Beta function formalism for Grushin operators (1)
Applicable Nonlinear Analysis, Volume 3, Issue 1, April 2026, Pages: 8–44
DER-CHEN CHANG
Department of Mathematics and Statistics, Georgetown University, Washington D.C. 20057, USA
Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City 242, Taiwan, ROC
JONATHAN RIESS
Department of Mathematics and Statistics and Department of Physics, Georgetown University, Washington DC, 20057, USA
Abstract
This is the first part of a series of articles. Here we we develop the Beta function formalism and apply it to the Grushin operator \(\Delta_G = \frac{1}{2}(\partial_x^2 + x^{2k}\partial_y^2)\) with \(k\geq 1\). We want to construct the heat kernel for the operator \(\Delta_G\). In other words, we want to construct the fundamental solution for the operator \(\frac{\partial}{\partial t}-\Delta_G\) for \(t>0\). We apply Hamiltonian mechanics to handle this problem and first need to solve the Hamiltonian system defined by the principal symbol \(H(x,y,\xi, \eta) =\frac 12(\xi^2+x^{2k}\eta^2)\) and given by Hamilton's equations: \[ \dot{x} \,=\, H_\xi=\xi,\,\,\, \dot{y}\, =\, H_\eta=x^{2k}\eta, \qquad \dot{\xi} \,= \, - H_x\,=\, -kx^{2k-1}\eta^2 ,\,\,\, \dot{\eta}\,= \, - H_y \,=\,0 \] Hence, from the Hamilton's equations, we know that \( \ddot{x}= -k\eta^2x^{2k-1}\) and \(\dot{y} = \eta x^{2k}\). Set \(-k\eta^2 x^{2k-1}\,=\, -V^\prime(x)\), therefore \(\frac 12\dot x^2+V(x)\,=\, E\). Here \(V(x) =\frac 12 \eta^2x^{2k}\) is the potenial and \(E\) is the total energy. This implies that \[ \begin{equation*} \begin{split} s\,=\, \int_{x_0}^{x_1} \frac{dx}{\sqrt{2(E -\frac{1}{2}\eta^2 x^{2k})}} \quad {\mbox{and}}\quad y(s)\,=\, \eta \int_0^s x^{2k}(\tau) d\tau. \end{split} \end{equation*} \]
\((1)\). When \(k=1\), the equation is a linear ODE and we can write down its solutions in the form of $$ x(s)=C_1\sin(\eta s+C_2), $$ where \(C_1\) and \(C_2\) are two constants. So this case can be studied by a careful analysis of the trigonometric functions.
\((2).\) The integrand in the above integral is not integrable for \(k\neq 1\). We can express this Abelian integral as a (Jacobi's) Elliptic functions: but this only really works for \(k=2\): $$ \int R(t,\sqrt{P(t)}) dt= \text{Elementary Functions} + c_1I_0+c_2I_1+c_3J_1, $$ where \(P(t)\) is degree \(3\) or \(4\).
\((3)\). When \(k\ge 3\), this allows us to find the parametrisations in terms of the incomplete Beta function of the (normal) geodesics solutions given by the Hamiltonian system for the associated Hamiltonian of the Grushin operator, i.e., its principal symbol. The Beta functions arise as a result of expressing the standard energy integral \(t= \int \frac{dx}{\sqrt{2E- \theta^2x^{2k}}}\), which is elliptic for \(k>1\), in terms of Beta functions.
Given these parametrisations, we can then count the number of geodesics connecting a given starting and ending point, which is done by finding the intersections of the \(x\)- and \(y\)-parametrisations. The existence of such curves joining any given two points on the underlying sub-Riemannian manifold, which is defined by the principal symbol of the Grushin operator, we justify by the Chow-Rashevskii theorem. In this article, we develop the motivation and mathematical background necessary to understand how and why we are interested in studying subelliptic and hypoelliptic operators such as Grushin and Kohn's operator. We also introduce the basics of the elliptic function theory and elliptic integrals, which we will use throughout the entire article.Cite this Article as
Der-Chen Chang and Jonathan Riess, Beta function formalism for Grushin operators (1), Applicable Nonlinear Analysis, 3(1), 8–44, 2026