A transform for the Grushin operator with applications
Krzysztof Stempak
TL;DR
The paper analyzes the Grushin operator $G=-\Delta_{x'}-|x'|^2\Delta_{x''}$ on $\mathbb{R}^d$ by constructing a transform $\mathcal{G}$ that blends a partial Fourier transform with a scaled Hermite expansion, yielding a unitary intertwining with a nonnegative multiplication operator. This provides a simple self-adjoint extension $\mathbb{G}=\mathcal{G}^{-1}\mathbb{M}_\Theta\mathcal{G}$ and a concrete Sobolev-type domain $\mathrm{Dom}\,\mathbb{G}=W^2_G(\mathbb{R}^d)$, along with an alternative form-based description that confirms the same extension. The authors derive a closed formula for the heat kernel $p_t(x,y)$ of $\mathbb{G}$ and establish the corresponding heat semigroup, including group-invariance and scaling properties. Overall, the work provides a transparent operator-theoretic framework for spectral analysis and heat-kernel formulas of the Grushin operator, connecting harmonic analysis with subelliptic geometry through a Mehler-type spectral decomposition and explicit functional calculus.
Abstract
In the setting of the Grushin differential operator $G=-Δ_{x'}-|x'|^2Δ_{x''}$ with domain ${\rm Dom}\,G=C^\infty_c(\mathbb{R}^d)\subset L^2(\mathbb{R}^d)$, 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 $L^2(Γ)$. 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.