2019, Vol.22, No.1, pp.84 - 92
The solution of the one-dimensional Schrödinger equation for a potential involving an attractive ∼ x 2/3 and a repulsive centrifugal-barrier ∼ x−2 terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five biconfluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed to 91ℏ2/(72m). The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.
Key words: integrable Schrödinger potential, energy spectrum, bi-confluent Heun equation
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