Este trabajo estudia los sistemas integrables y superintegrables en bajas dimensiones proponiendo un método para su obtención. Se particulariza dicho análisis a la MASA nilpotente de su(1,1), la cual nos lleva al bien conocido potencial Morse. Formulado este, se plantean las factorizaciones tanto clásica como cuántica del Hamiltoniano asociado a dicho potencial.
The aim of the study we are about to conduct is to go in depth in the properties
of integrable low dimensional systems by the construcction, factorization, and
resolution of the Morse 1-dimensional one. In this way, we will begin with a review
of the basics that justify and define this analysis. Here, we will discuss about the
Algebra, the Lie groups, and other mathematical notions closely linked with the
symmetries that characterize this systems.We will remember also, the standart elements
typical of the hamiltonian mechanic, from the classic point of view as well as
the quantum one, which we will work with in our following factorization. Then, we
will delve into this integrable systems, showing his properties, relating them with
the superintegrable ones and suggesting a general method to build them. After this
introductory parragraph, we will fix this notions to our Morse potential offer. In the
next sections, we will use the Hamiltonian built to split it with a classical and quantum
factorization, considering the interchange operators, the obtaining and ploting
of eigenstates, and the determination of the symmetry relations and their group
behavior. Finally, we will conclude our study with a few considetations about the
whole proces, the techniques applied, etc.
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