Alexander V. Turbiner

An infinite family of solvable and integrable quantum systems on a plane

An infinite family of exactly solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.

Quantal problems with partial algebraization of the spectrum

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (e.g., harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that a part of the eigenvalues, and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the hamiltonian. For one-dimensional motion this hidden symmetry isSL(2,R). It is shown that other groups lead to a partial algebraization in multidimensional quantal problems. In particular,SL(2,R)×SL(2,R),SO(3) andSL(3,R) are relevant to two-dimensional motion inducing a class of quasi-exactly-solvable two-dimensional hamiltonians. Typically they correspond to systems in a curved space, but sometimes the curvature turns out to be zero. Graded algebras open the possibility of constructing quasi-exactlysolvable hamiltonians acting on multicomponent wave functions. For example, with a (non-minimal) superextension ofSL(2,R) we get a hamiltonian describing the motion of a spinor particle.

QUASI-EXACTLY-SOLVABLE QUANTAL PROBLEMS: ONE-DIMENSIONAL ANALOGUE OF RATIONAL CONFORMAL FIELD THEORIES

The class of quasi-exactly-solvable problems in ordinary quantum mechanics discovered recently shows remarkable parallels with rational two-dimensional conformal field theories. This fact suggests that investigation of the quasi-exactly-solvable models may shed light on rational conformal field theories. We discuss a relation between these two theoretical schemes and propose a mathematical formulation for the procedure of constructing quasi-exactly solvable systems. This discussion leads us to a kind of generalization of the Sugawara construction.

Exact solvability of the Calogero and Sutherland models

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 Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra slN in finitedimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.

Analytic continuation of eigenvalue problems

In this paper we consider the dependence of Schrodinger equation eigenvalue problems on the coupling-constant parameters in the potential. We show that unless great care is taken, analytic continuation in these parameters can lead to surprising and paradoxical conclusions. Careful analysis of the analytical properties shows that such eigenvalue problems can have elaborate internal structure; these problems can incorporate several different eigenvalue problems joined together. For example, the anharmonic oscillator, whose potential is V(x)=a2x6−3ax2, is actually four different eigenvalue problems combined into one.

Periodic orbits for an infinite family of classical superintegrable systems

We show that all bounded trajectories in the two-dimensional classical system with the potential are closed for all integer and rational values of k. The period is and does not depend on k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

Hidden algebras of the (super)Calogero and Sutherland models

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 that in this parametrization the Hamiltonians of the AN, BCN, BN, CN and DN Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed—for arbitrary values of the coupling constants—as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N+1) or the Lie superalgebra gl(N+1|N) for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by Turbiner, and implies that the Calogero and Jack–Sutherland polynomials, as well as their supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N+1) and the Lie superalgebra gl(N+1|N).

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, R) acting on RP1. As examples, some polynomials connected to projectivized representations of sl2(R), sl2(R)q, osp(2,2), and so3 are briefly discussed.

Two electrons in an external oscillator potential: The 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

QUASI-EXACTLY-SOLVABLE MANY-BODY PROBLEMS

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