Algebraic Spectral Relations for Elliptic Quantum Calogero-Moser Problems

Abstract

Explicit algebraic relations between the quantum integrals of the elliptic Calogero– Moser quantum problems related to the root systems A ₂ and B ₂ are found.     

Quantum completely integrable systems connected with semi-simple Lie algebras

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators _k introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom.     

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Open quantum systems and Feynman integrals: Some problems

A few open problems of mathematical physics are presented. They concern open quantum systems and Feynman path integrals; some of them are technical, while others are of conceptual importance.Dedicated to the 30th anniversary of the Joint Institute for Nuclear Research.     

Sur les intégrales de mouvement du système de sinus-Gordon discret – On Integrals of Motion of the discrete sine-Gordon System

We give explicit formulas for some densities of integrals of motion for the discrete sine-Gordon system (quantum or not). The generating function for the densities of integrals of motion may be seen as the expansion of the logarithm of a certain continued fraction (possibly quantum). In the case of q root of the unity, we show that these integrals of motion can be identified to the classical integrals of motion.     

Quantum dynamics of tight-binding networks coherently controlled by external fields

With some reviews on the investigations on the schemes for quantum state transfer based on spin systems, we discuss the quantum dynamics of magnetically-controlled networks for Bloch electrons. The networks are constructed by connecting several tight-binding chains with uniform nearest-neighbor hopping integrals. The external magnetic field and the connecting hopping integrals can be used to control the intrinsic properties of the networks. For several typical networks, rigorous results are shown for some specific values of external magnetic field and the connecting hopping integrals: a complicated network can be reduced into a virtual network, which is a direct sum of some independent chains with uniform nearest-neighbor hopping integrals. These reductions are due to the fermionic statistics and the Aharonov-Bohm effects. In application, we study the quantum dynamics of wave packet motion of Bloch electrons in such networks. For various geometrical configurations, these networks can function as some optical devices, such as beam splitters, switches and interferometers. When the Bloch electrons as Gaussian wave packets input these devices, various quantum coherence phenomena can be observed, e.g., the perfect quantum state transfer without reflection in a Y-shaped beam, the multi-mode entanglers of electron wave by star-shaped network, magnetically controlled switches, and Bloch electron interferometer with the lattice Aharonov-Bohm effects. With these quantum coherent features, the networks are expected to be used as quantum information processors for the fermion system based on the possible engineered solid state systems, such as the array of quantum dots that can be implemented experimentally.      

On some integrable systems in the extended lobachevsky space

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrödinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.     

Path integration in non-relativistic quantum mechanics

A new method for computing path integrals explicitly is developed and applied to problems in non-relativistic quantum mechanics, such as: wave functions, propagators on configuration spaces and on phase space, caustic problems, bound states. Path integrals for paths on curved spaces and for paths on multiply-connected spaces are computed.     

Monte Carlo studies of nuclear many-particle systems

Stochastic evaluation of path integrals provides a useful tool for the study of a variety of nuclear systems which are otherwise not amenable to definitive analysis through perturbative, variational, or stationary-phase approximations. Ground state properties of potential models, such as quantum fluctuations in the density, are examined. Tunneling problems in quantum many-particle systems, such as spontaneous fission and the ground state structure of systems with degenerate vacuua are treated by incorporating one's physical understanding of the essential collective degrees of freedom in the stochastic algorithm. The role of subnuclear degrees of freedom is studied by comparing the exact solution of a simple confining quark model with the solution to a phase-shift equivalent hadronic potential model. This work is supported in part through funds provided by the U.S. Department of Energy (D.O.E.) under contract number DE-AC02-76ERO3069.     

On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

By use of complete orthonormal sets of ψ^α exponential-type orbitals (ψ^α-ETOs,α=1,0,-1,-2,...) the series expansion formulas for the noninteger n Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) are derived. These formulas enable us to express the overlap integrals with NISTOs through the overlap integrals over ISTOs with the same and different screening constants. By calculating concrete cases the convergence of the series for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances is tested. The accuracy of the results is quite high for quantum numbers, screening constants and location of STOs.     

Effective classical potential for fluctuating field systems of finite size

We extend a previously proposed variational method in which quantum mechanical path integrals over fluctuating orbits are approximated by single integrals over an effective classical potential, to field systems of finite size and point the way for higher order corrections.     

Lacunae of Hyperbolic Riesz Kernels and Commutative Rings of Partial Differential Operators

The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov) and the theory of lacunae for hyperbolic Riesz kernels. As an application, we give a simple and explicit construction of additional quantum integrals for the generalized Calogero–Moser problem associated with a finite reflection group W and W-invariant integer root multiplicities m. These quantum integrals are not W-invariant and cannot be constructed directly using the standard technique of Dunkl operators.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Two-dimensional supersymmetry: From SUSY quantum mechanics to integrable classical models

Two known two-dimensional SUSY quantum mechanical constructions—the direct generalization of SUSY with first-order supercharges and higher-order SUSY with second-order supercharges—are combined for a class of 2-dim quantum models, which are not amenable to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained—the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related “flipped” potentials are established.     

Statistical Theory of Bremsstrahlung in Plasma-Molecular Systems

Different approaches to the statistical theory of bremsstrahlung in nonequilibrium plasma-molecular systems are compared. The spectra of bremsstrahlung, as well as collision integrals and effective dissipation constants are calculated for such systems within the framework of both classical and quantum theory.     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

Complete reduction of integrals in two-loop five-light-parton scattering amplitudes

We reduce all the most complicated Feynman integrals in two-loop five-light-parton scattering amplitudes to basic master integrals, while other integrals can be reduced even easier. Our results are expressed as systems of linear relations in the block-triangular form, very efficient for numerical calculations. Our results are crucial for complete next-to-next-to-leading order quantum chromodynamics calculations for three-jet, photon, and/or hadron production at hadron colliders. To determine the block-triangular relations, we develop an efficient and general method, which may provide a practical solution to the bottleneck problem of reducing multiloop multiscale integrals.     

Integrals of the motion for a nonlinear quantum oscillator

We construct and discusss explicitly time dependent integrals of the motion of non-autonomous quantum systems. Such integrals may exist even when the classical limit of the dynamics is non-integrable.     

The Toda hierarchy and the KdV hierarchy

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins are the degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions of classical systems are motivated by Conformal Field Theory, and their quantum counterparts can be thought of as being the degenerations of the critical level Knizhnik-Zamolodchikov-Bernard equations.