Heat coefficients for magnetic Laplacians on the complex projective space $\mathbf{P}^{n}(\mathbb{C})$

Denoting by $Δ_ν$ the Fubini-Study Laplacian perturbed by a uniform magnetic field strength proportional to $ν$, this operator has a discrete spectrum consisting on eigenvalues $β_m, \ m\in\mathbb{Z}_+$, 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 $Δ_ν$. Using a suitable polynomial decomposition of the multiplicity of each $β_m$, we write down a trace formula for the heat operator associated with $Δ_ν$ 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 $Δ_ν$.

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