L^p norms of non-critical Schrödinger semigroups

Abstract

We consider Schrödinger semigroups e^(−tH), H = −Δ + V on R^n with V ~ −c❘x❘^(−2), as ❘x❘ → ∞, 0 < c < [(12)(n−2)]^2, with H ⩾ 0. We determine the exact power law divergence of ∥e^(−tH)∥_(p,p) and of some ∥e^(−tH)∥_(q,p) as maps from L^p to L^q. The results are expressed most naturally in terms of the power α for which there exists a positive resonance η such that Hη = 0, η(x) ~ ❘x❘^(−α).

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