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Threshold resonance in geometric scattering. (English) Zbl 1159.58310
Summary: We study the properties of the resonant states at 0 for Schrödinger operators of the form \(P=-\Delta_g+ \frac{q(\theta)}{r^2}+ V_0(x)\) on a Riemannian manifold, where \(x=r\theta\) are some polar coordinates and \(g\) is a perturbation of a Riemannian metric \(g_0= dr^2+ r^2h\) with \(h\) independent of \(r\). A characterization of zero energy resonant states is given in terms of small eigenvalues of \(-\Delta_h+q(\theta)\).

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
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