Lemma 21 ([Uniform exterior form convergence]). Let V=V_ c +V_ b be as in Lemma 19 ([Compactness under escaping translations]). Suppose every bounded-potential limit along escaping translates is a fixed constant c_∞ . For every ε>0 there …

Fix a ball B_L . We first claim _ u∈H^1_0(B_L); \|u\|_ H^1(B_L) ≤1 |∫_ B_L (V(y+x)-c_∞)|u(x)|^2\,dx| ⟶0 (|y|→∞). Sobolev's inequality and the vanishing local L^ 3/2 norm prove the claim for the critical part. If the bounded part failed, choose escaping centers and tests violating it. Extract a weak-star subsequence of the bounded multipliers and a subsequence of the tests converging strongly in L^2(B_L) . Their squares converge in L^1 , because \||u_j|^2-|u|^2\|_1 ≤(\|u_j\|_2+\|u\|_2)\|u_j-u\|_2. Uniform boundedness of the multipliers handles this changing test, and their weak-star limit is c_∞ . This contradicts the selected violation and proves. Choose once and for all a smooth, locally finite quadratic partition ∑_ℓχ_ℓ^2=1 with uniformly bounded overlap, each χ_ℓ supported in a ball of one fixed radius, and ∑_ℓ|∇χ_ℓ|^2≤K . For any prescribed η>0 , applies with error η to all these balls whose centers are sufficiently distant. If u is supported sufficiently far out, every nonzero χ_ℓu has such a center. Summing the local potential estimates gives |∫(V-c_∞)|u|^2| , ≤η∑_ℓ\|χ_ℓu\|_ H^1 ^2; , ≤η\|∇u\|_2^2+η(1+K)\|u\|_2^2. Here the cross terms cancel because ∑_ℓχ_ℓ∇χ_ℓ=0 . Choosing η≤ε/(1+K) proves . This argument uses a uniform local form estimate, not a pointwise limit of V .