On a Spectral Property of Jacobi Matrices
نویسنده
چکیده
Let J be a Jacobi matrix with elements bk on the main diagonal and elements ak on the auxiliary ones. We suppose that J is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of J coincides with [−2, 2], and its discrete spectrum is a union of two sequences {xj }, x + j > 2, x − j < −2, tending to ±2. We denote sequences {ak+1 − ak} and {ak+1 + ak−1 − 2ak} by ∂a and ∂2a, respectively. The main result of the note is the following theorem. Theorem. Let J be a Jacobi matrix described above and σ be its spectral measure. Then a− 1, b ∈ l4, ∂2a, ∂2b ∈ l2 if and only if i) ∫ 2 −2 log σ′(x)(4 − x) dx > −∞, ii) ∑
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