The confluent algorithm in second-order supersymmetric quantum mechanics

Supersymmetry Transformations for Delta Potentials

“…In this section it will be illustrated, by means of the delta-well potential, the advantages for manipulating spectra of the second-order SUSY transformations [56,57,58,59] compared with the first-order ones. It is nowadays known that the second-order SUSY partners H 2 of the initial Hamiltonian H 0 can be generated either by employing two eigenfunctions u 1 (x), u 2 (x) of H 0 , not necessarily physical, associated to two different factorization energies ǫ 1,2 , ǫ 1 = ǫ 2 [37,41] or by an appropriate eigenfunction u 1 (x) in the limit when ǫ 2 → ǫ 1 (the so called confluent case [61,62]). In both situations the two Hamiltonians H 0 , H 2 are intertwined by a secondorder operator in the way…”

“…4.1 Confluent case [61,62] Let us consider in the first place the limit ǫ 2 → ǫ 1 ≡ ǫ = −k 2 /2 < 0, taking as seed the Schrödinger solution u + (x) vanishing as x −→ −∞, which means to take the u(x) given in equation (9) with D = 0, namely:…”

Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials

“…An independent proposal was made wherein the terminology "confluent" algorithm was introduced and some partners of the free particle and the harmonic oscillator were exhibited [37]. The confluent algorithm was then studied in more general terms and applied to the free particle, one-soliton well, and harmonic oscillator [19]. It was also considered in a general construction of all possible first-and second-order partners of the TDPT potential [12].…”