The asymptotic of eigenvalues for difference operator with growing potential

We consider : 𝐷(𝐴) ⊂ 𝑙2(Z) → 𝑙2(Z), (𝐴𝑥)(𝑛) = 𝑎(𝑛)𝑥(𝑛), ∈ Z, ∈ 𝐷(𝐴), and (𝐵𝑥)(𝑛) = -2𝑥(𝑛) + 𝑥(𝑛 - 1) + 𝑥(𝑛 + 1). Let : Z → C be a sequence with property: 1) 𝑎(𝑖) ̸= 𝑎(𝑗), 𝑖 ̸= 𝑗; 2) lim |𝑛|→∞ |𝑎(𝑛)| = ∞; 3) 0 0, such that the spectrum (𝒜) of operator has form (𝒜) = (𝑘)⋃︁(︂ ⋃︁|𝑖|>𝑘 𝑖)︂, where (𝑘) is a finite set with number of points not exceeding 2𝑘 + 1 and = = { 𝑖}, |𝑖| > 𝑘, are singleton sets. The asymptotic formulas of eigenvalues have the following form: = 𝑎(𝑖) + 2 + 𝑂(𝑑-1 ), = 𝑎(𝑖) + 2 - 𝑎(𝑖 + 1) - 2𝑎(𝑖) + 𝑎(𝑖 - 1) (𝑎(𝑖 + 1) - 𝑎(𝑖))(𝑎(𝑖 - 1) - 𝑎(𝑖)) + 𝑂(𝑑-3 ), |𝑖| > 𝑘. Theorem 2. Let the sequence : Z → C satisfies the condition Re 𝑎(𝑛) 6 6 for all ∈ Z and a ∈ R. Then the operator is the generator of the semigroup operators : R+ → End 𝑙2(Z) and this semigroup is similar to ̃︀𝑇 : R+ → End 𝑙2(Z) type ̃︀𝑇(𝑡) = ̃︀𝑇(𝑘)(𝑡) ⊕ ̃︀𝑇(𝑘)(𝑡), ∈ R+, acting in 𝑙2(Z) = ℋ(𝑘) ⊕ℋ(𝑘), where ℋ(𝑘) = Im𝑄𝑘 and ℋ(𝑘) = Im (𝐼 -𝑄𝑘). The semigroup ̃︀𝑇(𝑘) : R+ → Endℋ(𝑘) determined by the formula ̃︀𝑇(𝑘)(𝑡)𝑥 = Σ︁|𝑛|>𝑘 𝑛𝑡𝑃𝑛𝑥, ∈ ℋ(𝑘), ∈ R+, where the numbers 𝑛, |𝑛| > 𝑘, are defined by Theorem 1. Theorem 3. Let 6 Re 𝑎(𝑛) 6, ∈ R, for every ∈ Z. Then the operator : 𝐷(𝒜) ⊂ 𝑙2(Z) → 𝑙2(Z) is generator of group : R → End 𝑙2(Z). This group is similar to ̃︀𝑇 : R → End 𝑙2(Z), where ̃︀𝑇(𝑡) = ̃︀𝑇(𝑘)(𝑡)⊕ ̃︀𝑇(𝑘)(𝑡), ∈ R and ̃︀𝑇(𝑘)(𝑡)𝑥 = Σ︁|𝑛|>𝑘 𝑛𝑡𝑃𝑛𝑥, ∈ ℋ(𝑘), ∈ R. Theorem 4. Let the operator : 𝐷(𝒜) ⊂ 𝑙2(Z) → 𝑙2(Z) be self-adjoint. Then is a generator of isometric group : R → End 𝑙2(Z). This group is similar to ̃︀𝑇(𝑡) = ̃︀𝑇(𝑘)(𝑡) ⊕ ̃︀𝑇(𝑘)(𝑡), ∈ R. and ̃︀𝑇(𝑘)(𝑡)𝑥 = Σ︁|𝑛|>𝑘 𝑛𝑡𝑃𝑛𝑥, ∈ ℋ(𝑘), ∈ R.