An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces

Abstract

Explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi-dimensional spaces in a continuous sequence of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also given for SO(N)-reduced matrix elements of basic orbital observables. These developments make it possible to determine the matrix elements of polynomial and a other Hamiltonians analytically, to within SO(N) Clebsch-Gordan coefficients, and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent.

19 pages, 8 figures

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• Bohr Hamiltonian with an energy dependent $\gamma$-unstable Coulomb-like potential
• Construction of SO(5)>SO(3) spherical harmonics and Clebsch-Gordan coefficients
• Phonon and multi-phonon excitations in rotational nuclei by exact diagonalization of the Bohr Hamiltonian
• The emergent dynamical symmetry at the triple point of nuclear deformations
• Euclidean Dynamical Symmetry in Nuclear Shape Phase Transitions
• Bohr Hamiltonian with a deformation-dependent mass term: physical meaning of the free parameter
• A computer code for calculations in the algebraic collective model of the atomic nucleus
• Quadrupole collective variables in the natural Cartan-Weyl basis
• Algebraic evaluation of matrix elements in the Laguerre function basis
• Microscopic shell-model counterpart of the Bohr-Mottelson model
• The quadrupole collective model from a Cartan-Weyl perspective
• Spectral properties of a tractable collective Hamiltonian
• An equations-of-motion approach to quantum mechanics: application to a model phase transition
• Towards a geometrical classification of statistical conservation laws in turbulent advection