Exact solutions for the Schrödinger equation with a conditionally integrable potential of x^{2/3} attractive plus fixed x^{-2} repulsive terms are given in terms of non-integer Hermite functions.
A singular Lambert-W Sc hrödinger potential exactly solvable in terms of the confluent hypergeometric functions
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A Schr\"odinger potential involving $x^\frac{2}{3}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions
Exact solutions for the Schrödinger equation with a conditionally integrable potential of x^{2/3} attractive plus fixed x^{-2} repulsive terms are given in terms of non-integer Hermite functions.