2024/11/15 by Suresh Eswarathasan, Eswarathasan, Suresh, Allan Greenleaf +3 · 2 citations
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2411.10401
openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, P=(P1,P2,…, Pn), where Pi are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold Mn, with ∑ Pi2 elliptic and [Pi,Pj]=0 for 1≤ i,j≤ n. A particularly important case is when (M,g) is Riemannian and P1=(-Δ)^\frac12. We illustrate our result with several examples, including surfaces of revolution.