Uniform individual asymptotics for the eigenvalues and eigenvectors of large Toeplitz matrices

The asymptotic behavior of the spectrum of large Toeplitz matrices has been studied for almost one century now. Among this huge work, we can nd the Szeg\H{o} theorems on the eigenvalue distribution and the asymptotics for the determinants, as well as other theorems about the individual asymptotics for the smallest and largest eigenvalues. Results about uniform individual asymptotics for all the eigenvalues and eigenvectors appeared only ve years ago. The goal of the present lecture is to review this area, to talk about the obtained results. This review is based on joint works with Manuel Bogoya, Albrecht B\ ottcher, and Egor Maximenko

Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. We here use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.Comment: 28 pages, 7 figure

Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-\alpha$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|\alpha|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,\pi]$. The situation changes drastically when $|\alpha|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $\alpha$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.Comment: 22 pages, 4 figure

The inverse scattering transform for weak Wigner-von Neumann type potentials

In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.Comment: To appear in Nonlinearit