Title: On a class of Grushin operators: a geometric mechanics approach
We study a class of Grushin operators ¢G = @2 @x2 + x2k @2 @y2 ; k 2 N via the geometric mechanics approach, which is based on analysis of the geometry induced by the operators. By using the shooting method, we reduce the problem of determining the number of geodesics connecting any two points in the induced geometry to a problem of finding how many solutions there are to the Hamilton-Jacobi equation. After a detailed analysis, we obtain the number and length of geodesics connecting any two points. It is shown that there may be more than one geodesics connecting two points arbitrarily near to each other, in some cases, the number of geodesics is even infinite. The most interesting and important case is k = 2, which is related to the quadric harmonic oscillators. In the final part, we discuss some open problems and future work in this direction.
Contents
1 Introduction
1.1 Subelliptic operators
1.2 Euclidean Laplacian and Laplace-Beltrami operators
1.3 Carnot-Carath´ eodory geometry
1.4 Heisenberg group and Heisenberg geometry
2 Quartic Oscillator and Grushin Operators
2.1 Quartic oscillator
2.2 Grushin operators
3 Analysis of the Geodesics
3.1 Hamiltonian mechanics and geometry induced by ∆G
3.2 Case θ = 0
3.3 Case θ 6
= 0 I: geodesics with two endpoints in y-axis
3.4 Case θ 6
= 0 II: x0 = 0 and x1 6= 0
4 Discussion and FutureWork
4.1 Case θ 6
= 0 III: geodesics both starting and ending outside the y-axis
4.2 Inverse kernels to the Grushin operators
4.3 Complex Hamilton-Jacobi theory
Author: Li, Yutian
Source: City University of Hong Kong
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