Threshold resonance in geometric scattering. (English) Zbl 1159.58310

Wang, Xue Ping

Mat. Contemp. 26, 135-164 (2004).

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