We study aspects of the quantum and classical dynamics of a 3-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h (3) typical for the H 3 Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schrödinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part S(θ) does not satisfy any linear equation. In this case S(θ) satisfies a nonlinear ODE that has the Painlevé property and its solutions can be expressed in terms of the Painlevé transcendent P 6 . We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.

We develop a new semiclassical approach, which starts with the density matrix given by the Euclidean time path integral with fixed coinciding endpoints, and proceed by identifying classical (minimal Euclidean action) path, to be referred to as flucton, which passes through this endpoint. Fluctuations around flucton path are included, by standard Feynman diagrams, previously developed for instantons. We calculate the Green function and evaluate the one loop determinant both by direct diagonalization of the fluctuation equation, and also via the trick with the Green functions. The two-loop corrections are evaluated by explicit Feynman diagrams, and some curious cancellation of logarithmic and polylog terms is observed. The results are fully consistent with large-distance asymptotics obtained in quantum mechanics. Two classic examples -quartic double-well and sineGordon potentials -are discussed in detail, while power-like potential and quartic anharmonic oscillator are discussed in brief. Unlike other semiclassical methods, like WKB, we do not use the Schrödinger equation, and all the steps generalize to multi-dimensional or quantum fields cases straightforwardly.

The general description of superintegrable systems with one polynomial integral of order N in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean plane. We consider classical and quantum Hamiltonian systems allowing separation of variables in polar coordinates. The potentials can be classified into two major classes and their main properties are described. We conjecture that a new infinite family of superintegrable potentials in terms of the sixth Painlevé transcendent P 6 exists and demonstrate this for the first few cases.