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.

Excited state quantum phase transitions in many-body systems

“…For physical reasons, the SO(n)-invariant "multipolemultipole" Hamiltonian To relate the U(n + 1) and SU(1, 1) descriptions, let us first observe that the Casimir operators of SO(n) and SU b (1, 1) are related, and that the SO(n) an-gular momentum quantum number v and the SU b (1, 1) seniority v b are actually identical. This is an example of a general correspondence between the algebras SO(n) and SU(1, 1) [67,68]. According to the basic quasispin relations above, Thus, for L even, the Hamiltonians (Ĥ M M ) ± and (Ĥ P P ) ∓ differ only by a constant (i.e., a function of conserved quantum numbers), while, for L odd, it is (Ĥ M M ) ± and (Ĥ P P ) ± which differ only by a constant.…”

Construction of spherical harmonics and Clebsch–Gordan coefficients

“…The Bohr collective model in the resulting calculational scheme is thus genuinely an algebraic collective model. The availability of SO(5) ⊃ SO(3) Clebsch-Gordan coefficients, in conjunction with a large body of analytic expressions for β matrix elements [3,4], makes it possible to algebraically construct the matrix elements for any collective model Hamiltonian expressible as a polynomial in the collective quadrupole moments and canonical momenta. Algebraic expressions for matrix elements of a variety of other operators, including the term β −2 occurring in the Davidson potential [40,41], have also been derived [3,4].…”

“…The SO(5) spherical harmonics are eigenfunctions of the Laplace-Beltrami operatorΛ 2 on the four-sphere S 4 , that is, the angular part of the Laplacian in five dimensions. This operatorΛ 2 is also the second order Casimir invariant of SO (5).…”

“…Matrix elements of an essentially unlimited set of potential and kinetic energy operators, often in analytic form, can easily be constructed in an SU(1, 1) × SO(5) basis [3,4], in terms of SO(5) ⊃ SO(3) Clebsch-Gordan coefficients. The resulting calculational scheme, the so-called algebraic collective model (ACM), is described in Refs.…”

“…The resulting calculational scheme, the so-called algebraic collective model (ACM), is described in Refs. [2][3][4]8]. Examples of physical applications may be found in Ref.…”