Lp-analyticity of Schrödinger semigroups on Riemannian manifolds

We address the problems of extrapolation, analyticity, and Lp-spectral independence for C0-semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lp-analyticity of angle π 2 of Schrödinger semigroups on Riemannian manifolds with Ricci curvature bounded below, under the condition of form smallness of the negative part of the potential. MSC 2000: 47D06, 47N20, 35J10 The final aim of the present paper is the following result. Let M be a Riemannian manifold with Ricci curvature bounded below. Then a Schrödinger semigroup with form small negative part of the potential is analytic of angle π2 on Lp(M), for p from a certain subinterval of [1,∞) that contains 2 and is determined by the form bound (see Theorem 7 and Remark 8(a) below). First we put the problem in a more abstract context. Let (M,μ) be a measure space, Ω ⊆ M a measurable subset, s ∈ [1,∞), and Ts = ( Ts(t); t > 0 ) a C0-semigroup on Ls(Ω). Assume that ||Ts(t) Ls∩Lq : Lq(Ω) → Lq(Ω)|| 6 Me ωt for all t > 0, for some s 6= q ∈ [1,∞). Then the operators Ts(t) Ls∩Lq extend to bounded operators Tq(t), which define a semigroup on Lq(Ω). We say that Ts extrapolates to the semigroup Tq on Lq(Ω). By Riesz-Thorin interpolation we obtain a family of C0-semigroups Tp on Lp(Ω), with p between s and q. The semigroups Tp are consistent in the sense that Tp1(t) Lp1∩Lp2 = Tp2(t) Lp1∩Lp2 for all t > 0 and all p1, p2 between s and q. If the initial semigroup Ts is known to be analytic, e.g. if s = 2 and Ts is symmetric, then all the semigroups Tp for p strictly between q and s are analytic by Stein interpolation. But in general, the angle of analyticity tends to 0 as p→ q, and Tq is not analytic in general (see, e.g., [Dav89; Thm. 4.3.6], [Voi96] for the case q = 1). We are thus led to the following problem: under which conditions does the semigroup Ts extrapolate to a consistent family of C0-semigroups Tp on Lp(Ω) with p-independent angle of analyticity, for p from some subinterval of [1,∞) containing s? The conditions on both the space M and the semigroup Ts will be formulated in terms of a measurable semi-metric on M . We will also address the problem of p-independence of the spectra of the Lp-generators. The main results will only be stated; for the proofs we refer the reader to [Vog01]. Most of the known results concerning the problem of analyticity are about semigroups acting on the whole Lp-scale. Then the question of analyticity in L1 is of particular interest. Starting from [Ama83], there are several specific results on certain classes of elliptic operators on domains of R stating that the semigroup on L1 is analytic, but not giving the optimal angle ([Kat86], [CaVe88], [ArBa93], only to mention a few). E.-M. Ouhabaz was the first to establish analyticity of angle π2 in L1(R N ). In his thesis ([Ouh92a]) he observed that a Gaussian upper bound on the semigroup kernel for