Self-adjoint second-order difference equations with unbounded coefficients in the double root case

Spectra of some selfadjoint Jacobi operators in the double root case

“…We wanted to find the largest class of Jacobi operators in the double root case which contain the operators inducted by (n α ) and (ln n). In [12] we described the asymptotic behavior of the generalized eigenvectors of such operators. Here we are interested in their spectral properties.…”

“…The above Jacobi operator induces second order difference equation (1.3) considered in [12]. We will use the asymptotic formulas of the generalized eigenvectors found in [12] to prove mixed nature of the spectrum of the Jacobi operator J(λ n , q n ).…”

“…We will use the asymptotic formulas of the generalized eigenvectors found in [12] to prove mixed nature of the spectrum of the Jacobi operator J(λ n , q n ). In [12, Section 2] the reader will find some properties and analysis of regularity of the sequences (λ n ) and (q n ).…”

“…We wrote the asymptotic formulas in the exponential form instead of the product form, as it is in [12], because it is suitable for us. This notation simplifies some calculations and allows us to use the Euler summation formula.…”

“…Unfortunately, we are in the critical situation and the vectors e ± are linearly dependent, even more they are equal! To verify this see [12,Section 5].…”