4,372 research outputs found

### Tridiagonal realization of the anti-symmetric Gaussian $\beta$-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary
ensemble gives an anti-symmetric tridiagonal matrix with all independent
elements. The random variables permit the introduction of a positive parameter
$\beta$, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
$\{q_i\}$, the first components of the eigenvectors. Three proofs are given.
One involves an inductive construction based on bordering of a family of random
matrices which are shown to have the same distributions as the anti-symmetric
tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg
integral theory. A second proof involves the explicit computation of the
Jacobian for the change of variables between real anti-symmetric tridiagonal
matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the
anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples
of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some
simple properties of the shooting eigenvector and associated Pr\"ufer phases of
the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal
transformation proof for both cases (Method III

### Exact calculation of the ground state single-particle Green's function for the $1/r^2$ quantum many body system at integer coupling

The ground state single particle Green's function describing hole propagation
is calculated exactly for the $1/r^2$ quantum many body system at integer
coupling. The result is in agreement with a recent conjecture of Haldane.Comment: Late

### Normalization of the wavwfunction for the Calogero-Sutherland model with internal degrees of freedom

The exact normalization of a multicomponent generalization of the ground
state wavefunction of the Calogero-Sutherland model is conjectured. This result
is obtained from a conjectured generalization of Selberg's $N$-dimensional
extension of the Euler beta integral, written as a trigonometric integral. A
new proof of the Selberg integral is given, and the method is used to provide a
proof of the mulicomponent generalization in a special two-component case.Comment: 16 pgs, latex, no figures, submitted Int. J. of Mod. Phys.

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