A transform for the Grushin operator with applications

Abstract

In the setting of the Grushin differential operator $G=-\hat{I}{”}_{x^{\prime }}-|x^{\prime }{|}^2\hat{I}{”}_{x^{\prime \prime }}$ with domain $\mathrm{Dom}G=C_c^{\infty }({\mathbb{R}}^d)\subset L^2({\mathbb{R}}^d)$, we define a scalar transform which is a mixture of the partial Fourier transform and a transform based on the scaled Hermite functions. This transform unitarily intertwines $G$ with a multiplication operator by a nonnegative real-valued function on an appropriately associated `dual' space $L^2(\hat{I}“)$. This allows to construct a self-adjoint extension $\mathbb{G}$ of $G$ as a simple realization of this multiplication operator. Another self-adjoint extensions of $G$ are defined in terms of sesquilinear forms and then these extensions are compared. Aditionally, a closed formula for the heat kernel that corresponds to the heat semigroup $\{\exp (-t\mathbb{G}){\}}_{t>0}$ is established.

19 pages