Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity

The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix T-n(a) as n goes to infinity, with a continuous and real-valued symbol a having a power singularity of degree gamma with 1 < gamma < 2, at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple-loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues lambda(j)(T-n(a)) with j ? epsilon n, where the eigenvalues are arranged in nondecreasing order and epsilon is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case j < epsilon n. (iii) We numerically show that the main term of the asymptotics for eigenvalues with j < epsilon n, formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter.