  Denoting by $\Delta_\nu$ the Fubini-Study Laplacian perturbed by a uniform
magnetic field strength proportional to $\nu$, this operator has a discrete
spectrum consisting on eigenvalues $\beta_m, \ m\in\mathbb{Z}_+$, when acting
on bounded functions of the complex projective $n$-space. For the corresponding
eigenspaces, we give a new proof for their reproducing kernels by using
Zaremba's expansion directly. These kernels are then used to obtain an integral
representation for the heat kernel of $\Delta_\nu$. Using a suitable polynomial
decomposition of the multiplicity of each $\beta_m$, we write down a trace
formula for the heat operator associated with $\Delta_\nu$ in terms of Jacobi's
theta functions and their higher order derivatives. Doing so enables us to
establish the asymptotics of this trace as $t\searrow 0^+$ by giving the
corresponding heat coefficients in terms of Bernoulli numbers and polynomials.
The obtained results can be exploited in the analysis of the spectral zeta
function associated with $\Delta_\nu$.
