Higher Derivative Supersymmetry, a Modified Crum–darboux Transformation and Coherent State

Abstract

Inspired by the possibility of factorizing the second derivative interwining operators using a modified form of the well-known Crum–Darboux transformation, we present a scheme for generating a new pair of isospectral potentials. We also consider the interesting problem of constructing coherent states for such factorizable operators.

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…”

RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM

“…The orthonormality and completeness of ψ Going from first-to kth-order SUSYQM [28,29,30,31,32,33] is achieved by replacing the first-order differential operators A, A † by some kth-order ones A, A † in such a way that the SUSY partner Hamiltonians…”

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{

“…and H (+) , H (−) can be glued together so as to get a reducible SSUSY one h (1) , h (2) [10,11,12,13]. In such a framework, the two Hamiltonians h (1) and h (2) with corresponding potentials V (1) (x) and V (2) (x), respectively, intertwine with some second-order differential operators…”

“…Interestingly one can look for extensions to higher-order SUSY theories by resorting to higher-derivative versions of the factorization operators. These offer the scope of relating to non-trivial quantum systems as has been found in a number of recent works [10,11,12,13].…”