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 $q-$deformation of the canonical equivalence leading to a dilatation-covariant discretization preserving polynomiality of eigenfunctions is also presented.Comment: 29 pages, LaTe

He23+_2^{3+} and HeH2+^{2+} molecular ions in a strong magnetic field: the Lagrange mesh approach

Accurate calculations for the ground state of the molecular ions He$_2^{3+}$ and HeH$^{2+}$ placed in a strong magnetic field $B\gtrsim 10^{2}$ a.u. ($\approx 2.35 \times 10^{11}$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$_2^{3+}$ and HeH$^{2+}$ exist at $B > 100$\,a.u. and $B > 1000$\,a.u., respectively, as predicted in \cite{Tu:2007} while a saddle point in the potential curve appears for the first time at $B \sim 80$ a.u. and $B \sim 740$ a.u., respectively.Comment: 8 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:0912.104

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 $sl_2$-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 $sl_2$-algebraic structure in atomic physics.Comment: 7 pages (plus one figure avaliable via direct request), LaTeX, Preprint IFUNAM FT 94-4

Hidden algebra of the NN-body Calogero problem

A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-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 $SL(2,{\bf R})$ acting on ${\bf RP_1}$. As examples, some polynomials connected to projectivized representations of $sl_2 ({\bf R})$, $sl_2 ({\bf R})_q$, $osp(2,2)$ and $so_3$ 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 $(li, e)$ and molecular type $(li, li, e)$, $(li, \alpha, e)$ and $(li, p, e)$ are studied in the presence of a strong magnetic field $B \leq 10^{7}$ 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 ${\rm LiH}^{3+}$, ${\rm LiHe}^{4+}$ and ${\rm Li}_{2}^{5+}$ can exist for sufficiently strong magnetic fields $B \gtrsim 10^{4}$ a.u. and that ${\rm Li}_{2}^{5+}$ 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 $G_2, F_4$ is re-examined and for $E_{6,7,8}$ 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