Pavel Exner and Vladimir Lotoreichik
Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer
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ABSTRACT. We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $H$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $H$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $H$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schr\"odinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer $\mathcal{P}$ at infinity.