2025/03/10 by Krzysztof Stempak, Stempak, Krzysztof · 1 citation
Mathematics · #math.FA
paper · pdf · doi:10.48550/arxiv.2503.07073
published as Journal of Differential Equations 481 (2026) 114630 · 19 pages
arxiv created 2025/03/10 · arxiv updated 2026/07/30
In the setting of the Grushin differential operator G=-Δx'-|x'|2Δx'' with domain \rm Dom G=C^∞c(ℝd)⊂ L2(ℝd), 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 L2(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.