Lower bounds in cones for solutions to the Schrödinger equation (Q1092320)

The author investigates the asymptotic behaviour at infinity of square integrable generalized solutions \(\psi\) of the Schrödinger equation under the conditions that the real valued potential \(V=V_ 1+V_ 2\) where (i) \(\lim_{\alpha \to \infty}\| \rho^{1/2}V_ 1(-\Delta +\alpha^ 2)^{-3/4}\| =0\), \(\rho^ 2=| x|^ 2+1;\) (ii) \(\forall \epsilon >0\), \(i=1,2,..\). \(\exists C_{\epsilon}>0\) such that \(\pm \rho \partial_ 1V_ 2<-\epsilon \Delta\) in the distribution sense; (iii) \(V_ 2(-\Delta +1)^{-1}\) is compact; (iv) \((-\Delta +1)^{-1}\rho \partial_ iV_ 2(-\Delta +1)^{-1}\) is compact for any \(i=1,...,n.\) If \(\psi\) is a nonzero solution in the distribution sense of the Schrödinger equation \((-\Delta +V)\psi =E\psi,\) \(E<0\), \(\psi\in D(\Delta)\) and C is nonempty open cone in \({\mathbb{R}}^ n\) then \[ - \lim_{R\to \infty}R^{-1}\ell n(\int_{S^{n-1}\cap C}(\psi (R\omega)|^ 2 d\omega)^{1/2}=\sqrt{-E}. \] This result is a consequence of more general ones, giving the precise lower bounds in certain directions for a solution of the Schrödinger equation in an open cone \(\Omega \subset {\mathbb{R}}^ n\).