2,283 research outputs found
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
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
Forward Analysis and Model Checking for Trace Bounded WSTS
We investigate a subclass of well-structured transition systems (WSTS), the
bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete
deterministic ones, which we claim provide an adequate basis for the study of
forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth.
Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered
previously for the termination of forward analysis, boundedness is decidable.
Boundedness turns out to be a valuable restriction for WSTS verification, as we
show that it further allows to decide all -regular properties on the
set of infinite traces of the system
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
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
Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton - Jacobi Formalism
We obtain the band edge eigenfunctions and the eigenvalues of solvable
periodic potentials using the quantum Hamilton - Jacobi formalism. The
potentials studied here are the Lam{\'e} and the associated Lam{\'e} which
belong to the class of elliptic potentials. The formalism requires an
assumption about the singularity structure of the quantum momentum function
, which satisfies the Riccati type quantum Hamilton - Jacobi equation, in the complex plane. Essential
use is made of suitable conformal transformations, which leads to the
eigenvalues and the eigenfunctions corresponding to the band edges in a simple
and straightforward manner. Our study reveals interesting features about the
singularity structure of , responsible in yielding the band edge
eigenfunctions and eigenvalues.Comment: 21 pages, 5 table
Proving Safety with Trace Automata and Bounded Model Checking
Loop under-approximation is a technique that enriches C programs with
additional branches that represent the effect of a (limited) range of loop
iterations. While this technique can speed up the detection of bugs
significantly, it introduces redundant execution traces which may complicate
the verification of the program. This holds particularly true for verification
tools based on Bounded Model Checking, which incorporate simplistic heuristics
to determine whether all feasible iterations of a loop have been considered.
We present a technique that uses \emph{trace automata} to eliminate redundant
executions after performing loop acceleration. The method reduces the diameter
of the program under analysis, which is in certain cases sufficient to allow a
safety proof using Bounded Model Checking. Our transformation is precise---it
does not introduce false positives, nor does it mask any errors. We have
implemented the analysis as a source-to-source transformation, and present
experimental results showing the applicability of the technique
Zero-temperature generalized phase diagram of the 4d transition metals under pressure
We use an accurate implementation of density functional theory (DFT) to
calculate the zero-temperature generalized phase diagram of the 4 series of
transition metals from Y to Pd as a function of pressure and atomic number
. The implementation used is full-potential linearized augmented plane waves
(FP-LAPW), and we employ the exchange-correlation functional recently developed
by Wu and Cohen. For each element, we obtain the ground-state energy for
several crystal structures over a range of volumes, the energy being converged
with respect to all technical parameters to within meV/atom. The
calculated transition pressures for all the elements and all transitions we
have found are compared with experiment wherever possible, and we discuss the
origin of the significant discrepancies. Agreement with experiment for the
zero-temperature equation of state is generally excellent. The generalized
phase diagram of the 4 series shows that the major boundaries slope towards
lower with increasing for the early elements, as expected from the
pressure induced transfer of electrons from states to states, but are
almost independent of for the later elements. Our results for Mo indicate a
transition from bcc to fcc, rather than the bcc-hcp transition expected from
- transfer.Comment: 28 pages and 10 figures. Submitted to Phys. Rev.
Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation
In this paper we classify Weingarten surfaces integrable in the sense of
soliton theory. The criterion is that the associated Gauss equation possesses
an sl(2)-valued zero curvature representation with a nonremovable parameter.
Under certain restrictions on the jet order, the answer is given by a third
order ordinary differential equation to govern the functional dependence of the
principal curvatures. Employing the scaling and translation (offsetting)
symmetry, we give a general solution of the governing equation in terms of
elliptic integrals. We show that the instances when the elliptic integrals
degenerate to elementary functions were known to nineteenth century geometers.
Finally, we characterize the associated normal congruences
Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
Inozemtsev models are classically integrable multi-particle dynamical systems
related to Calogero-Moser models. Because of the additional q^6 (rational
models) or sin^2(2q) (trigonometric models) potentials, their quantum versions
are not exactly solvable in contrast with Calogero-Moser models. We show that
quantum Inozemtsev models can be deformed to be a widest class of partly
solvable (or quasi-exactly solvable) multi-particle dynamical systems. They
posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A
new method for identifying and solving quasi-exactly solvable systems, the
method of pre-superpotential, is presented.Comment: LaTeX2e 28 pages, no figure
- …