12,549 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
Quasi Exactly Solvable NxN-Matrix Schroedinger Operators
New examples of matrix quasi exactly solvable Schroedinger operators are
constructed. One of them constitutes a matrix generalization of the quasi
exactly solvable anharmonic oscillator, the corresponding invariant vector
space is constructed explicitely. Also investigated are matrix generalizations
of the Lame equation.Comment: 16 latex pages, new results adde
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
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