Function spaces associated with Schroedinger operators: the Poeschl-Teller potential

We address the function space theory associated with the Schroedinger operator H. The discussion is featured with the Poeschl-Teller potential in quantum physics. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces (including Sobolev spaces) associated with H. We then use interpolation method to identify these spaces with the classical ones for a certain range of p,q> 1. A physical implication is that the corresponding wave function $\psi (t,x)=e^{-itH}f(x)$ admits appropriate time decay in the Besov space scale.