2020/08/28 by M. W. Geis, Geis, Michael
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2008.12482
openalex publication_date 2020/08/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let (S2,g) be a convex surface of revolution and H \⊂ S2 the\nunique rotationally invariant geodesic. Let \φ^\ℓm be the orthonormal\nbasis of joint eigenfunctions of \Δg and \∂_\θ, the\ngenerator of the rotation action. The main result is an explicit formula for\nthe weak-* limit of the normalized empirical measures, \Σm = -\ℓ^\ℓ\n||\φ^\ℓm||2L2(H) \δ\(m)/(\ℓ)(c) on [-1,1]. The\nexplicit formula shows that, asymptotically, the L2 norms of restricted\neigenfunctions are minimal for the zonal eigenfunction m = 0, maximal for\nGaussian beams m = \± 1, and exhibit a (1 - c2)-\(1)/(2) type\nsingularity at the endpoints. For a pseudo-differential operator B we also\ncompute the limits of the normalized measures \∑m = -\ℓ^\ℓ \⟨ B\n\φ^\ℓm , \φ^\ℓm \⟩ \δ\(m)/(\ℓ)(c).\n