Spectral measures of Jacobi operators with random potentials
نویسندگان
چکیده
Let Hω be a self-adjoint Jacobi operator with a potential sequence {ω(n)}n of independently distributed random variables with continuous probability distributions and let μωφ be the corresponding spectral measure generated by Hω and the vector φ. We consider sets A(ω) which are independent of two consecutive given entries of ω and prove that μωφ(A(ω)) = 0 for almost every ω. This is applied to show equivalence relations between spectral measures for random Jacobi matrices and to study the interplay of the eigenvalues of these matrices and their submatrices. Mathematics Subject Classification(2000): 47B36, 47A25, 39A12.
منابع مشابه
Spectral and Dynamical Properties of Certain Random Jacobi Matrices with Growing Parameters
Abstract. In this paper, a family of random Jacobi matrices, with off-diagonal terms that exhibit power-law growth, is studied. Since the growth of the randomness is slower than that of these terms, it is possible to use methods applied in the study of Schrödinger operators with random decaying potentials. A particular result of the analysis is the existence of operators with arbitrarily fast t...
متن کاملMass Transportation Proofs of Free Functional Inequalities, and Free Poincaré Inequalities
This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put f...
متن کاملSparse Potentials with Fractional Hausdorff Dimension
Abstract. We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimensi...
متن کاملChoquet order for spectra of higher Lamé operators and orthogonal polynomials
We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated w...
متن کاملExtended Jacobi and Laguerre Functions and their Applications
The aim of this paper is to introduce two new extensions of the Jacobi and Laguerre polynomials as the eigenfunctions of two non-classical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive de nite inner product de ned over the compact intervals [-1, 1] and [0,1), respectively and also th...
متن کامل