Factorization Method

In order 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 problem in quantum mechanics consists in using the factorization method, where a certain differential operator is factorized in terms of different operators. The first ideas about this method were proposed by Dirac [1930] and Fock [1931] in order to solve the one-dimensional harmonic oscillator and later on they were exploited by Schrödinger [Schrödinger, 1940a,b, 1941] to solve different problems.

 

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

 

After many years, and contrary to the common belief that the factorization method was completely explored, Mielnik [1984] made an important 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 and he used it 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 rather a whole family of new solvable potentials were obtained by the factorization method. In this classic work, Mielnik obtained a family of potentials isospectral to the harmonic oscillator.

 

Many years later to Infeld and Hull’s article, and from a totally different area of physics, Witten [1981] proposed a mechanism to form hierarchies of isospectral Hamiltonians, which are now called supersymmetric partners. In this work, a toy model for supersymmetry in quantum field theory is considered. It turns out that this technique is closely related with the generalization of the factorization method proposed by Mielnik. In terms of the now completely developed theory, we would say that 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 gathered now in an area of science that is commonly called supersymmetric quantum mechanics, or SUSY QM, and there is a big community of scientists working on this topic nowadays.

 

Almost immediately after Mielnik’s work, Fernández [1984a] applied the same technique to the hydrogen atom and he also obtained a new one-parameter family of potentials with the same spectrum. In the mean time, Nieto [1984]; Andrianov, Borisov, and Ioffe [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 in order to obtain new solvable potentials in quantum mechanics by generalizing the process used by Mielnik and Fernández. Now we say that 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 first-order. This is natural, being the Hamiltonian a second-order differential operator, it is expected to be factorized in terms of lower-order operators. Nevertheless, Andrianov, Ioffe, and Spiridonov [1993] proposed to use higher-order operators (see also Andrianov et al. [1995]). An alternative point of view was also proposed by Bagrov and Samsonov [1995].

 

After many years away from these developments, it is worth to notice 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, in order 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 1980’s. 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 important 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 under 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, Bermudez et al. [2013] developed new exactly-solvable potentials from the inverted oscillator (also known as repulsive oscillator) using higher-order techniques. This work has caused some further developments to generalize this case and obtain more solutions. Nowadays, the factorization method is very 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 for the inverted oscillator and several theoretical developments (see Bermudez et al. [2011] and [2012]).