Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4999510.22363/2658-4670-2026-34-1-139-144UNJXACPhysics and AstronomyФизикаResearch ArticleSolution of the one-dimensional Schrödinger equation for a heterostructure with a triangular potential function by the power series methodРешение одномерного уравнения Шредингера для гетероструктур с треугольной потенциальной функцией методом степенных рядовhttps://orcid.org/0000-0002-7674-1716BelyaevaIrina N.БеляеваИ. Н.

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of department of Mathematics of Faculty of Mathematics and Science of Institute of Pedagogy

ibelyaeva@bsuedu.ruhttps://orcid.org/0000-0003-1131-3195ChekanovNikolay A.ЧекановН. А.

Professor, Doctor of Sciences in Physics and Mathematics, Professor of department of Mathematics of Faculty of Mathematics and Science of Institute of Pedagogy

nikchek137@gmail.comhttps://orcid.org/0009-0002-6353-3552KorotenkoRoman V.КоротенкоР. В.

Student of department of Mathematics of Faculty of Mathematics and Science of Institute of Pedagogy

1474589@bsuedu.ruhttps://orcid.org/0000-0001-9134-2951ChekanovaNatalia N.ЧекановаН. Н.

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department Information Technology and Mathematic Modeling 

natchek1976@gmail.comBelgorod State National Research UniversityБелгородский государственный национальный исследовательский университетKharkov National University named after V.N. KarazinХарьковский национальный университет имени В. Н. Каразина30042026341Vol 34, No 1 (2026)ТОМ 34, № 1 (2026)13914429042026Copyright ©; 2026, Belyaeva I.N., Chekanov N.A., Korotenko R.V., Chekanova N.N.Copyright ©; 2026, Беляева И.Н., Чеканов Н.А., Коротенко Р.В., Чеканова Н.Н.2026Belyaeva I.N., Chekanov N.A., Korotenko R.V., Chekanova N.N.Беляева И.Н., Чеканов Н.А., Коротенко Р.В., Чеканова Н.Н.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/49995

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.

В работе методом степенных рядов решается одномерное уравнение Шрёдингера с треугольной потенциальной функцией, которая применяется в различных современных гетероструктурах, в частности для GaAs и других. Варьируя доступные параметры, можно получить желаемую точность численного решения уравнения Шрёдингера с любым типом потенциальной функции для современных гетероструктур. Для исходного уравнения Шрёдингера получены волновые функции в виде функций Эйри и аналитическая формула для уровней энергии с помощью нулей функции Эйри. Значения энергетических уровней из этой аналитической формулы согласуются с результатами, полученными методом прямых степенных рядов, с точностью до $10^{-4}$ процентов, то есть до 5 десятичных знаков. Однако рациональнее и проще использовать решение уравнения Шрёдингера. Но для достижения высокой точности вычислений необходимо установить флажок Digits на несколько десятков значащих цифр и увеличить количество членов степенного ряда, что приводит увеличению времени счета на компьютере.

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