An action function for a higher step Grushin operator

• Published in: Journal of geometry and physics Vol. 62; no. 9; pp. 1949 - 1976
• Main Authors: Furutani, Kenro, Iwasaki, Chisato, Kagawa, Toshinao
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2012
• Subjects: Action function; Complex Hamilton–Jacobi method; Grushin operator; Heat kernel; Higher order oscillator; Volume form; 22E25; Grushin operator; Volume form; Higher order oscillator; Heat kernel; 35K05; Complex Hamilton–Jacobi method; Action function
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/j.geomphys.2012.04.004

The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.