{
  "id": "2202.02160",
  "version": "v1",
  "published": "2022-02-04T14:39:20.000Z",
  "updated": "2022-02-04T14:39:20.000Z",
  "title": "Heat coefficients for magnetic Laplacians on the complex projective space $\\mathbf{P}^{n}(\\mathbb{C})$",
  "authors": [
    "K. Ahbli",
    "A. Hafoud",
    "Z. Mouayn"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "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 $\\Delta_\\nu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-04T14:39:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex projective space",
      "heat coefficients",
      "magnetic laplacians",
      "uniform magnetic field strength proportional",
      "higher order derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}