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Exact solution of the position-dependent mass Schr\"odinger equation with the completely positive oscillator-shaped quantum well potential

E.I. JafarovS.M. Nagiyev
Dec 2022
摘要
Two exactly-solvable confined models of the completely positiveoscillator-shaped quantum well are proposed. Exact solutions of theposition-dependent mass Schr\"odinger equation corresponding to the proposedquantum well potentials are presented. It is shown that the discrete energyspectrum expressions of both models depend on certain positive confinementparameters. The spectrum exhibits positive equidistant behavior for the modelconfined only with one infinitely high wall and non-equidistant behavior forthe model confined with the infinitely high wall from both sides. Wavefunctionsof the stationary states of the models under construction are expressed throughthe Laguerre and Jacobi polynomials. In general, the Jacobi polynomialsappearing in wavefunctions depend on parameters $a$ and $b$, but the Laguerrepolynomials depend only on the parameter $a$. Some limits and special cases ofthe constructed models are discussed.
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