Some identities for determinants of structured matrices

Perturbation determinants, the spectral shift function, trace identities, and all that

We discuss applications of the M. G. Kre\uın theory of the spectral shift function to the multi-dimensional Schrödinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace identities for the Schrödinger operator are derived.

结构化矩阵行列式渐近性的综述

A survey of asymptotics of determinants for structured matrices

本综述介绍如何利用算子理论方法生成结构化矩阵行列式的渐近性，涵盖有限 Toeplitz 矩阵、Toeplitz 加 Hankel 矩阵及其推广，并首次提出部分新结果。

On the asymptotics of a Toeplitz determinant with singularities

We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hanke…

Operator identities corresponding to inverse problems

The structured operators and corresponding operator identities, which appear in inverse problems for the self-adjoint and skew-self-adjoint Dirac systems with rectangular potentials, are studied in detail. In particular, it is shown that operators with the close to displacement kernels are included in this class. A special case of positive and factorizable operators is dealt with separately.