Let us consider the Schrodinger operators $H$ with decaying potentials $V$ in $\R^d$. For the free Schrodinger operator(i.e.,potential $V=0$ ), there is no positive eigenvalue.   So  it is expected that  the Schrodinger operators keep such property for small potentials. In  this  Seminar, I will prove that $H$ does not have any positive eigenvalue  if $V(x)=\frac{o(1)}{|x|}$ for $d=1$. In the next Seminar, I will prove the result for higher dimension (i.e. $d>1$). This result is based on a classical paper of Kato[Growth Properties of Solutions of the Reduced Wave Equation With a Variable Cofficient]. 

Actually $V(x)=\frac{o(1)}{|x|}$  is optimal by Wigner-von Neumann type potential. Thus $V(x)=\frac{o(1)}{|x|}$ is a spectral transition for eigenvalue.  We can also get a spectral transition for singular continuous spectrum in some sense, which has been done by Agmon. Similar  results hold  for Laplacian on Riemannian manifold (especially for asymptotic flat and hyperbolic cases) which is characterized  by radial curvature or metric sturcture.  In this quarter, I plan to choose some specific topics among them to present.