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Self-adjointness criteria and self-adjoint extensions of the Laplace-Beltrami operator on $α$-Grushin manifolds

Ivan Beschastnyi, Hadrian Quan

Abstract

The Grushin plane serves as one of the simplest examples of a sub-Riemannian manifold whose distribution is of non-constant rank. Despite the fact that the singular set where this distribution drops rank is itself a smoothly embedded submanifold, many basic results in the spectral theory of differential operators associated to this geometry remain open, with the question of characterizing self-adjoint extensions being a recent question of interest both in sub-Riemannian geometry and mathematical physics. In order to systematically address these questions, we introduce an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity, closely related to the 0-calculus of Mazzeo arising in asymptotically hyperbolic geometry. Extending results of arXiv:2011.03300, arXiv:1105.4687, arXiv:1609.01724, this calculus allows us to give criterion for essential self-adjointness of the Curvature Laplacian, $Δ-cS$ for $c>0$ (here $S$ is the scalar curvature). When this operator is not essentially self-adjoint, we determine several natural self-adjoint extensions. Our results generalize to a broad class of differential operators which are elliptic in this calculus.

Self-adjointness criteria and self-adjoint extensions of the Laplace-Beltrami operator on $α$-Grushin manifolds

Abstract

The Grushin plane serves as one of the simplest examples of a sub-Riemannian manifold whose distribution is of non-constant rank. Despite the fact that the singular set where this distribution drops rank is itself a smoothly embedded submanifold, many basic results in the spectral theory of differential operators associated to this geometry remain open, with the question of characterizing self-adjoint extensions being a recent question of interest both in sub-Riemannian geometry and mathematical physics. In order to systematically address these questions, we introduce an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity, closely related to the 0-calculus of Mazzeo arising in asymptotically hyperbolic geometry. Extending results of arXiv:2011.03300, arXiv:1105.4687, arXiv:1609.01724, this calculus allows us to give criterion for essential self-adjointness of the Curvature Laplacian, for (here is the scalar curvature). When this operator is not essentially self-adjoint, we determine several natural self-adjoint extensions. Our results generalize to a broad class of differential operators which are elliptic in this calculus.
Paper Structure (21 sections, 39 theorems, 316 equations, 3 figures)

This paper contains 21 sections, 39 theorems, 316 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement and previous results
  3. Main tools and results
  4. The structure of the paper
  5. Principal results of the $\alpha$-calculus
  6. Manifolds with corners and quasi-homogeneous blow-up
  7. The double stretched product
  8. Polyhomogeneous asymptotics
  9. $\alpha$-calculus and existence of a parametrix
  10. Natural domains for the Laplacian on $\alpha$-Grushin manifolds
  11. Closure of $\Delta - cS$
  12. Adjoint of $\Delta - cS$
  13. Self-adjoint extensions
  14. Asymmetry forms method and the symplectic linear algebra of self-adjoint extensions
  15. Construction of self-adjoint extensions
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $G$ be the $\alpha$-Grushin plane of dimension two. Depending on $\alpha$ the following statements hold:

Figures (3)

  • Figure 1: Example of a $p$-filtration $V_1 \subset V_2 = \mathbb{R}^2_2$ (right) and a non-example (left)
  • Figure 2: The blow-down map $\beta$ of the $\alpha$-stretched product space $M\times_\alpha M$
  • Figure 3: The triple space $M_\alpha^3$

Theorems & Definitions (86)

  • Definition 1.1
  • Theorem 1.1: EU
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • ...and 76 more