Function Spaces Associated with Schrödinger Operators: The Pöschl-Teller Potential

Abstract

We address the function space theory associated with the Schrödinger operator H = −d 2 /dx 2 + V . The discussion is featured with potential V (x) = −n(n + 1) sech 2 x, which is called in quantum physics Pöschl-Teller potential. Using a dyadic system, we introduce TriebelLizorkin 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 −itH f (x) admits appropriate time decay in the Besov space scale.

On the Wave Equation Associated to the Hermite and the Twisted Laplacian

“…This kind of estimate is quite standard; we sketch a proof for the sake of completeness. It is well known ( [3], [14]) that the integral kernel of φ 0 (λ…”

“…Thus in particular the distorted homogeneous Besov norm coincides with the non-homogeneous one B [14], [3], [2]. A natural question is to compare such distorted norms with the standard ones (1.10); we shall not pursue this problem here.…”

Time decay for Schrödinger equation with rough potentials

“…The proof of the main theorem is a careful modification of that of the one dimensional result for a special potential in [6].…”

Littlewood-Paley theorem for Schrödinger operators

“…A special example is the one dimensional Pöschl-Teller model V (x) = −ν(ν + 1) sech 2 x, ν ∈ N, cf. [14]. We will discuss the problem in more detail in [23] where V is assumed to have only polynomial decay at infinity.…”

“…Hardy-Littlewood maximal function by a standard argument; see [20] or [9,14] for some simple details.…”