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

Sobre el género Spergula L. [Incl. Spergularia (pers.) pers. ex J. Presl &C. Presl, nom. cons.] (Caryohyllaceae) y sus especies en la península ibérica e Islas Baleares

Se enumeran las razones por las que es necesario incluir sin más dilación el género Spergularia (Pers.) Pers. ex J. Presl & C. Presl, nom. cons., en Spergula L., tal como había sido sugerido ya por numerosos autores Se relacionan las especies del género Spergula L., s.l., presentes en la Península Ibérica e Islas Baleares –o mencionadas para dicho territorio–, con su nombre correcto, basiónimo y principales sinónimos. Se proponen las nuevas combinaciones: Spergula sect. Lepigonum (Fr.) G. López, Spergula rupicola (Lebel ex Le Jol.) G. López, Spergula australis (Samp.) G. López, Spergula tangerina (P. Monnier) G. López, Spergula capillacea (Kindb.) G. López, Spergula nicaeensis (Sarato ex Burnat) G. López, Spergula heldreichii (Foucaud) G. López.There are not consistent reasons to maintain Spergularia (Pers.) Pers. ex J. Presl & C. Presl, nom. cons., as an independent genus and it should be included in Spergula L., as already proposed by several botanists. The differential characters between both genera completely break down if considering all the species and particularly the South-American ones. The species of Spergula L., s.l., present or mentioned for the Iberian Peninsula and Balearic Islands are listed, with the correct name, basionym and main synonyms. The new combinations Spergula sect. Lepigonum (Fr.) G. López, Spergula rupicola (Lebel ex Le Jol.) G. López, Spergula australis (Samp.) G. López, Spergula tangerina (P. Monnier) G. López, Spergula capillacea (Kindb.) G. López, Spergula nicaeensis (Sarato ex Burnat) G. López, Spergula heldreichii (Foucaud) G. López, are proposed

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 $2\times 2$ 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

Quasi-Exactly Solvable Spin 1/2 Schr\"odinger Operators

The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave functions with polynomial components to be equivalent to a \sch\ operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of several new examples of multi-parameter QES spin 1/2 Hamiltonians in one dimension.Comment: 32 pages, LaTeX2e using AMS-LaTeX packag

On form-preserving transformations for the time-dependent Schr\"odinger equation

In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schr\"odinger equation (TDSE). In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of time-independent potentials related by a usual Darboux transformation for the stationary Schr\"odinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving transformation of the TDSE to a time-independent potential. The preeminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials.Comment: LaTeX2e (with amsmath, amssymb, amscd, cite packages), 11 page

Strong asymptotics of multi-level Hermite-Pad\'e polynomials

We obtain the strong asymptotics of multiple orthogonal polynomials which arise in a mixed type Hermite-Pad\'e approximation problem defined on a Nikishin system of functions. The results obtained allow to give exact estimates of the rate of convergence of the approximating rational functions and the strong asymptotics of Cauchy biorthogonal polynomials.Comment: 34 page