Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps

Abstract

We study the eigenvalue asymptotics of a Neumann Laplacian −Δ_N^Ω in unbounded regions Ω of R^2 with cusps at infinity (a typical example is Ω = {(x, y) ϵR^2: x > 1, ¦y¦< e^(−x)^2}) and prove that N_E(−Δ_N^Ω) ~ N_E(H_v) +E2Vol(Ω), where H_v is the canonical one-dimensional Schrödinger operator associated to the problem. We establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a Dirichlet Laplacian −Δ_D^Ω for a class of cusp-type regions of infinite volume.

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