{
  "id": "2011.11571",
  "version": "v1",
  "published": "2020-11-23T17:33:38.000Z",
  "updated": "2020-11-23T17:33:38.000Z",
  "title": "Fourier coefficients of restrictions of eigenfunctions",
  "authors": [
    "Emmett L. Wyman",
    "Yakun Xi",
    "Steve Zelditch"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Let $\\{e_j\\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \\subset M$ be a submanifold and let $\\{\\psi_k\\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \\[ \\langle \\gamma_H e_j, \\psi_k \\rangle_{L^2(H)} = \\int_H e_j \\overline \\psi_k \\, dV_H, \\] of restrictions $\\gamma_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\\{(\\mu_k, \\lambda_j)\\}_{j,k - 0}^{\\infty}$ of the (square roots of the) Laplacian $\\Delta_M$ on $M$ and the Laplacian $\\Delta_H$ on $H$ in a family of suitably `thick' regions in $\\mathbb R^2$. Thick regions include (1) the truncated cone $\\mu_k/\\lambda_j \\in [a,b] \\subset (0,1)$ and $\\lambda_j \\leq \\lambda$, and (2) the slowly thickening strip $|\\mu_k - c\\lambda_j| \\leq w(\\lambda)$ and $\\lambda_j \\leq \\lambda$, where $w(\\lambda)$ is monotonic and $1 \\ll w(\\lambda) \\lesssim \\lambda^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-23T17:33:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Pxx",
      "35S30",
      "58J40"
    ],
    "keywords": [
      "fourier coefficients",
      "restrictions",
      "orthonormal basis",
      "laplace eigenfunctions",
      "compact riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}