Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

Spectral analysis of tridiagonal Fibonacci Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the spectrum is a Cantor set of zero Lebesgue measure, and discuss its fractal structure and Hausdorff dimension. We also extend some known results on the diagonal and the off-diagonal Fibonacci Hamiltonians.

Spectral distributions and isospectral sets of tridiagonal matrices

We analyze the correspondence between finite sequences of finitely supported probability distributions and finite-dimensional, real, symmetric, tridiagonal matrices. In particular, we give an intrinsic description of the topology induced on sequences of distributions by the usual Euclidean structure on matrices. Our results provide an analytical tool with which to study ensembles of tridiagonal matrices, important in certain inverse problems and integrable systems. As an application, we prove that the Euler characteristic of any generic isospectral set of symmetric, tridiagonal matrices is a t…

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, alth…

三对角Toeplitz矩阵的对角对称化

Diagonal symmetrisation of tridiagonal Toeplitz matrices

本文提出了一种将三对角Toeplitz矩阵对角相似变换为对称Toeplitz矩阵的方法，并给出了该矩阵的特征值、行列式和逆矩阵的显式表达式，同时探讨了其在极值特征值和条件数方面的应用。

一类非自伴带状矩阵的正谱问题与逆谱问题

Direct and inverse spectral problems for a class of non-selfadjoint band matrices

本文研究一类非自伴带状矩阵的正谱与逆谱问题，提出了从给定谱数据重构矩阵的充要条件，并推广了周期Jacobi矩阵等相关结论。

Mixed spectral regimes for square Fibonacci Hamiltonians

For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states measure (i.e. one whose absolutely continuous and singular continuous components are both nonzero). Using the methods developed in this paper, we also show existence of parameter regimes for the square continuum Fibonacci Schrodinger operator yielding mixed interval-Cantor spectra. These examples provide the first explicit examples of an interesting phenomenon…