9,006 research outputs found
On algebraic classification of quasi-exactly solvable matrix models
We suggest a generalization of the Lie algebraic approach for constructing
quasi-exactly solvable one-dimensional Schroedinger equations which is due to
Shifman and Turbiner in order to include into consideration matrix models. This
generalization is based on representations of Lie algebras by first-order
matrix differential operators. We have classified inequivalent representations
of the Lie algebras of the dimension up to three by first-order matrix
differential operators in one variable. Next we describe invariant
finite-dimensional subspaces of the representation spaces of the one-,
two-dimensional Lie algebras and of the algebra sl(2,R). These results enable
constructing multi-parameter families of first- and second-order quasi-exactly
solvable models. In particular, we have obtained two classes of quasi-exactly
solvable matrix Schroedinger equations.Comment: LaTeX-file, 16 pages, submitted to J.Phys.A: Math.Ge
A Novel Multi-parameter Family of Quantum Systems with Partially Broken N-fold Supersymmetry
We develop a systematic algorithm for constructing an N-fold supersymmetric
system from a given vector space invariant under one of the supercharges.
Applying this algorithm to spaces of monomials, we construct a new
multi-parameter family of N-fold supersymmetric models, which shall be referred
to as "type C". We investigate various aspects of these type C models in
detail. It turns out that in certain cases these systems exhibit a novel
phenomenon, namely, partial breaking of N-fold supersymmetry.Comment: RevTeX 4, 28 pages, no figure
Quasi-exactly Solvable Lie Superalgebras of Differential Operators
In this paper, we study Lie superalgebras of matrix-valued
first-order differential operators on the complex line. We first completely
classify all such superalgebras of finite dimension. Among the
finite-dimensional superalgebras whose odd subspace is nontrivial, we find
those admitting a finite-dimensional invariant module of smooth vector-valued
functions, and classify all the resulting finite-dimensional modules. The
latter Lie superalgebras and their modules are the building blocks in the
construction of QES quantum mechanical models for spin 1/2 particles in one
dimension.Comment: LaTeX2e using the amstex and amssymb packages, 24 page
On the families of orthogonal polynomials associated to the Razavy potential
We show that there are two different families of (weakly) orthogonal
polynomials associated to the quasi-exactly solvable Razavy potential V(x)=(\z
\cosh 2x-M)^2 (\z>0, ). One of these families encompasses the
four sets of orthogonal polynomials recently found by Khare and Mandal, while
the other one is new. These results are extended to the related periodic
potential U(x)=-(\z \cos 2x -M)^2, for which we also construct two different
families of weakly orthogonal polynomials. We prove that either of these two
families yields the ground state (when is odd) and the lowest lying gaps in
the energy spectrum of the latter periodic potential up to and including the
gap and having the same parity as . Moreover, we show
that the algebraic eigenfunctions obtained in this way are the well-known
finite solutions of the Whittaker--Hill (or Hill's three-term) periodic
differential equation. Thus, the foregoing results provide a Lie-algebraic
justification of the fact that the Whittaker--Hill equation (unlike, for
instance, Mathieu's equation) admits finite solutions.Comment: Typeset in LaTeX2e using amsmath, amssymb, epic, epsfig, float (24
pages, 1 figure
noise and integrable systems
An innovative test for detecting quantum chaos based on the analysis of the
spectral fluctuations regarded as a time series has been recently proposed.
According to this test, the fluctuations of a fully chaotic system should
exhibit 1/f noise, whereas for an integrable system this noise should obey the
1/f^2 power law. In this letter, we show that there is a family of well-known
integrable systems, namely spin chains of Haldane-Shastry type, whose spectral
fluctuations decay instead as 1/f^4. We present a simple theoretical
justification of this fact, and propose an alternative characterization of
quantum chaos versus integrability formulated directly in terms of the power
spectrum of the spacings of the unfolded spectrum.Comment: 5 pages, 3 figures, RevTe
A New Algebraization of the Lame Equation
We develop a new way of writing the Lame Hamiltonian in Lie-algebraic form.
This yields, in a natural way, an explicit formula for both the Lame
polynomials and the classical non-meromorphic Lame functions in terms of
Chebyshev polynomials and of a certain family of weakly orthogonal polynomialsComment: Latex2e with AMS-LaTeX and cite packages; 32 page
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