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Exact solutions of the schrodinger equation for a class of hyperbolic potential well

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Chinese Physics B 2022年第4期31卷 109-115页

作者： Xiao-Hua Wang Chang-Yuan Chen Yuan You Fa-Lin Lu Dong-Sheng Sun Shi-Hai Dong School of Physics and Electronic Engineering Yancheng Teachers UniversityYancheng 224007China Research Center for Quantum Physics Huzhou UniversityHuzhou 313000China Laboratorio de Informaci´on Cu´antica CIDETECInstituto Polit´ecnico NacionalUPALMC.P 07700CDMXMexico

We propose a new scheme to study the exact solutions of a class of hyperbolic potential *** first apply different forms of function transformation and variable substitution to transform the schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same *** then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential *** a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun *** linearly dependent relation between two eigenfunctions is also studied.

关键词： hyperbolic potential well schrodinger equation Wronskian determinant confluent Heun function

Dirichlet problem for schrodinger equation with the boundary value in the BMO space

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Science China Mathematics 2022年第7期65卷 1431-1468页

作者： Renjin Jiang Bo Li Center for Applied Mathematics Tianjin UniversityTianjin 300072China College of Data Science Jiaxing UniversityJiaxing 314001China

Let(X,d,μ)be a metric measure space satisfying a Q-doubling condition(Q>1)and an L^(2)-Poincaré*** L=L+V be a Schrödinger operator on X,where L is a non-negative operator generalized by a Dirichlet form,and V is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition RH_(q) for some q≥(Q+1)/*** show that a solution to(L−∂_(t)^(2))u=0 on X×R_(+) satisfies the Carleson condition,sup_(B(xB,rB))1/μ(B(xB,rB))∫_(0)^(rB)∫_(B(xB,rB))|t∇u(x,t)|^(2)dμdt/tschrodinger operator L with the tracein the BMO(bounded mean oscillation)space associated with L.

关键词： schrodinger equation BMO Carleson measure metric measure space

Efficient Splitting Methods Based on Modified Potentials:Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the schrodinger equation

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Communications in Computational Physics 2023年第4期33卷 937-961页

作者： Sergio Blanes Fernando Casas Cesáreo González Mechthild Thalhammer Universitat Politècnica de València Instituto de Matemática Multidisciplinar46022 ValenciaSpain Universitat Jaume I IMAC and Departament de Matemàtiques12071 CastellónSpain Universidad de Valladolid Departamento de Matemática Aplicada47011 ValladolidSpain Leopold–Franzens–Universität Innsbruck Institut für Mathematik6020 InnsbruckAustria.

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent schrodinger equation,both in real and imaginary *** are based on the use of a double commutator and a modified processor,and are more efficient than other widely used schemes found in the ***,for certain potentials,they achieve order *** examples in one,two and three dimensions clearly illustrate the computational advantages of the new schemes.

关键词： schrodinger equation imaginary time propagation parabolic equations operator splitting methods modified potentials

A Conservative SAV-RRK Finite Element Method for the Nonlinear schrodinger equation

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 583-601页

作者： Jun Yang Nianyu Yi School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

*** this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear schrodinger equation into an equivalent system and to transform the energy into a quadratic *** use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time *** mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta *** examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.

关键词： schrodinger equation mass conservation energy conservation finite element method relaxation Runge-Kutta scalar auxiliary variable

GLOBAL SUPERCONVERGENCE ESTIMATES OF FINITE ELEMENT METHOD FOR schrodinger equation

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Journal of Computational Mathematics 1998年第6期16卷 521-526页

作者： Lin, Q Liu, XQ Acad Sinica Inst Syst Sci Beijing 100080 Peoples R China Xiangtan Teachers Coll Dept Math Hunan 411201 Peoples R China

In this paper, we shall study the initial boundary value problem of schrodinger equation. The second order gradient superconvergence estimates for the problem are obtained solving by linear finite elements.

关键词： finite element superconvergence estimates interpolation schrodinger equation

Approximate analytical solutions and mean energies of stationary schrodinger equation for general molecular potential

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Chinese Physics B 2021年第7期30卷 116-131页

作者： Eyube E S Rawen B O Ibrahim N Department of Physics School of Physical SciencesModibbo Adama University of TechnologyP M B 2076 YolaAdamawa StateNigeria Directorate of Basic and Remedial Studies Abubakar Tafawa Balewa University(ATBU)P M B 750001BauchiBauchi StateNigeria Department of Physics Faculty of ScienceP M B 1069MaiduguriBorno StateNigeria

The schrodinger equation is solved with general molecular potential via the improved quantization *** for bound state energy eigenvalues,radial eigenfunctions,mean kinetic energy,and potential energy are obtained in compact *** modeling the centrifugal term of the effective potential,a Pekeris-like approximation scheme is ***,we use the Hellmann–Feynman theorem to derive the relation for expectation *** state energy eigenvalues,wave functions and meanenergies of Woods–Saxon potential,Morse potential,Mobius squared and Tietz–Hua oscillators are deduced from the general molecular *** addition,we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz.H_(2),CO,HF,and O_(2).Results obtained are in perfect agreement with the data available from the literature for the potentials and *** also show that as the vibrational quantum number increases,the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then *** in a Morse potential,the mean kinetic energy increases linearly with vibrational quantum number increasing.

关键词： general molecular potential schrodinger equation improved quantization rule

Modulation instability,rogue waves and conservation laws in higher-order nonlinear schrodinger equation

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Communications in Theoretical Physics 2021年第2期73卷 1-10页

作者： Min-Jie Dong Li-Xin Tian School of Mathematical Sciences Nanjing Normal UniversityNanjingJiangsu 210023China Nonlinear Scientific Research Center Jiangsu UniversityZhenjiangJiangsu 212013China

In this paper,the modulation instability(MI),rogue waves(RWs)and conseryation laws of the coupled higher-order nonlinear schrodinger equation are *** to MI and the 2×2 Lax pair,Darboux-dressing transformation with an asymptotic expansion method,the existence and properties of the one-,second-,and third-order RWs for the higher-order nonlinear schrodinger equation are *** addition,the main characteristics of these solutions are discussed through some graphics,which are draw widespread attention in a variety of complex systems such as optics,Bose-Einstein condensates,capillary fow,superfluidity,fluid dynamics,and *** addition,infinitely-many conservation laws are established.

关键词： higher-order nonlinear schrodinger equation modulation instability rogue waves conservation laws

REGULAR SOLUTIONS FOR schrodinger equation ON UNBOUNDED DOMAINS

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Acta Mathematica Scientia 1998年第4期18卷 387-393页

作者：任艳霞吴荣 Department of Mathematics Nankai University Tianjin 300071 China

The authors study a class of solutions,namely, regular solutions of the schrodinger equation (1/2 Delta + q)u = 0 on unbounded domains. They definite the regular solutions in terms of sample path properties of Brownian motion and then characterize them by analytic method. In Section 4, they discuss the regular solution to the stochastic Dirichlet problem for the equation (1/2 Delta + q)u = 0 having limit alpha at infinity.

关键词： schrodinger equation regular solution stochastic Dirichlet problem

Coordinate Transformation and Exact Solutions of schrodinger equation with Position-Dependent Effective Mass

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Communications in Theoretical Physics 2007年第6期47卷 1001-1009页

作者： JU Guo-Xing CAI Chang-Ying XIANG Yang REN Zhong-Zhou Department of Physics Nanjing University Nanjing 210093 China Department of Physics Jinggangshan University Ji＇an 343009 China Center of Theoretical Nuclear Physics National Laboratory of Heavy-Ion Accelerator Lanzhou 730000 China

Using the coordinate transformation method, we solve the one-dimensional schrodinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term Vm, which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of Vm for several mass functions are discussed.

关键词： coordinate transformation schrodinger equation position-dependent mass eigenfunction special functions