We will show that for any periodic potential, the spectrum contains at most finitely many gaps. Furthermore, for small enough periodic potential, the spectrum contains no gaps at all. 

This talk is based on a 1982 paper "A remark on two dimensional periodic potentials" by Dahlberg and Trubowitz.

My goal is to prove Agmon theorem in two talks. In the first talk, I will use the limiting absorption principle for the free Laplacian to prove Agmon theorem. Next Friday, Lili Yan will present the limiting absorption principle for free Laplacian.

 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. 

 I will continue  the last Seminar to present  an elementary   formula about the average expansion of certain products of  cocycles,  which allows us to reobtained some known results about Lyapunov exponent.  Those  results are  based on  a paper of  A.Avila and J.Bochi -A formula with some applications to the theory of Lyapunov exponent.