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In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantumsystem with purely continuous spectrum: the quantum free particle in aLobachevski space with constant negative curvature. The SGA contains thegeometrical symmetry algebra of the system plus a subalgebra of operators thatgive the spectrum of the system and connects the eigenfunctions of theHamiltonian 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 arepresentation of $\frak{so}(4,2)$ by functions on a realization of theLobachevski space given by a two sheeted hyperboloid, where the Lie algebracommutators are the usual Poisson-Dirac brackets. Then, introduce a quantizedversion of the representation in which functions are replaced by operators on aHilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of theHamiltonian are given and "naive" ladder operators are identified. Thepreviously defined "naive" ladder operators shift the eigenvalues by a complexnumber so that an alternative approach is necessary. This is obtained by a nonself-adjoint function of a linear combination of the ladder operators whichgives the correct relation among the eigenfunctions of the Hamiltonian. We givean eigenfunction expansion of functions over the upper sheet of two sheetedhyperboloid in terms of the eigenfunctions of the Hamiltonian.

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