Resolvent estimates in amalgam spaces and asymptotic expansions for Schrödinger equations

Abstract

We consider Schrödinger equations i∂_tu = (−Δ + V)u in ℝ³ with a real potential V such that, for an integer k ≥ 0, ⟨x⟩^kV(x) belongs to an amalgam space ℓ^p(L^q) for some 1 ≤ p < 3/2 < q ≤ ∞, where ⟨x⟩ = (1 + |x|²)^1/2. Let H = −Δ + V and let P_ac be the projector onto the absolutely continuous subspace of L²(ℝ³) for H. Assuming that zero is not an eigenvalue nor a resonance of H, we show that solutions u(t) = exp(−itH)P_acφ admit asymptotic expansions as t → ∞ of the form
$$\bigg\| \langle x \rangle^{-k-\varepsilon} \bigg( u(t)- \sum_{j=0}^{[k/2]}t^{-\frac32-j}P_j \varphi \bigg) \bigg\|_{\infty}\leq C |t|^{-\frac{k+3+\varepsilon}2} \big\| \langle x \rangle^{k+\varepsilon}\varphi \big\|_1$$
for 0 < ε < 3 (1/p − 2/3), where P₀, …, P_[k/2] are operators of finite rank and [k/2] is the integral part of k/2. The proof is based upon estimates of boundary values on the reals of the resolvent (−Δ − λ²)⁻¹ as an operator-valued function between certain weighted amalgam spaces.