Third-harmonic generation investigated by a short-range bottomless exponential potential well

In this paper, we introduce a short-range bottomless potential which can be exactly integrabled. And the exact solutions of Schrödinger equation of the potential have been derived through the hypergeometric function. After the results are brought into the system of third-harmonic generation (THG), the special phenomenons of third-harmonic generation under short-range exponential potential with inverse square root singularity will be exhibited, that is, the third-harmonic coefficient (THC) will present a mirror reverse state. At the same time, according to the varied tendency of energy intervals and matrix elements, the energy regions of the third-harmonic coefficient and the value of the THC peaks, as well as the numbers, are specifically elaborated.