It is shown that for the one-dimensional quantum anharmonic oscillator with potential $V(x)= x^2+g^2 x^4$ the Perturbation Theory (PT) in powers of $g^2$ (weak coupling regime) and the semiclassical expansion in powers of $hbar$ for energies coincide
. It is related to the fact that the dynamics in $x$-space and in $(gx)$-space corresponds to the same energy spectrum with effective coupling constant $hbar g^2$. Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in $g^2$ for the logarithmic derivative of wave function leads to PT (with polynomial in $x$ coefficients) for the RB equation and to the true semiclassical expansion in powers of $hbar$ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy $sim 10^{-6}$ locally and unprecedented accuracy $sim 10^{-9}-10^{-10}$ in energy for any $g^2 geq 0$. A generalization to the radial quartic oscillator is briefly discussed.
It is shown that planar quantum dynamics can be related to 3-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the $O(d)$ symmetry reduction from $d$ degrees of freedom to one degre
e is presented. A link between two-dimensional (super-integrable) systems and 3-body (super-integrable) systems is revealed. As illustration we present number of examples. We demonstrate that the celebrated Calogero-Wolfes 3-body potential has a unique property: two-dimensional quantum dynamics coincides with 3-body quantum dynamics on the line at $d=1$; it is governed by the Tremblay-Turbiner-Winternitz potential for parameter $k=3$.
It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in ${mathbb R}^3$ along the Remarkable Figure-8-shape trajectory discovered by Moore-Chenciner-Montg
omery. It is shown they become constants of motion along this trajectory. Hence, 3-body choreographic motion on Figure-8-shape trajectory in ${mathbb R}^3$ Newtonian gravity (Moore, 1993), as well as in ${mathbb R}^2$ modified Newtonian gravity by Fujiwara et al, 2003, is maximally superintegrable. It is conjectured that any 3-body potential theory which admit Figure-8-shape choreographic motion is superintegrable along the trajectory.
Based on Generalized Bloch equation the trans-series expansion for the phase (exponent) of the ground state density for double-well potential is constructed. It is shown that the leading and next-to-leading semiclassical terms are still defined by th
e flucton trajectory (its classical action) and quadratic fluctuations (the determinant), respectively, while the the next-to-next-to-leading correction (at large distances) is of non-perturbative nature. It comes from the fact that all flucton plus multi-instanton, instanton-anti-instanton classical trajectories lead to the same classical action behavior at large distances! This correction is proportional to sum of all leading instanton contributions to energy gap.
As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 we study aspects of the quantum and classical dynamics of a $3$-body system with equal masses, each body with $d$ degrees of freedom, with interaction
depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. It corresponds to a three-dimensional quantum particle moving in a curved space with special $d$-dimension-independent metric in a certain $d$-dependent singular potential, while at $d=1$ it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is $d$-independent, it has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model as in the $d=3$ case. We find an exactly-solvable three-body $S^3$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.
