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.
We consider the blowup rate for blowup solutions to L 2 -critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "log-log law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.
Let H be a Schrödinger operator on R n . Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.
Abstract. Let H = − + V be a Schrödinger operator on the real line, where V = ε 2 χ [−1,1] . We define the Besov spaces for H by developing the associated Littlewood-Paley theory. This theory depends on the decay estimates of the spectral operator ϕ j (H) for the high and low energies. We also prove a Mihlin-Hörmander type multiplier theorem on these spaces, including the L p boundedness result. Our approach has potential applications to other Schrödinger operators with short-range potentials, as well as in higher dimensions.
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