Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I

Abstract

Abstract. Let Ω ⊂ R N be a compact imbedded Riemannian manifold of dimension d ≥ 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)ω) : x ∈ R, ω ∈ Ω} with r > 0 and smooth, and the natural metric ds 2 = (1 + r (x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends:

Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II

“…In Part I we proved that, see Lemma 2.2 and Corollary 2.3 of [21], It has a smooth potential V with the following asymptotic behavior.…”

“…The reader should compare W ν in (2.10) with the Wronskian for n = 0, d = 1 derived in [21]: The reader should compare W ν in (2.10) with the Wronskian for n = 0, d = 1 derived in [21]:…”

“…Our goal is to develop the scattering theory of the following class of operators (ν = 0 is treated in Part I, see [21]): Definition 3.1. Our goal is to develop the scattering theory of the following class of operators (ν = 0 is treated in Part I, see [21]): Definition 3.1.…”

“…In our previous paper [21] we dealt with the case d = 1, n = 0 and proved (1.2) and (1.3) for that case. In other words, −∆ Ω Y n = µ 2 n Y n where 0 = µ 2 0 < µ 2 1 ≤ µ 2 2 ≤ .…”

Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

“…First, smooth potentials which are inverse square at least one end arise in several contexts in physics and geometry, for example in general relativity in connection with Schwarzschild and de-Sitter spaces, see [6]. Second, this paper is part of the program initiated in [22,23]. In fact, the analysis carried out here is an essential part in the solution of the "large angular momentum" problem from [23].…”