324 research outputs found
Quantum mechanical study of molecules - Eigenvalues and eigenvectors of real symmetric matrices
Computer methods for calculating eigenvalue and eigenvectors of real symmetric matrices arising in problems of molecular quantum mechanic
Tridiagonal realization of the anti-symmetric Gaussian -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
, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
, 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 . The third proof maps matrices from the
anti-symmetric Gaussian -ensemble to those realizing particular examples
of the Laguerre -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
Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations
International audienceFirst, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation
A family of tridiagonal pairs and related symmetric functions
A family of tridiagonal pairs which appear in the context of quantum
integrable systems is studied in details. The corresponding eigenvalue
sequences, eigenspaces and the block tridiagonal structure of their matrix
realizations with respect the dual eigenbasis are described. The overlap
functions between the two dual basis are shown to satisfy a coupled system of
recurrence relations and a set of discrete second-order difference
equations which generalize the ones associated with the Askey-Wilson orthogonal
polynomials with a discrete argument. Normalizing the fundamental solution to
unity, the hierarchy of solutions are rational functions of one discrete
argument, explicitly derived in some simplest examples. The weight function
which ensures the orthogonality of the system of rational functions defined on
a discrete real support is given.Comment: 17 pages; LaTeX file with amssymb. v2: few minor changes, to appear
in J.Phys.A; v3: Minor misprints, eq. (48) and orthogonality condition
corrected compared to published versio
Explicit inverse of nonsingular Jacobi matrices
We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm–Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provide the entries of the inverse matrixPeer ReviewedPostprint (author's final draft
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