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Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds

Riccardo Adami, Ugo Boscain, Dario Prandi, Lucia Tessarolo

Abstract

Let $M$ be a 3-dimensional contact sub-Riemannian manifold and $S$ a surface embedded in $M$. Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation $\mathscr{F}$. In this paper we study the Schrödinger evolution of a particle constrained on $\mathscr{F}$. In particular, we relate the self-adjointness of the Schrödinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.

Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds

Abstract

Let be a 3-dimensional contact sub-Riemannian manifold and a surface embedded in . Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation . In this paper we study the Schrödinger evolution of a particle constrained on . In particular, we relate the self-adjointness of the Schrödinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.

Paper Structure

This paper contains 16 sections, 20 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Example: Hyperbolic paraboloid in the Heisenberg case
  3. The general case and the Laplacian on the foliation
  4. Further remarks
  5. Heat diffusion
  6. Other intrinsic laplacians
  7. Structure of the paper
  8. The geometric structure
  9. The characteristic foliation
  10. Local structure near characteristic points
  11. The Laplacian on leaves
  12. Local expression of $\Delta_\ell$
  13. Essential self-adjointess of $\Delta_\ell$
  14. The foliated Laplacian on $S$
  15. The case of half-line leaves via Von Neumann's theory of self-adjoint extensions
  16. ...and 1 more sections

Key Result

Proposition 2

Let $a\ge 0$ and $a\neq 1/2$. Consider $\Delta_x$ and $\Delta_y$ with domain $\mathcal{C}^\infty_0(\mathbb{R}_+)$. Then, the operator $\Delta_y$ is not essentially self-adjoint in $L^2(\mathbb{R}_+)$. Moreover,

Figures (2)

  • Figure 1: Qualitative picture for the characteristic foliation at an isolated characteristic point, with the corresponding values for $\widehat{K}_p$. From left to right, we recognise a saddle, a proper node, a star node and a focus.
  • Figure 2: Heteroclinic cycle between 4 saddles. Even if the operator is not essentially self-adjoint on the black leaves composing the cycle, it is nevertheless essentially self-adjoint at infinity on every leaf in the interior of the cycle.

Theorems & Definitions (43)

  • Remark 1
  • Proposition 2
  • Theorem 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Remark 8
  • Proposition 9
  • Remark 10
  • ...and 33 more