  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.
