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

TL;DR

This paper derives explicit formulas for the heat kernel and trace of magnetic Laplacians on complex projective spaces, revealing their spectral properties and asymptotic behavior using advanced mathematical techniques.

Contribution

It provides a new proof for the reproducing kernels of eigenspaces and derives heat coefficients in terms of Bernoulli numbers, advancing spectral analysis of magnetic Laplacians.

Findings

Explicit heat kernel integral representation

Trace formula involving Jacobi theta functions

Asymptotic heat coefficients in terms of Bernoulli numbers

Abstract

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 $\Delta_\nu$.