Pointwise Weyl Laws for Quantum Completely Integrable Systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916482736193536 |
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| author | Eswarathasan, Suresh Greenleaf, Allan Keeler, Blake |
| author_facet | Eswarathasan, Suresh Greenleaf, Allan Keeler, Blake |
| contents | The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of Hörmander, following important prior contributions by Gärding, Levitan, Avakumović, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator.
Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-Δ)^\frac12$. We illustrate our result with several examples, including surfaces of revolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise Weyl Laws for Quantum Completely Integrable Systems Eswarathasan, Suresh Greenleaf, Allan Keeler, Blake Analysis of PDEs Mathematical Physics Spectral Theory The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of Hörmander, following important prior contributions by Gärding, Levitan, Avakumović, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-Δ)^\frac12$. We illustrate our result with several examples, including surfaces of revolution. |
| title | Pointwise Weyl Laws for Quantum Completely Integrable Systems |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2411.10401 |