Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space

TL;DR

This paper constructs a Spectrum Generating Algebra for a free quantum particle in Lobachevski space, enabling the connection and expansion of eigenfunctions in a continuous spectrum setting.

Contribution

It develops a novel SGA framework for continuous spectra in hyperbolic space using Lie algebra representations and operator techniques.

Findings

Constructed the SGA as so(4,2) for the free particle in Lobachevski space.

Identified ladder operators and addressed their complex eigenvalue shifts.

Provided an eigenfunction expansion over the hyperboloid surface.

Abstract

In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is $\frak{so}(3,1)$ and the SGA is $\frak{so}(4,2)$. We start with a representation of $\frak{so}(4,2)$ by functions on a realization of the Lobachevski space given by a two sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and "naive" ladder operators are identified. The previously defined "naive" ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non self-adjoint function of a linear combination of the ladder operators which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheet of two sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.