On discreteness of spectrum of Schro"dinger operator with bounded potential

Let’s consider a complete noncompact Riemannian manifold without boundary which is representable as ∪ 𝐷, where is a compact set and is isometric to the product R0 × S1 × S2 × · · · × S𝑘 (где R0 = (𝑟0,+∞), а S𝑖 are compact Riemannian manifolds without boundary) with metric 𝑑𝑠2 = 𝑑𝑟2 + 𝑞2 1(𝑟)𝑑 21 + · · · + 𝑞2 𝑘(𝑟)𝑑 2 𝑘, where 2 is the metric on S𝑖 and 𝑞𝑖(𝑟) is a smooth positive function on R0. We assume dim S𝑖 = and denote 𝑠(𝑟) = 𝑞𝑛1 1 (𝑟) · · · 𝑘 (𝑟). The manifold is called a manifold with end. Since its end is a simple warped product, is the simplest case of a quasimodel manifold. On the manifold we study the Laplace-Beltrami operator -Δ = -div∇ and the Schr¨odinger operator -Δ = -div∇ + 𝑐(𝑟, ). We denote 𝐹(𝑟) = (︂𝑠′(𝑟) 2𝑠(𝑟))︂′ + (︂𝑠′(𝑟) 2𝑠(𝑟))︂2. Theorem 1. Let’s 𝑐(𝑟, ) ≥ 0. The spectrum of the Schr¨odinger operator on the manifold is discrete if one of the following conditions is satisfied: (𝐷) 0 and lim 𝑟→∞ (𝐵(𝑟)) cap𝐵(𝑟) = 0. We can note that the conditions of the theorem 1 are not just sufficient, but necessary for discreteness of the Laplacian spectrum. Theorem 2. If there is a function ˜𝑐(𝑟) on manifold such that 𝑐(𝑟, ) ≥ ≥ ˜𝑐(𝑟) and ˜𝑐(𝑟) + 𝐹(𝑟) > -𝐶 (𝐶 = const > 0), then the spectrum of the Schr¨odinger operator on the manifold is discrete if ∀𝜔 > 0 lim 𝑟→∞ 𝑟+𝜔 w (˜𝑐(𝑟) + 𝐹(𝑟)) = +∞.