Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Stempak, Krzysztof
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979314525
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