CAS Seminar

Anton Kutsenko

(Jacobs University)

"Sharp spectral estimates for periodic Jacobi matrices"

Date: Wed, March 20, 2019
Time: 15:45
Place: Research I Meeting Room

Abstract: A part of approximation theory, the theory of infinite 3-diagonal Jacobi matrices, is rich and simultaneously deep. On the one hand, Jacobi matrices correspond to the discrete approximation of 1D wave equations and weighted Schrodinger equations. On the other hand, the Jacobi matrices are related to the orthogonal polynomials on various subsets of the real line. It is well-known that the spectrum of self-adjoint \(p\)-periodic scalar Jacobi matrices consists of \(p\) continuous bands separated by \(p-1\) gaps. If the Jacobi matrix is homogeneous then all the gaps are closed. The inhomogeneity of the Jacobi matrix, or of the coefficients of the corresponding wave equation, allows the gaps to be open. We give sharp estimates of the width of gaps and bands in terms of the elements of Jacobi matrix.

The first results about estimates of spectral bands appear in arXiv:1007.5412 in 2010. While the estimates improve some well-known spectral estimates, it was very hard to publish it somewhere. Only in 2015, it was published in a conference proceedings. I forgot about the result for a long time until, recently, I found that the bands estimates can be useful for refining some of the gap estimates obtained by L. Golinski in his nice paper arXiv:1704.03679 accepted in JST since 2017. Now, the refined estimates are published here.