Factorization: little or great algorithm?

Abstract

The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Bäcklund approaches.

Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % pepucqWFtepvaaa!46A4! $$ \mathcal{P}\mathcal{

“…(x) of the extended GPT potential can be determined by using the standard intertwining relations [3,4,5,6], satisfied by the partner Hamiltonians…”

“…Over the years supersymmetric quantum mechanics (SUSYQM) has emerged as one of the most insightful tools towards construction of Hamiltonians with a prescribed spectrum starting from some given exactly solvable form [3,4,5,6]. The key technique such as the factorization method (sometimes couched in the language of intertwining relationships amongst operators inducing factorization) has enabled one to uncover many useful properties of quantum mechanics.…”

Supersymmetric quantum mechanics and Painlevé equations

“…These equations immediately lead to the higher-order SUSY QM [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In this treatment, the standard SUSY algebra with two generators…”

Complete Analytical Solutions of the Mie-Type Potentials in N-Dimensions

“…It is well known that the factorization operators are not, in general, the ladder operator of the system (for details see Refs. [35][36][37][38]). …”