Heat coefficients for magnetic Laplacians on the complex projective space Pⁿ(ℂ)

Abstract

We denote by ${\mathrm{\Delta }}_{\nu }$ the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues ${\text{β}}_m,m\in {\mathbb{Z}}_{+}$. 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 ${\mathrm{\Delta }}_{\nu }$. Using a suitable polynomial decomposition of the multiplicity of each ${\text{β}}_m$, we write down a trace formula for the heat operator associated with ${\mathrm{\Delta }}_{\nu }$ 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 ${\mathrm{\Delta }}_{\nu }$.

Keywords:

• Heat coefficients
• magnetic Laplacians
• complex projective space
• reproducing kernels

AMS Subject Classifications:

• 58Jxx
• 33C45
• 35A08
• 35C15
• 46E22

Acknowledgements

The authors would like to thank the Moroccan Association of Harmonic Analysis and Spectral Geometry.

Disclosure statement

No potential conflict of interest was reported by the author(s).