  It is well known that the cohomology groups of a closed manifold $M$ can be
reconstructed using the gradient dynamical of a Morse-Smale function $f\colon
M\to \R$. A direct result of this construction are Morse inequalities that
provide lower bounds for the number of critical points of $f$ in term of Betti
numbers of $M$. These inequalities can be deduced through a purely analytic
method by studying the asymptotic behaviour of the deformed Laplacian operator.
This method was introduced by E. Witten and has inspired a numbers of great
achievements in Geometry and Topology in few past decades. In this paper,
adopting the Witten approach, we provide an analytic proof for; the so called;
equivariant Morse inequalities when the underlying manifold is acted on by the
Lie group $G=S^1$ and the Morse function $f$ is invariant with respect to this
action.
