Mathematics > General Mathematics
[Submitted on 24 May 2023 (v1), last revised 1 May 2024 (this version, v2)]
Title:Self-adjoint Laplace operator with translation invariance on infinite-dimensional space $\mathbb R^\infty$
View PDF HTML (experimental)Abstract:The standard Laplacian $-\triangle_{\mathbb R^n}$ in $L^2(\mathbb R^n)$ is self-adjoint and translation invariant on the finite-dimensional vector space $\mathbb R^n$. In this paper, using some quadratic form, we define a translation invariant operator $-\triangle_{\mathbb R^\infty}$ on $\mathbb R^\infty$ as a non-negative self-adjoint operator in some Hilbert space $L^2(\mathbb R^\infty)$, which is a subset of the set $CM(\mathbb R^\infty)$ of all complex measures on the product measurable space $\mathbb R^\infty$. Furthermore, we show that for any $f\in L^2(\mathbb R^n)$ and any $u\in L^2(\mathbb R^\infty)$, $e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}(f\otimes u) =(e^{\sqrt{-1}\triangle_{\mathbb R^n}t}f)\otimes (e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}u) \ (t\in (-\infty,+\infty))$ and $e^{\triangle_{\mathbb R^\infty}t}(f\otimes u)=(e^{\triangle_{\mathbb R^n}t}f)\otimes (e^{\triangle_{\mathbb R^\infty}t}u) \ (t\in [0,+\infty))$ hold. This clearly shows that $-\triangle_{\mathbb R^\infty}$ is an analog of $-\triangle_{\mathbb R^n}$. The starting point for the discussion in this paper is to naturally introduce a translation invariant Hilbert space structure into $CM(\mathbb R^\infty)$.
Submission history
From: Hiroki Yagisita [view email][v1] Wed, 24 May 2023 22:14:46 UTC (12 KB)
[v2] Wed, 1 May 2024 23:43:36 UTC (12 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
Connected Papers (What is Connected Papers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.