Abstract
We analyze the quantum mechanics of anyons on the sphere in the presence of a constant magnetic field. We introduce an operator method for diagonalizing the Hamiltonian and derive a set of exact anyon energy eigenstates, in partial correspondence with the known exact eigenstates on the plane. We also comment on possible connections of this system with integrable systems of the Calogero type.
Highlights
In the ω → 0 thermodynamic limit, Z2d should become the infinite volume partition function, i.e., Z2d → S/(2πβ) where S is the infinite area of the twodimensional box (looking at (1.1) this implies that when βω → 0 the thermodynamic limit prescription 1/(βω)2 → S/(2πβ) should hold)
The Calogero model on a circle, known as the Calogero-Sutherland model, is again an integrable model with exact N-body energy eigenstates expressed in terms of Jack polynomials
One should be able to identify a relevant confined anyon model, with confining parameter R, for which a class of exact N-body anyon eigenstates could be mapped, through a certain N-body kernel, on the N-body Calogero-Sutherland eigenstates, as it is the case with an harmonic confinement
Summary
The presence of the measure factor in (2.2) modifies the hermiticity properties of operators. Taking it into account we derive, for ∂ ≡ ∂/∂z, ∂ ≡ ∂/∂z. H uj = j(j + 1) uj Single-valuedness and regularity of φ near z = 0 requires j to be a non-negative integer All such states for j = 0, 1, 2, . Degenerate higher angular momentum states are found by acting with J+. Top angular momentum states with J3 = j are annihilated by J+ Such states are functions of z − z2u only, and to be eigenstates of J3 with eigenvalue j is they must be of the form (z − z2u)j, so z appears with highest power z2j.
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