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

Symmetries of differential equations. IV

By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form [x^(r_i)]_i = F_i(where r_i > 1 for every i = 1 , ... ,n) cannot admit an infinite number of pointlike symmetry vectors. When r_i = r for every i = 1, ... ,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n_ 2 + nr + 2 (when r> 2), and by 2n_2 + 4n + 2 (when r = 2). The group of symmetries of ͞x^r = ͞0 (r> 1) has also been computed, and the result obtained shows that when n > 1 and r> 2 the number of independent symmetries of these equations does not attain the upper bound 2n _2 + nr + 2, which is a common bound for all systems of differential equations of the form ͞x^r = F[t, ͞x, ... , ͞x^(r - 1 )] when r> 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n^2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system ͞x= ͞0, and is also a common bound for all systems of the form ͞x = ͞F (t ,͞x, ‾̇x)

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

Estudio poblacional del Oso de las Cavernas (Ursus spelaeus Rosenmüller-Heinroth) de cuevas gallegas, NW de la Península Ibérica

[Abstract] A population study of two contemporary sites of Ursus spelaeus from Serra do Courel (Galicia, NW of Iberian Peninsula) has been carried out. The different morphology of the sites, as well as the taphonomical processes that affected the deposits produce a biased preservation of the bone remains that is interpreted in this paper

Data about Cervus elaphus (Cervidae, Artiodactyla, Mammalia) in carstic caves from Galicia (NW Spain)

[Abstract] A review of the data about Cervus elaphus from galician caves is carried out in this paper. Some of these caves present a deposit of bone remains with anthropic origin, but one of them does not seen to have been originated by the human action. The abundance and the good preservation of the material in this cave allow to study the structure of the population of Cervus elaphus

Una aproximación paleobiológica a los Osos de las Cavernas de Liñares y Eirós (Galicia, España)

[Abstract] The sites of Liñares and Eirós are closely situated in the NW of Spain and both contains a large number of Ursus spelaeus remains. However, the chronology of these sites is different and correspond to different climatic conditions. Once considered the effects of the preservational biass in the deposit, the demographic particularities of each population can be explained in terms of the different climatic conditions suffered by the studied cave bear populations