We calculate the autocorrelation function for the characteristic polynomial
of a random matrix in the microscopic scaling regime. While results fitting
this description have be proved before, we will cover all values of inverse
temperature $\beta \in (0,\infty)$. The method also differs from prior work,
relying on matrix models introduced by Killip and Nenciu

Killip, Rowan

Ryckman, Eric

arXiv.org e-Print Archive

CiteSeerX

English

Autocorrelations of the characteristic polynomial of a random matrix
  under microscopic scaling

We describe an ensemble of (sparse) random matrices whose eigenvalues follow
the Gibbs distribution for n particles of the Coulomb gas on the unit circle at
inverse temperature beta. Our approach combines elements from the theory of
orthogonal polynomials on the unit circle with ideas from recent work of
Dumitriu and Edelman. In particular, we resolve a question left open by them:
find a tri-diagonal model for the Jacobi ensemble.Comment: 28 page

Killip, R.

Nenciu, I.

Matrix models for circular ensembles

We construct $C^\infty$ solutions to the one-dimensional nonlinear wave
equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with}
\quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\infty$
hypersurface. As a corollary, we show that smooth solutions can blow up (at the
first instant) on an arbitrary compact set.
  We also construct solutions that blow up on general space-like $C^k$
hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$

Visan, Monica

Smooth solutions to the nonlinear wave equation can blow up on Cantor
  sets

We discuss the proof of and systematic application of Case's sum rules for
Jacobi matrices. Of special interest is a linear combination of two of his sum
rules which has strictly positive terms. Among our results are a complete
classification of the spectral measures of all Jacobi matrices J for which
J-J_0 is Hilbert--Schmidt, and a proof of Nevai's conjecture that the Szego
condition holds if J-J_0 is trace class.Comment: 69 pages, published versio

Simon, Barry

Caltech Authors

Absolutely continuous spectrum for one-dimensional Schro¨dinger operators with slowly decaying potentials: Some optimal results,

Absolutely continuous spectrum of Jacobi matrices,

Absolutely continuous spectrum of one-dimensional Schro¨dinger operators and Jacobi matrices with slowly decreasing potentials,

An Introduction to Orthogonal Polynomials, Mathematics and its Applications,

An upper bound on the number of eigenvalues of an infinite-dimensional Jacobi matrix,

Analysis with weak trace ideals and the number of bound states of Schro¨dinger operators,

Analytic properties of the scattering matrix,

Analytic Theory of Continued Fractions,

Assche, Orthogonal polynomials with asymptotically periodic recurrence coefficients,

Beitra¨ge zue Theorie der Toeplitzschen Formen,

Bound for the kinetic energy of fermions which proves the stability of matter,

Bounded Analytic Functions, Pure and Applied Mathematics 96,

Distribution of poles for scattering on the real line,

Expansions in eigenfunctions of selfadjoint operators, Translated from the Russian by

First KdV integrals and absolutely continuous spectrum for 1-D Schro¨dinger operator,

Geronimus, Orthogonal polynomials: Estimates, Asymptotic Formulas,

Handbook of Mathematical Functions with Formulas, Graphs,

Ideals and Their Applications, London Mathematical Society Lecture Note Series 35,

Imbedded singular continuous spectrum for Schro¨dinger operators,

Inequalities for the moments of the eigenvalues of the Schro¨dinger Hamiltonian and their relation to Sobolev inequalities,

Introduction to the Theory of Linear Nonselfadjoint Operators,

it is a compact perturbation of the free

Jacobi matrices with decaying potential and dense point spectrum,

Jacobi Operators and Completely Integrable Nonlinear Lattices,

Lieb-Thirring inequalities for Jacobi matrices,

Linear Transformations in Hilbert Spaces and Their Applications to Analysis,

m-functions and inverse spectral analysis for finite and semiinfinite Jacobi matrices,

On a generalization of some investigations of

on infinite determinants of Hilbert space operators,

On information and sufficiency,

On the absolutely continuous spectrum of one-dimensional Schro¨-dinger operators with square summable potentials,

On the bound state of Schro¨dinger operators in one dimension,

On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potentials,

On the spectra of infinite-dimensional Jacobi matrices,

On the Toda lattice.

Orthogonal polynomials and measures with finitely many point masses,

Orthogonal polynomials defined by a recurrence relation,

Orthogonal polynomials from the viewpoint of scattering theory,

Orthogonal polynomials whose distribution functions have finite point spectra,

Orthogonal Polynomials,

Perturbations of one-dimensional Schro¨dinger operators preserving the absolutely continuous spectrum,

Polynomials, Fourth edition,

polynomials, recurrences, Jacobi matrices, and measures,

Quantum Entropy and Its Use, Texts and Monographs in Physics,

Real and Complex Analysis, Third edition,

Recherches sur les fractions continues,

Scattering theory, orthogonal polynomials, and q-series,

Stationary Sequences in Hilbert ’s Space (Russian), Bolletin Moskovskogo Gosudarstvenogo Universiteta.

Statistical mechanics of quantum spin systems.

Sur les polynomes de Tchebicheff,

Szego˝’s extremum problem on the unit circle,

The absolutely continuous spectrum of one-dimensional Schro¨dinger operators with decaying potentials,

The bound state of weakly coupled Schrod¨inger operators in one and two dimensions,

The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians,

The classical moment problem as a self-adjoint finite difference operator,

The Statistical Mechanics of Lattice Gases. Vol . I, Princeton Series in Physics,