Perturbations of integrable systems and Dyson-Mehta integrals (2003)
We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it...
Solvability of $F_4$ quantum integrable systems (2003)
Vieyra, Juan C. Lopez, Turbiner, Alexander
It is shown that the $F_4$ rational and trigonometric integrable systems are exactly-solvable for {\it arbitrary} values of the coupling constants. Their spectra are found explicitly while...
Canonical Discretization. I. Discrete faces of (an)harmonic oscillator (2000)
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...
Different faces of harmonic oscillator (1999)
Harmonic oscillator in Fock space is defined. Isospectral as well as polynomiality-of-eigenfunctions preserving, translation-invariant discretization of the harmonic oscillator is presented....
Solutions of Non-linear Differential and Difference Equations with Superposition Formulas (1999)
Turbiner, Alexander, Winternitz, Pavel
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their...
$H_3^{++}$ molecular ions can exist in strong magnetic fields (1998)
Turbiner, Alexander, V., Juan Carlos Lopez, H, Ulises Solis
Using the variational method it is shown that for magnetic fields $B\geq 10^{11}$ G there can exist a molecular ion $H_3^{++}$.
Hidden Algebra of Three-Body Integrable Systems (1998)
It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial...
Lie Algebras in Fock Space (1997)
A catalogue of explicit realizations of representations of (super) Lie algebras and quantum algebras in Fock space is presented.
Two-body Elliptic Model in proper variables: Lie-algebraic forms and their discretizations (1997)
Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented. Translation-invariant and dilatation-invariant discretizations of the model are obtained.
Solvability of the G_2 Integrable System (1997)
Rosenbaum, Marcos, Turbiner, Alexander, Capella, Antonio
It is shown that the 3-body trigonometric G_2 integrable system is exactly-solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary...
Hidden Algebras of the (super) Calogero and Sutherland models (1997)
Brink, Lars, Turbiner, Alexander, Wyllard, Niclas
We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown...
Interesting Relations in Fock Space (1996)
Certain non-linear relations between the generators of the (q-deformed) Heisenberg algebra are found. Some of these relations are invariant under quantization and $q$-deformation.
Hidden $sl_2$-algebra of finite-difference equations (1995)
Smirnov, Yuri, Turbiner, Alexander
The connection between polynomial solutions of finite-difference equations and finite-dimensional representations of the $sl_2$-algebra is established (the talk given at the Wigner Symposium,...
Exact Solvability of the Calogero and Sutherland Models (1995)
Ruhl, Werner, Turbiner, Alexander
Translationally invariant symmetric polynomials as coordinates for $N$-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland $N$-body...
Lie-algebraic discretization of differential equations (1995)
Smirnov, Yuri, Turbiner, Alexander
A certain representation for the Heisenberg algebra in finite-difference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is...
Invariant Indentities in the Heisenberg Algebra (1994)
Polynomial relations between the generators of $q$--deformed Heisenberg algebra invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the...
Quasi-Exactly-Solvable Differential Equations (1994)
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any...
Two electrons in an external oscillator potential: hidden algebraic structure (1994)
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...
Operator Identities, Representations of Algebras and the Problem of Normal Ordering (1993)
Turbiner, Alexander, Post, Gerhard
Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum)...
Hidden algebra of the $N$-body Calogero problem (1993)
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...
Classification of linear differential operators with an invariant subspace of monomials (1993)
Post, Gerhard, Turbiner, Alexander
A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.
On some operator identities and representations of algebras (1993)
Turbiner, Alexander, Post, Gerhard
Certain infinite families of operator identities related to powers of positive root generators of (super) Lie algebras of first-order differential operators and $q$-deformed algebras of first-order...
Lie-algebras and linear operators with invariant subspaces (1993)
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given....
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on...
A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is...
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result...