Tunneling for the ∂-operator

2023/03/10 by Johannes Sjöstrand, Sjöstrand, Johannes, Vogel, Martin
Mathematics · Physics and Astronomy · #FOS: Mathematics #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2303.06096

openalex publication_date 2023/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the small singular values of the 2-dimensional semiclassical differential operator P = 2 e^-ϕ/h∘ hD_z∘ e^ϕ/h on S¹+iS¹ and on S¹+iℝ where ϕ is given by sin y and by y³/3, respectively. The key feature of this model is the fact that we can pinpoint precisely where in phase space the Poisson bracket \p,p\=0, where p is the semiclassical symbol of P. We give a precise asymptotic description of the exponentially small singular values of P by studying the tunneling effects of an associated Witten complex. We use these asymptotics to determine a Weyl law for the exponentially small singular values of P.

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