arXiv subjects

Willard Miller, Jr.

Publications and source records attributed to Willard Miller, Jr..

3 recordsLinked to original sources

arXiv2018

Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability

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.

math-ph↗

arXiv2017

The quantum n-body problem in dimension $d\ge n-1$: ground state

We employ generalized Euler coordinates for the $n$ body system in $d \geq n-1$ dimensional space, which consists of the centre-of-mass vector, relative (mutual), mass-independent distances $r_{ij}$ and angles as remaining coordinates. We prove that the kinetic energy of the quantum $n$-body problem for $d \geq n-1$ can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator $Δ_{rad}$ which depends on relative distances alone and (iii) the differential operator $Ω$ which annihilates any angle-independent function. The operator $Δ_{rad}$ has a large reflection symmetry group $Z_2^{\oplus \frac{n(n-1)}{2}}$ and in $ρ_{ij}=r_{ij}^2$ variables is an algebraic operator, which can be written in terms of generators of their {\it hidden} algebra $sl(\frac{n(n-1)}{2}+1, R)$. Thus, $Δ_{rad}$ makes sense of the Hamiltonian of a quantum Euler-Arnold $sl(\frac{n(n-1)}{2}+1, R)$ top in a constant magnetic field. It is conjectured that for any $n$, the similarity-transformed $Δ_{rad}$ is the Laplace-Beltrami operator plus (effective) potential; thus, it describes a $\frac{n(n-1)}{2}$-dimensional quantum particle in curved space. This was verified for $n=2,3,4$. After de-quantization the similarity-transformed $Δ_{rad}$ becomes the Hamiltonian of the classical top with variable tensor of inertia in an external potential. This approach allows a reduction of the $dn$-dimensional spectral problem to a $\frac{n(n-1)}{2}$ -dimensional spectral problem if the eigenfunctions depend only on relative distances. We prove that the ground state function of the $n$ body problem depends on relative distances alone.

math-ph↗

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↗