A transform for the Grushin operator with applications
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916648192049152 |
|---|---|
| author | Stempak, Krzysztof |
| author_facet | Stempak, Krzysztof |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A transform for the Grushin operator with applications Stempak, Krzysztof Functional Analysis 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. |
| title | A transform for the Grushin operator with applications |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2503.07073 |