592 research outputs found
Canonical Discretization. I. Discrete faces of (an)harmonic oscillator
A certain notion of canonical equivalence in quantum mechanics is proposed.
It is used to relate quantal systems with discrete ones. Discrete systems
canonically equivalent to the celebrated harmonic oscillator as well as the
quartic and the quasi-exactly-solvable anharmonic oscillators are found. They
can be viewed as a translation-covariant discretization of the (an)harmonic
oscillator preserving isospectrality. The notion of the deformation of the
canonical equivalence leading to a dilatation-covariant discretization
preserving polynomiality of eigenfunctions is also presented.Comment: 29 pages, LaTe
Two electrons in an external oscillator potential: hidden algebraic structure
It is shown that the Coulomb correlation problem for a system of two
electrons (two charged particles) in an external oscillator potential possesses
a hidden -algebraic structure being one of recently-discovered
quasi-exactly-solvable problems. The origin of existing exact solutions to this
problem, recently discovered by several authors, is explained. A degeneracy of
energies in electron-electron and electron-positron correlation problems is
found. It manifests the first appearence of hidden -algebraic structure
in atomic physics.Comment: 7 pages (plus one figure avaliable via direct request), LaTeX,
Preprint IFUNAM FT 94-4
He and HeH molecular ions in a strong magnetic field: the Lagrange mesh approach
Accurate calculations for the ground state of the molecular ions He
and HeH placed in a strong magnetic field a.u.
(G) using the Lagrange-mesh method are presented.
The Born-Oppenheimer approximation of zero order (infinitely massive centers)
and the parallel configuration (molecular axis parallel to the magnetic field)
are considered. Total energies are found with 9-10 s.d. The obtained results
show that the molecular ions He and HeH exist at \,a.u. and \,a.u., respectively, as predicted in \cite{Tu:2007}
while a saddle point in the potential curve appears for the first time at a.u. and a.u., respectively.Comment: 8 pages, 1 figure, 2 tables. arXiv admin note: text overlap with
arXiv:0912.104
Hidden algebra of the -body Calogero problem
A certain generalization of the algebra of first-order
differential operators acting on a space of inhomogeneous polynomials in is constructed. The generators of this (non)Lie algebra depend on
permutation operators. It is shown that the Hamiltonian of the -body
Calogero model can be represented as a second-order polynomial in the
generators of this algebra. Given representation implies that the Calogero
Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces
with explicit bases, which are closely related to the finite-dimensional
representations of above algebra. This representation is an alternative to the
standard representation of the Bargmann-Fock type in terms of creation and
annihilation operators.Comment: 10pp., CWRU-Math, October 199
On polynomial solutions of differential equations
A general method of obtaining linear differential equations having polynomial
solutions is proposed. The method is based on an equivalence of the spectral
problem for an element of the universal enveloping algebra of some Lie algebra
in the "projectivized" representation possessing an invariant subspace and the
spectral problem for a certain linear differential operator with variable
coefficients. It is shown in general that polynomial solutions of partial
differential equations occur; in the case of Lie superalgebras there are
polynomial solutions of some matrix differential equations, quantum algebras
give rise to polynomial solutions of finite--difference equations.
Particularly, known classical orthogonal polynomials will appear when
considering acting on . As examples, some
polynomials connected to projectivized representations of ,
, and are briefly discussed.Comment: 12p
One-electron atomic-molecular ions containing Lithium in a strong magnetic field
The one-electron Li-containing Coulomb systems of atomic type and
molecular type , and are studied in
the presence of a strong magnetic field a.u. in the
non-relativistic framework. They are considered at the Born-Oppenheimer
approximation of zero order (infinitely massive centers) within the parallel
configuration (molecular axis parallel to the magnetic field). The variational
and Lagrange-mesh methods are employed in complement to each other. It is
demonstrated that the molecular systems , and
can exist for sufficiently strong magnetic fields a.u. and that can even be stable at
magnetic fields typical of magnetars.Comment: 22 pages, 9 figures, 4 table
Solvability of the Hamiltonians related to exceptional root spaces: rational case
Solvability of the rational quantum integrable systems related to exceptional
root spaces is re-examined and for is established in the
framework of a unified approach. It is shown the Hamiltonians take algebraic
form being written in a certain Weyl-invariant variables. It is demonstrated
that for each Hamiltonian the finite-dimensional invariant subspaces are made
from polynomials and they form an infinite flag. A notion of minimal flag is
introduced and minimal flag for each Hamiltonian is found. Corresponding
eigenvalues are calculated explicitly while the eigenfunctions can be computed
by pure linear algebra means for {\it arbitrary} values of the coupling
constants. The Hamiltonian of each model can be expressed in the algebraic form
as a second degree polynomial in the generators of some infinite-dimensional
but finitely-generated Lie algebra of differential operators, taken in a
finite-dimensional representation.Comment: 51 pages, LaTeX, few equations added, one reference added, typos
correcte
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