# Factorization method

To describe a system in quantum mechanics, one must solve either an eigenvalue problem for the matrix formulation or a second-order differential equation with boundary conditions for the wave formulation. An elegant procedure to solve this quantum mechanics problem involves using the factorization method, where a particular differential operator is factorized in terms of different operators. Dirac [1930] and Fock [1931] proposed the first ideas about this method to solve the one-dimensional harmonic oscillator. Later, Schrödinger [1940a,b, 1941] exploited them to solve different problems.

The first generalization of this technique was given by Infeld [1941]. After several other contributions, Infeld and Hull finished it ten years later with their seminal paper [Infeld, Hul, 1951], where they performed an exhaustive classification of all the systems solvable through the factorization method. This work includes the harmonic oscillator, the hydrogen-atom potential, the free particle, the radial oscillator, some spin systems, the Pöschl-Teller potential, and Lamé potentials. For many years this work was considered to be the culmination of the technique, i.e., if someone wanted to see the viability of using the factorization method, they would check the paper by Infeld and Hull. This fact also meant that people thought this method was essentially finished.

After many years, contrary to the common belief that the factorization method was completely explored, Mielnik [1984] made a significant contribution. In his work, Mielnik did not consider the particular solution used in the factorization method of Infeld and Hull, but rather the general solution. Mielnik used this solution to find a family of new factorizations of the harmonic oscillator that also lead to related new solvable potentials. In this way, after 33 years, not only one but a whole family of new solvable potentials was obtained by the factorization method. Mielnik obtained a family of potentials isospectral to the harmonic oscillator in this classic work.

Many years later to Infeld and Hull's article, and from a different area of physics, Witten [1981] proposed a mechanism to form hierarchies of isospectral Hamiltonians, now called supersymmetric partners. This work considers a toy model for supersymmetry in quantum field theory. This technique is closely related to the generalization of the factorization method proposed by Mielnik. Regarding the now completely developed approach, Mielnik found the first-order SUSY partner potentials of the harmonic oscillator for the specific factorization energy -1/2. As a result, the study of analytically solvable Hamiltonians was reborn. This generalization of the factorization method or intertwining technique is now gathered in an area of science commonly called supersymmetric quantum mechanics, or SUSY QM. There is a big community of scientists working on this topic nowadays.

Almost immediately after Mielnik's work, Fernandez [1984a] applied the same technique to the hydrogen atom and obtained a new one-parameter family of potentials with the same spectrum. In the meantime, Nieto [1984]; Andrianov, Borisov, and loffe [1984]; and Sukumar [1985a] developed the formal connection between SUSY QM and the factorization method. They were the first to understand the full power of the technique to obtain new solvable potentials in quantum mechanics by generalizing the process used by Mielnik and Fernandez. They generalized the factorization method to a general solvable potential with an arbitrary factorization energy. All these developments caused a new interest in the algebraic methods of solution in quantum mechanics and the search for new exactly-solvable potentials.

Until that moment, the factorization operators were always of the first order. This is the natural approach: given that the Hamiltonian is a second-order differential operator, it is expected to be factorized in terms of lower-order operators. Nevertheless, Andrianov, loffe, and Spiridonov [1993] proposed to use higher-order operators (see also Andrianov et al. [1995]). Later, Bagrov and Samsonov [1995] proposed an alternative point of view.

After many years away from these developments, it is worth noticing that the group of Cinvestav returned to the study of SUSY QM. In a remarkable work, Fernández et al. [1998a] generalized the factorization method from Mielnik's point of view to obtain new families of potentials isospectral to the harmonic oscillator, using second-order differential intertwining operators. Soon after, Rosas-Ortiz [1998a,b] applied the same techniques to the hydrogen atom. This generalization was achieved using two iterative first-order transformations, as viewed by Mielnik and Sukumar in the 1980s. With this theory, it was possible to obtain an energy spectrum with spectral gaps, i.e., the regularity of the spectrum was lost. A review of SUSY QM from the point of view of a general factorization method can be found in the works by Mielnik and Rosas-Ortiz [2004] and by Fernández and Fernández-García [2005].

Furthermore, it is essential to mention that even when most of the papers on this theory are gathered under the keyword of SUSY QM, there is a lot of work on this topic from different points of view. We can mention, for example, Darboux transformations [Matveev and Salle, 1991; Fernández-García and Rosas-Ortiz, 2008], intertwining technique [Cariñena, Ramos, and Fernández, 2001], factorization method [Mielnik and Rosas-Ortiz, 2004], N-fold supersymmetry [Aoyama et al., 2001; Sato and Tanaka, 2002; González-López and Tanaka, 2001; Bagchi and Tanaka, 2009], and non-linear hidden supersymmetry [Leiva and Plyushchay, 2003; Plyushchay, 2004; Correa et al., 2007, 2008a].

Furthermore, using higher-order techniques, Bermudez et al. [20131 developed new exactly-solvable potentials from the inverted oscillator (also known as the repulsive oscillator). This work has caused further developments to generalize this case and obtain more solutions. Nowadays, the factorization method is significantly developed, and it is harder to expand the set of systems that can be solved through this technique. Nevertheless, this research area is still very active, as in the case of the inverted oscillator and several theoretical developments (see Bermudez et al. [2011 and 2012].