Conditional Stability Of Coefficients Inverse Problem For Strongly Coupled Schr(?)dinger Equations

Quantum mechanics,one of the two fundamental pillars of modern physics,is the theory that describes the microscopic world.Schr(?)dinger equation is the basic equation of quantum mechanics,and it is a non-relativistic second-order partial differential equation established by combining the concept of matter waves with the wave equation.Each microscopic system has a corresponding Schr(?)dinger equation or Schr(?)dinger equations.Most of the Schr(?)dinger equations derived from the microscopic system are coupled.In the coupled partial differential equations,objects are interrelated and influence each other.We can understand the properties of the microscopic system by studying the corresponding Schr(?)dinger equation.However,some coefficients of the equation are unknown,for that reason,the equation is underdetermined,and we need the additional observation data to identify the unknown coefficients in the equation and further study the solution of the equation.Therefore,the research on how to utilize the locally observable measurement data to determine the required coefficients has a profound application background.In this thesis,we retrieve a stationary potential in the strongly coupled Schr(?)dinger equations from either boundary or internal measurements.Conditional stabilities are obtained by means of new Carleman estimates for the system with different observations,respectively.The first chapter briefly introduces the background and current development of the relevant research work of this thesis,and describes the specific form of the equations and the main work of this thesis.Chapter 2 researches the coefficients inverse problem of the strongly coupled Schr(?)dinger equations from the internal measurement.The internal observation data used in this chapter is the measurement of the solution of the Schr(?)dinger equations on any subdomain in domain.By converting the strongly coupled terms in the strongly coupled Schr(?)dinger equations to the derivatives of the solution of the Schr(?)dinger equations with respect to time variables when the coefficient matrix of the equations satisfies certain conditions,we establish the Carleman estimate for the equations from the internal measurement.Based on the Carleman estimation established above,the conditional stability of the coefficient inverse problem of the strongly coupled Schr(?)dinger equation from the internal measurement is obtained.Chapter 3 studies the coefficients inverse problem of the strongly coupled Schr(?)dinger equations from the boundary measurement.We established the Carleman estimate of the strongly coupled Schr(?)dinger equations from the boundary measurement when the solutions of the equations are observed on the partial boundary which satisfies certain geometric conditions,and we obtained the conditional stability of the coefficient inverse problem of the strongly coupled Schr(?)dinger equation from the boundary measurement.