Numerical study of the SWKB condition of novel classes of exactly solvable systems

Abstract

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the the exceptional orthogonal polynomial, even the system possesses the shapeinvariance.In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein--Adler Hermite, Laguerre and Jacobipolynomials.For all of them, one can always remove the $\hbar$-dependency from the condition, and it is satisfied with a certain degree of accuracy.