A Jacobi Algorithm in Phase Space: Diagonalizing (skew-) Hamiltonian and Symplectic Matrices with Dirac-Majorana Matrices

Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.

Discrete orthogonal ensemble on the exponential lattices

Inspired by Aomoto's $q$-Selberg integral, the orthogonal ensemble in the exponential lattice is considered in this paper. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus, corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. Examples including Al-Salam & Carlitz, $q$-Laguerre, little $q$-Jacobi and big $q$-Jacobi cases are considered.

The spectral geometry of the Riemann curvature tensor

Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. It deals with results for the Jacobi operator, the higher order Jacobi operator, the Szabo operator, and the skew-symmetric curvature operator. Some results in the complex setting are presented as well for the skew-symmetric curvature operator.

Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials

We show that skew-orthogonal functions, defined with respect to Jacobi weight $w_{a,b}(x)={(1-x)}^a{(1+x)}^b$, $a$, $b>-1$, including the limiting cases of Laguerre ($w_{a}(x)=x^{a}e^{-x}$, $a > -1$) and Gaussian weight ($w(x)=e^{-x^2}$), satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formulæ for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random $2N\times 2N$ matrices. Using the GCD formulæwe calculate the level-densities and prove that in the bulk of the spectrum, …

Exchange operator formalism for N-body spin models with near-neighbors interactions

We present a detailed analysis of the spin models with near-neighbors interactions constructed in our previous paper [Phys. Lett. B 605 (2005) 214] by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.

准周期 Jacobi 块矩阵的非微扰局域化

Non-perturbative localization for quasi-periodic Jacobi block matrices

本文证明了在所有 Lyapunov 指数非零且基础动力学为 Diophantine 旋转的假设下，准周期 Jacobi 块矩阵算子的非微扰 Anderson 局域化，并探讨了其在 $b=1$ 时的算术局域化及相关物理模型中的应用。