provides us with fairly accurate approximations of the $\lambda_n(c)$ with low
computational load, even for very large values of the parameters $c$ and $n.$
Finally, we provide the reader with some numerical examples that illustrate the
different results of this work.
