Gegenbauer Spectral Method For Two-Dimensional Schrodinger Equation

Schr?dinger equation is one of the basic equations in the physical system.It describes the change of quantum state of the quantum system against time,which plays an important role in quantum mechanics.While solving the Schr?dinger equation of the microscopic system,the wave function and corresponding energy can be obtained,and in addition,by calculating the probability distribution of the particles,more and further properties of it can be found and understood.The Schr?dinger equation is widely used in many fields.For example,many scientists use it to demonstrate some phenomena in physics,chemistry,and other fields,and the results are consistent with the reality.In recent years,scholars tend to do research on the Schr?dinger equation with more complex potential function in order to solve more practical issues.Thus,it is significant to establish an effective numerical scheme for solving two-dimensional Schr?dinger equation.In this paper,Gegenbauer spectral method is used to solve the two-dimensional Schr?dinger equation.According to the processing requirements of spectral method,the equation should be first processed.By a specific mapping,the complex region is transformed into a standard region,and the general boundary value condition is inverted into a zero boundary value condition.Next step is to approximate the spatial derivation by calculating the spatial inner product,discretizing the two-dimensional Schr?dinger equation,and transforming the original equation into a set of linear ordinary differential equations in the complex domains.Furthermore,the time direction can be discretized by using the Gegenbauaer-Gauss-Lobatto(GGL)configuration method,which is to select an appropriate time step by the length of T,divide it into several intervals,and use the method to discretize on each interval step by step.The numerical solutions of the system of equations can be obtained,which is also the solution of the original problem.We analyze the error of the real and imaginary parts of the solution separately,and then obtain the error estimate results.Finally,we use programming to establish the numerical simulations and plot graphs of error surface and convergence curve,which demonstrate that the method has high precision and good stability.