The Liouville-type theorems for solution of stationary Schro"dinger equation with finite Dirichlet integral

In this article we learn some property of solutions of stationary Shrodinger equation = Δ𝑢 - 𝑐(𝑥)𝑢 = 0, (1) where 𝑐(𝑥) ≥ 0 smooth function, with finite Dirichlet integral w |𝑢|2 + 𝑐(𝑥)𝑢2𝑑𝑥 (2) on non-compact Riemannian manifolds. We prove an analog of Ahlfors’s theorem on existence of non-trivial boundary harmonic function with finite energy integral. Main result of this article is the next theorem. Let be non-compact Riemannian manifold. Theorem 1. If non-trivial solution of equation (1) with finite integral (2) exists on (this solution may be not bounded), then there exists bounded solution of equation (1) with finite energy integral (2). To prove this theorem we use the following lemmas. Lemma 1. (Maximum principle) Let be precompact open set in with smooth boundary. If = 0, ∈ 𝐵, then sup |𝑢| = sup |𝑢|. Lemma 2. Let ⊂ precompact open subset on 𝑀, { 𝑖}∞𝑖 =1 is uniformly bounded on family of solutions (1), ∈ 𝐶2, (𝐵). Then the family { 𝑖}∞𝑖 =1 is compact in class 𝐶2(𝐵′), where 𝐵′ ⊂ 𝐵. Let be set of functions from class 𝐶2(𝐵) with finite Dirichlet integral w |∇𝑦|2 + 𝑐(𝑥)𝑦2𝑑𝑥. Lemma 3. is linear space, also on can be defined dot product as ⟨𝑎, = w (⟨∇𝑎,∇𝑏⟩ + 𝑐(𝑥)𝑎𝑏) 𝑑𝑥, ∀𝑎, ∈ 𝐹. and norm for this dot product as ‖𝑎‖ = ⟨𝑎, 1 2 = (︃w |∇𝑎|2 + 𝑐(𝑥)𝑎2𝑑𝑥)︃1 2. Lemma 4. (Dirichlet principle). Let ⊂ - precompact open subset on with smooth boundary. If for functions 𝑢, ∈ 𝐶2(𝐵) {︂ Δ𝑢 - 𝑐(𝑥)𝑢 = 0, ∈ 𝐵, 𝑢|𝜕𝐵 = 𝑣|𝜕𝐵, then w |∇𝑢|2 + 𝑐(𝑥)𝑢2𝑑𝑥 ≤ w |∇𝑣|2 + 𝑐(𝑥)𝑣2𝑑𝑥.