arXiv subjects

Alexander V. Turbiner

Publications and source records attributed to Alexander V. Turbiner.

34 records · Page 2Linked to original sources

arXiv2014

(Quasi)-exact-solvability on the sphere $S^n$

An Exactly-Solvable (ES) potential on the sphere $S^n$ is reviewed and the related Quasi-Exactly-Solvable (QES) potential is found and studied. Mapping the sphere to a simplex it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is $gl_n$ in a finite-dimensional representation realized by first order differential operators acting on $RP^n$. It is shown that variables in the Schrödinger eigenvalue equation can be separated in spherical coordinates and a number of the integrals of the second order exists assuring the complete integrability. The QES system is completely-integrable for $n=2$ and non-maximally superintegrable for $n\ge 3$. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Wigner-Inönü Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our $S^n$ system to a QES non-maximal superintegrable system on Euclidean space $E^n$, an extension of the Smorodinsky-Winternitz potential.

math-ph↗

arXiv2014

$\mathrm H_2^+$ in a weak magnetic field

The electronic energy of $\mathrm H_2^+$ in magnetic fields of up to $B=0.2B_0$ (or 4.7 $\times 10^4$ Tesla) is investigated. Numerical values of the magnetic susceptibility for both the diamagnetic and paramagnetic contributions are reported for arbitrary orientations of the molecule in the magnetic field. It is shown that both diamagnetic and paramagnetic susceptibilities grow with inclination, while paramagnetic susceptibility is systematically much smaller than the diamagnetic one. Accurate two-dimensional Born-Oppenheimer surfaces are obtained with special trial functions. Using these surfaces, vibrational and rotational states are computed and analysed for the isotopologues $\mathrm H_2^+$ and $\mathrm D_2^+$.

quant-ph↗

arXiv2013

From Quantum $A_N$ to $E_8$ Trigonometric Model: Space-of-Orbits View

A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for $(A-B-C{-D)$-models, both rational and trigonometric, is related to the universal enveloping algebra $U_{gl_n}$. For the exceptional $(G-F-E)$-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with $BC_1\equiv(\mathbb{Z}_2)\oplus T$ symmetry. In particular, the $BC_1$ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra $sl(2)\oplus sl(2)$.

math-ph↗

arXiv2012

Particular Integrability and (Quasi)-exact-solvability

A notion of a particular integrability is introduced when two operators commute on a subspace of the space where they act. Particular integrals for one-dimensional (quasi)-exactly-solvable Schroedinger operators and Calogero-Sutherland Hamiltonians for all roots are found. In the classical case some special trajectories for which the corresponding particular constants of motion appear are indicated.

math-ph↗

arXiv2011

From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).

math-ph↗

arXiv2011

The H$_2^+$ molecular ion: a solution

Combining the WKB expansion at large distances and Perturbation Theory at small distances it is constructed a compact uniform approximation for eigenfunctions. For lowest states $1s\si_{g}$ and $2p\si_{u}$ this approximation provides the relative accuracy $\lesssim 10^{-5}$ (5 s.d.) for any real $x$ in eigenfunctions and for total energy $E(R)$ it gives 10-11 s.d. for internuclear distances $R \in [0,50]$. Corrections to proposed approximations are evaluated. Separation constants and the oscillator strength for the transition $1s\si_{g} \rar 2p\si_{u}$ are calculated and compared with existing data.

quant-ph↗

arXiv2010

The quantum $H_3$ integrable system

The quantum $H_3$ integrable system is a 3D system with rational potential related to the non-crystallographic root system $H_3$. It is shown that the gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group $H_3$, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector $\vec \al\ =\ (1,2,3)$. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the $H_3$ Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of $H_3$-invariants "polynomially"-isospectral to the quantum $H_3$ model is defined.

math-ph↗

arXiv2009

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 $V(r,ϕ)=ω^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k ϕ}}+ \frac{βk^2}{r^2 \sin^2 {k ϕ}}$ are closed for all integer and rational values of $k$. The period is $T=\fracπ{2ω}$ and does not depend on $k$. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

math-ph↗

arXiv2009

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.

math-ph↗

arXiv2005

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.

hep-th↗

arXiv2004

The conformally invariant Laplace-Beltrami operator and factor ordering

In quantum mechanics the kinetic energy term for a single particle is usually written in the form of the Laplace-Beltrami operator. This operator is a factor ordering of the classical kinetic energy. We investigate other relatively simple factor orderings and show that the {\it only} other solution for a conformally flat metric is the conformally invariant Laplace-Beltrami operator. For non-conformally-flat metrics this type of factor ordering fails, by just one term, to give the conformally invariant Laplace-Beltrami operator.

quant-ph↗

arXiv2004

Quartic anharmonic many-body oscillator

Two quantum quartic anharmonic many-body oscillators are introduced. One of them is the celebrated Calogero model (rational $A_n$ model) modified by quartic anharmonic two-body interactions which support the same symmetry as the Calogero model. Another model is the three-body Wolfes model (rational $G_2$ model) with quartic anharmonic interaction added which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by the variational method.

hep-th↗

arXiv2001

Quantum Many--Body Problems and Perturbation Theory

We show that the existence of algebraic forms of exactly-solvable $A-B-C-D$ and $G_2, F_4$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.

hep-th↗

arXiv2001

Solvability of the $F_4$ Integrable System

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 eigenfunctions by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of $F_4$ root system and can be obtained by averaging over an orbit of the Weyl group. Alternative way of finding these variables exploiting a property of duality of the $F_4$ model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational $F_4$ model depending on two continuous and one discrete parameters is found.

hep-th↗

arXiv2000

Exact Solvability of Superintegrable Systems

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators.

hep-th↗

arXiv2000

Ion $H_2^+$ can dissociate in a strong magnetic field

In framework of a variational method the molecular ion $H_2^+$ in a magnetic field is studied. An optimal form of the vector potential corresponding to a given magnetic field (gauge fixing) is chosen variationally. It is shown that for any magnetic field strength as well as for any orientation of the molecular axis the system $(ppe)$ possesses a minimum in the potential energy. The stable configuration always corresponds to elongation along the magnetic line. However, for magnetic fields $B \gtrsim 5\times 10^{11} G$ and some orientations the ion $H_2^+$ becomes unstable decaying to $H{-atom} + p$.

astro-ph↗