Solution of the one-dimensional Schrödinger equation for a heterostructure with a triangular potential function by the power series method

In the work by the power series method the one-dimensional Schrödinger equation is solved with a triangular potential function which is applied in various modern heterostructures, in particular for GaAs and the others. By varying available parameters it is possible to obtain the desired precision of the numerical solution of the Schrödinger equation with any type of potential function for modern heterostructures. For the original Schrödinger equation are obtained wave functions in the form Airy functions and the analytical formula for the energy levels through the zeros of the Airy function. The values energy levels from this analytical formula agree with its results obtained by direct power series method with precision up to $10^{{-4}}$ percents, that is, up to 5 decimal signs. However, it is more rational and easier to use the Schrödinger equation solution, because the numerical calculations zeros of Airy function present separate complex and complicated numerical problem. But in order to achieve high numerical accuracy, it is necessary to set the Digits flag to several dozen significant digits and increasing the number of power series, that leads to an increasing in the time spent on the computer.