Discrete and Continuous Models and Applied Computational Science最新文献_第4页

Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics Kuryshkin-Wodkiewicz量子力学中的复特征值

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-139-148

A. Zorin, M. Malykh, L. Sevastianov

{"title":"Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics","authors":"A. Zorin, M. Malykh, L. Sevastianov","doi":"10.22363/2658-4670-2022-30-2-139-148","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-139-148","url":null,"abstract":"One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state ψ{psi} the quantity AA is equal to λ{lambda} if ψ{psi} is an eigenfunction of the operator O^(A){hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{hat{O}((A-lambda)2)psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkins mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){hat{O}((A-lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{hat{O}(A2)-2hat{O}(A)lambda + lambda 2 hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{hat{D}(A) = hat{O}(A2) - hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue λ{lambda} of the operator O^(A){hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by ±i⟨D^⟩{pm i sqrt{langle hat{D} rangle}}.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44963993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Investigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguides 光滑不规则集成光波导的绝热波导模模型研究

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-149-159

A. L. Sevastyanov

{"title":"Investigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguides","authors":"A. L. Sevastyanov","doi":"10.22363/2658-4670-2022-30-2-149-159","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-149-159","url":null,"abstract":"The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49199231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order 一阶常微分方程近似解的多级伪谱法(配位法)

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-127-138

K. Lovetskiy, D. Kulyabov, Ali Weddeye Hissein

{"title":"Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order","authors":"K. Lovetskiy, D. Kulyabov, Ali Weddeye Hissein","doi":"10.22363/2658-4670-2022-30-2-127-138","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-127-138","url":null,"abstract":"The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47444156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Numerical solution of Cauchy problems with multiple poles of integer order 整数阶多极Cauchy问题的数值解

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-105-114

A. Belov, N. Kalitkin

引用次数: 0

Optimization of an isotropic metasurface on a substrate 基板上各向同性超表面的优化

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-115-126

Zhanna O. Dombrovskaya

引用次数: 0

Analysis of queuing systems with threshold renovation mechanism and inverse service discipline 具有门限更新机制和反向服务规则的排队系统分析

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-05-03 DOI: 10.22363/2658-4670-2022-30-2-160-182

I. Zaryadov, Hilquias C. C. Viana, T. Milovanova

引用次数: 0

The quantization of the classical two-dimensional Hamiltonian systems 经典二维哈密顿系统的量子化

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-02-25 DOI: 10.22363/2658-4670-2022-30-1-39-51

I. Belyaeva

{"title":"The quantization of the classical two-dimensional Hamiltonian systems","authors":"I. Belyaeva","doi":"10.22363/2658-4670-2022-30-1-39-51","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-39-51","url":null,"abstract":"The paper considers the class of Hamiltonian systems with two degrees of freedom. Based on the classical normal form, according to the rules of Born-Jordan and Weyl-MacCoy, its quantum analogs are constructed for which the eigenvalue problem is solved and approximate formulas for the energy spectrum are found. For particular values of the parameters of quantum normal forms using these formulas, numerical calculations of the lower energy levels were performed, and the obtained results were compared with the known data of other authors. It was found that the best and good agreement with the known results is obtained using the Weyl-MacCoy quantization rule. The procedure for normalizing the classical Hamilton function is an extremely time-consuming task, since it involves hundreds and even thousands of polynomials for the necessary transformations. Therefore, in the work, normalization is performed using the REDUCE computer algebra system. It is shown that the use of the Weyl-MacCoy and Born-Jordan correspondence rules leads to almost the same values for the energy spectrum, while their proximity increases for large quantities of quantum numbers, that is, for highly excited states. The canonical transformation is used in the work, the quantum analog of which allows us to construct eigenfunctions for the quantum normal form and thus obtain analytical formulas for the energy spectra of different Hamiltonian systems. So, it is shown that quantization of classical Hamiltonian systems, including those admitting the classical mode of motion, using the method of normal forms gives a very accurate prediction of energy levels.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43583067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Finite-difference methods for solving 1D Poisson problem 求解一维Poisson问题的有限差分方法

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-02-25 DOI: 10.22363/2658-4670-2022-30-1-62-78

Serge Ndayisenga, L. Sevastianov, K. Lovetskiy

{"title":"Finite-difference methods for solving 1D Poisson problem","authors":"Serge Ndayisenga, L. Sevastianov, K. Lovetskiy","doi":"10.22363/2658-4670-2022-30-1-62-78","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-62-78","url":null,"abstract":"The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42424511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

On methods of building the trading strategies in the cryptocurrency markets 论加密货币市场交易策略的构建方法

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-02-25 DOI: 10.22363/2658-4670-2022-30-1-79-87

E. Shchetinin

引用次数: 1

On the many-body problem with short-range interaction 关于具有短程相互作用的多体问题

Discrete and Continuous Models and Applied Computational Science Pub Date : 2022-02-25 DOI: 10.22363/2658-4670-2022-30-1-52-61

M. M. Gambaryan, M. Malykh

{"title":"On the many-body problem with short-range interaction","authors":"M. M. Gambaryan, M. Malykh","doi":"10.22363/2658-4670-2022-30-1-52-61","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-52-61","url":null,"abstract":"The classical problem of the interaction of charged particles is considered in the framework of the concept of short-range interaction. Difficulties in the mathematical description of short-range interaction are discussed, for which it is necessary to combine two models, a nonlinear dynamic system describing the motion of particles in a field, and a boundary value problem for a hyperbolic equation or Maxwells equations describing the field. Attention is paid to the averaging procedure, that is, the transition from the positions of particles and their velocities to the charge and current densities. The problem is shown to contain several parameters; when they tend to zero in a strictly defined order, the model turns into the classical many-body problem. According to the Galerkin method, the problem is reduced to a dynamic system in which the equations describing the dynamics of particles, are added to the equations describing the oscillations of a field in a box. This problem is a simplification, different from that leading to classical mechanics. It is proposed to be considered as the simplest mathematical model describing the many-body problem with short-range interaction. This model consists of the equations of motion for particles, supplemented with equations that describe the natural oscillations of the field in the box. The results of the first computer experiments with this short-range interaction model are presented. It is shown that this model is rich in conservation laws.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41850883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0