Spectral properties of Neumann Laplacian of horns

Abstract

We study the Neumann Laplacian of unbounded regions in ℝ^n with cusps at infinity so that the corresponding Dirichlet Laplacian has compact resolvent. Typical of our results is that of the region {(x, y)_∈ℝ^2 ∥xy|<1} the Neumann Laplacian has absolutely continuous spectrum [0, ∞) of uniform multiplicity four and an infinity of eigenvalues E_0 < E_ 1 ≤... → ∞ and that for the region {(x, y)_∈ℝ^2∥y|≤ e^-^(∣x∣)}, it has absolutely continuous spectrum [1/4, ∞) of uniform multiplicity 2 and an infinity of eigenvalues E_0 = 0 < E_1 ≤... → ∞. We use the Enss theory with a suitable asymptotic dynamics.

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