Disreteness of the spectrum for the Schr"odinger operator and metric transformation on manifold

In this paper we prove the conservation property for the discreteness of the spectrum for the Schr¨odinger operator on the simple warped products of order with the special kind of quasi-isometric transformation of the metric. Let’s consider a complete noncompact Riemannian manifold 𝐷, which is isometric to the product R+ × S1 × S2 × · · · × S𝑘 (где R+ = (0,+∞), 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 R+. We assume dim S𝑖 = and denote 𝑠(𝑟) = 𝑞𝑛1 1 (𝑟) · · · 𝑘 (𝑟). Metric transformation on this manifold is determined by the following matrix (𝑟). ‖ (𝑟)‖ =⃦⃦⃦⃦⃦⃦⃦⃦⃦ 20 (𝑟) 0... 0 0 21 (𝑟)𝐸𝑛1... 0......... 0 0... 2 𝑘(𝑟)𝐸𝑛𝑘 ⃦⃦⃦⃦⃦⃦⃦⃦⃦. The coefficients of this matrix are 𝐶1-smooth, and let’s Σ(𝑟) will stand for its determinant. Actually, we can easily calculate it: Σ(𝑟) = det ‖ (𝑟)‖ = 20 (𝑟) 2𝑛1 1 (𝑟) · · · 2𝑛𝑘 (𝑟). On the manifold we study the Laplace - Beltrami operator -Δ = -div∇ and the Schr¨odinger operator -Δ = -div∇ + 𝑐(𝑟). With the mentioned metric transformation the Laplace - Beltrami operator will change to -̃︀Δ = - 1 √Σ div(√Σ -1∇). Transformed Schr¨odinger operator we write as ̃︀𝐿 = -̃︀Δ+𝑐(𝑟). Also we put 𝐹(𝑟) = 𝑐(𝑟) + (︂𝑠′(𝑟) 2𝑠(𝑟))︂′ + (︂𝑠′(𝑟) 2𝑠(𝑟))︂2, Φ(𝑟) = (︂ ′(𝑟) 2 (𝑟))︂′ + 𝑠′(𝑟) ′(𝑟) 2𝑠(𝑟) (𝑟) + (︂ ′(𝑟) 2 (𝑟))︂2, where (𝑟) = √Σ(𝑟) 0(𝑟). Then we get the following theorem. Theorem. Let’s 𝐹(𝑟) + Φ(𝑟) > -𝐶 (𝐶 = const > 0). The spectrum of the Schr¨odinger operator ̃︀𝐿 on the manifold is discrete if and only if ∀𝜔 > 0 lim 𝑟→∞ 𝑟+𝜔 w (𝐹(𝑟) + Φ(𝑟))𝑑𝑟 = +∞. And next we come to the following corollary. Corollary. If the Schrodinger operator on manifold has discrete spectrum, and we transform the metric of with some diagonal matrix ‖ (𝑟)‖, and Φ(𝑟) > const, then the Schr¨odinger operator ̃︀𝐿 has discrete spectrum too. The same way non-discrete spectrum holds this characteristic.