An action function for a higher step Grushin operator

作者：Furutani Kenro^*; Iwasaki Chisato; Kagawa Toshinao

来源：Journal of Geometry and Physics, 2012, 62(9): 1949-1976.

DOI：10.1016/j.geomphys.2012.04.004

• Heat kernel
• Grushin operator
• Complex Hamilton-Jacobi method
• Action function
• Volume form
• Higher order oscillator

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 %26quot;step 2%26quot; 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.

• 出版日期2012-9