Function spaces associated with Schrödinger operators: The Pöschl-Teller potential

We address the function space theory associated with the Schrödinger operator H = -d²/dx² + V. The discussion is featured with potential V (x) = -n(n + 1) sech²x, which is called in quantum physics Pöschl-Teller potential. Using a dyadic system, we introduce Triebel-Lizorkin spaces and Besov 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 ψ(t, x) = e^-itHf(x) admits appropriate time decay in the Besov space scale.