najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-1-15-21najo-373Research ArticleMATHEMATICSМАТЕМАТИКАDiscrete spectrum Analysis using Laplace transform and Volterra equations (DALV-method)Анализ дискретного спектра с использованием преобразования Лапласа и уравнений Вольтерра (DALV-метод)KuidinV. V.KuidinV. V.

St. Petersburg 197101

St. Petersburg 197101

vladislav.kujdin@metalab.ifmo.ruZalipaevV. V.ZalipaevV. V.

St. Petersburg 197101

St. Petersburg 197101

GulevichD. R.GulevichD. R.

St. Petersburg 197101

St. Petersburg 197101

ITMO UniversityITMO University2021280720251211521Copyright © Kuidin V.V., Zalipaev V.V., Gulevich D.R., 20252025Kuidin V.V., Zalipaev V.V., Gulevich D.R.Kuidin V.V., Zalipaev V.V., Gulevich D.R.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/373

The theory of excitons in two dimensional materials including graphene and transition metal dichalcogenides (TMD) is complicated, as there appears a screened interaction in equations. Such interaction can be represented as Keldysh potential. The exact solution does not seem to exist yet. The method of searching appropriate solutions to equations of quantum mechanics is believed to solve this problem by using Laplace transform of tempered distributions and Volterra equations. The method is to seek solution as a Laplace transform of some tempered distribution that satisfies the appropriate Laplace spectral equation which, under Laplace transform, gives us the initial equation. Due to Paly-Wigner-Schwarz theorem, the image functions behavior depends on the geometry of original one support. In addition, the homogenous Volterra equation does not have nontrivial continuous solution. These constraints together with the fact that the studied equations turn out to be Volterra equations of III kind lead to a method that seems to solve a wide class of quantum mechanics equations.

Теория экситонов в двумерных материалах, включая графен и дихалькогениды переходных металлов (ДМД), сложна, так как в уравнениях появляется экранированное взаимодействие. Такое взаимодействие можно представить как потенциал Келдыша. Точного решения пока не существует. Считается, что метод поиска подходящих решений уравнений квантовой механики решает эту проблему с использованием преобразования Лапласа, обобщенных функций и уравнений Вольтерра. Метод заключается в поиске решения в виде преобразования Лапласа некоторой обобщенной функции, которая удовлетворяет соответствующему спектральному уравнению Лапласа, которое при преобразовании Лапласа дает нам исходное уравнение. По теореме Пэли–Винера–Шварца поведение образов функций зависит от геометрии носителя исходной функции. Кроме того, однородное уравнение Вольтерра не имеет нетривиального непрерывного решения. Эти ограничения вместе с тем, что изучаемые уравнения оказываются уравнениями Вольтерра III рода, приводят к методу, который, по-видимому, решает широкий класс уравнений квантовой механики.

дискретный спектрпреобразование Лапласауравнение Вольтеррауравнение Шредингераdiscrete spectrumLaplace transformVolterra equationSchrödinger equationThe work is supported by the Russian Science Foundation under the Grant No. 18-12-00429.ReferencesTitchmarsh E.G., Teichmann T. Eigenfunction expansions associated with secondorder differential equations, part 2. PhT, 1958, 11(10), P. 34.Titchmarsh E.G., Teichmann T. Eigenfunction expansions associated with secondorder differential equations, part 2. PhT, 1958, 11(10), P. 34.Teschl G. Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators. Graduate studies in mathematics. American Mathematical Society, 2009.Teschl G. Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators. Graduate studies in mathematics. American Mathematical Society, 2009.Dunford N., Schwartz J.T. Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert Space. Wiley Classics Library, Wiley, 1988.Dunford N., Schwartz J.T. Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert Space. Wiley Classics Library, Wiley, 1988.Fedoryuk M.V., Maslov V.P. Quasiclassical approximation for quantum mechanics equations, 1976.Fedoryuk M.V., Maslov V.P. Quasiclassical approximation for quantum mechanics equations, 1976.Englefield M.J. Solution of Schrödinger equation by Laplace transform. Journal of The Australian Mathematical Society, 1968, 8(3), P. 557– 567.Englefield M.J. Solution of Schrödinger equation by Laplace transform. Journal of The Australian Mathematical Society, 1968, 8(3), P. 557– 567.Englefield M.J. Solution of coulomb problem by Laplace transform. Journal of Mathematical Analysis and Applications, 1974, 48(1), P. 270– 275.Englefield M.J. Solution of coulomb problem by Laplace transform. Journal of Mathematical Analysis and Applications, 1974, 48(1), P. 270– 275.Altuüg Arda and Ramazan Sever. Exact solutions of the Schrödinger equation via Laplace transform approach: pseudoharmonic potential and mietype potentials. Journal of Mathematical Chemistry, 2012, 50(4), P. 971–980.Altuüg Arda and Ramazan Sever. Exact solutions of the Schrödinger equation via Laplace transform approach: pseudoharmonic potential and mietype potentials. Journal of Mathematical Chemistry, 2012, 50(4), P. 971–980.Douglas R.M., Pimentel and Antonio S De Castro. A Laplace transform approach to the quantum harmonic oscillator. European Journal of Physics, 2012, 34(1), P. 199.Douglas R.M., Pimentel and Antonio S De Castro. A Laplace transform approach to the quantum harmonic oscillator. European Journal of Physics, 2012, 34(1), P. 199.Ginyih Tsaur and Jyhpyng Wang. A universal Laplace transform approach to solving Schrödinger equations for all known solvable models. European Journal of Physics, 2013, 35(1), P. 015006.Ginyih Tsaur and Jyhpyng Wang. A universal Laplace transform approach to solving Schrödinger equations for all known solvable models. European Journal of Physics, 2013, 35(1), P. 015006.Tapas Das and Altu¨g Arda. Exact analytical solution of the ndimensional radial Schrödinger equation with pseudoharmonic potential via Laplace transform approach. Advances in High Energy Physics, 2015, 2015.Tapas Das and Altu¨g Arda. Exact analytical solution of the ndimensional radial Schrödinger equation with pseudoharmonic potential via Laplace transform approach. Advances in High Energy Physics, 2015, 2015.Yangqiang Ran, Lihui Xue, Sizhu Hu, and RuKeng Su. On the coulombtype potential of the onedimensional Schrödinger equation. Journal of Physics A: Mathematical and General, 2000, 33(50), P. 9265.Yangqiang Ran, Lihui Xue, Sizhu Hu, and RuKeng Su. On the coulombtype potential of the onedimensional Schrödinger equation. Journal of Physics A: Mathematical and General, 2000, 33(50), P. 9265.Naber M. Time fractional Schrödinger equation. Journal of Mathematical Physics, 2004, 45(8), P. 3339–3352.Naber M. Time fractional Schrödinger equation. Journal of Mathematical Physics, 2004, 45(8), P. 3339–3352.Michael E. Clarkson and Huw O Pritchard. A Laplace transform solution of Schrödingers equation using symbolic algebra. International journal of quantum chemistry, 1992, 41(6), P. 829–844.Michael E. Clarkson and Huw O Pritchard. A Laplace transform solution of Schrödingers equation using symbolic algebra. International journal of quantum chemistry, 1992, 41(6), P. 829–844.Zemanian A.H. Generalized integral transformations, 1968.Zemanian A.H. Generalized integral transformations, 1968.Warnock R.L. Bart G.R. Linear integral equations of the third kind. SIAM Journal on Mathematical Analysis, 1973, 4(4), P. 609–622.Warnock R.L. Bart G.R. Linear integral equations of the third kind. SIAM Journal on Mathematical Analysis, 1973, 4(4), P. 609–622.Vainikko G. Cordial Volterra integral equations 1. Numerical Functional Analysis and Optimization, 2009, 30(910), P. 1145–1172.Vainikko G. Cordial Volterra integral equations 1. Numerical Functional Analysis and Optimization, 2009, 30(910), P. 1145–1172.Vainikko G. Cordial Volterra integral equations 2. Numrical Functional Analysis and Optimization, 2010, 31(2), P. 191–219.Vainikko G. Cordial Volterra integral equations 2. Numrical Functional Analysis and Optimization, 2010, 31(2), P. 191–219.Vladimirov V.S. Generalized functions in mathematical physics, Nauka, Moscow, 1979. English transl, Mir, Moscow, 1979.Vladimirov V.S. Generalized functions in mathematical physics, Nauka, Moscow, 1979. English transl, Mir, Moscow, 1979.

The authors declare that there are no conflicts of interest present.