On Krein's Example

On a generalisation of Krein's example

We generalise a classical example given by Krein in 1953. We compute the difference of the resolvents and the difference of the spectral projections explicitly. We further give a full description of the unitary invariants, i.e., of the spectrum and the multiplicity. Moreover, we observe a link between the difference of the spectral projections and Hankel operators.

The Krein spectral shift and rank one perturbations of spectra

We use recent results on the boundary behavior of Cauchy integrals to study the Krein spectral shift of a rank one perturbation problem for self-adjoint operators. As an application, we prove that all self-adjoint rank one perturbations of a self-adjoint operator are pure point if and only if the spectrum of the operator is countable. We also study pairs of pure point operators unitarily equivalent up to a rank one perturbation and give various examples of rank one perturbations of singular spectra.

An Example of J-unitary Operator. Solving a Problem Stated by M.G. Krein

Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal strictly positive and strictly negative \hat{V}^{\pm 1}-{invariant subspaces} L_{+}, L_{-}, such that they are mutually J-orthogonal and L_{+} + L_{-} is dense in the space. (d_1) if L_1 is non-zero \hat{V}-invariant subspace, then r(\hat{V}|L_1)=r(\hat{V}) (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, then r(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .

反谱问题练习曲

Etudes for the inverse spectral problem

本文研究半直线上典型 Hamiltonian 系统的反谱问题，通过引入截断 Toeplitz 算子扩展了经典结果，并用实例说明了求解算法与公式。

Krein-Feller算子的谱维数与$L^{q}$-谱

Spectral dimensions of Krein--Feller operators and $L^{q}$-spectra

本文研究了区间$(0,1)$上有限Borel测度的Krein-Feller算子谱维数，证明上谱维数对应$L^{q}$-谱不动点，给出谱维数存在充分条件并构造了反例。

高维情形下Krein-Feller算子的谱维数

Spectral dimensions of Krein-Feller operators in higher dimensions

该研究引入单调二进谱集函数推广$L^q$-谱至高维，刻画了高维Kreĭn-Feller算子的Dirichlet与Neumann谱维数的存在性、相等条件及临界阈值行为，并给出多类测度的应用与特征值界。