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The purpose of this communication is to point out the connection between a 1Dquantum Hamiltonian involving the fourth Painleve transcendent P$_{\rm IV}$,obtained in the context of second-order supersymmetric quantum mechanics andthird-order ladder operators, with a hierarchy of families of quantum systemscalled $k$-step rational extensions of the harmonic oscillator and related withmulti-indexed $X_{m_{1},m_{2},...,m_{k}}$ Hermite exceptionnal orthogonalpolynomials of type III. The connection between these exactly solvable modelsis established at the level of the equivalence of the Hamiltonians usingrational solutions of the fourth Painleve equation in terms of generalizedHermite and Okamoto polynomials. We also relate the different ladder operatorsobtained by various combinations of supersymmetric constructions involvingDarboux-Crum and Krein-Adler supercharges, their zero modes and thecorresponding energies. These results will demonstrate and clarify the relationobserved for a particular case in previous papers.

journal of mathematical physics

Connection between quantum systems involving the fourth Painleve  transcendent and $k$-step rational extensions of the harmonic oscillator  related to Hermite EOP