Uniqueness Theorem On The Inverse Spectral Problem Of Matrix-type Differential Operators

As an important branch of operator theory,inverse spectrum theory is closely related to other disciplines and has a wide range of applications in mathematical physics,quantum mechanics and other disciplines.In recent years,the matrix-valued Schrodinger operators and Dirac operators in the inverse spectrum theory have received widespread attention,and the research on such operators and related issues has become a hot issue in the inverse spectrum theory.Based on the existing theory,this paper defines the matrix-valued Weyl-Titchmarsh.function M±(z,x),Green's function g(z,x),with the help of the operators' Riccati equation,the high-energy asymptotic formula of M±(z,x).Discussed the uniqueness of matrix Schr?dinger operators and Dirac operators potential function in the case of local smoothness.The main research contents of this paper are as follows:Chapter 1:The research background and current situation of Schrodinger operators and Dirac operators,as well as the main work of this paper are described.Chapter 2:First introduced the basic solution of the matrix-valued Hamiltonian system of order 2m × 2m and the matrix-valued Weyl-Titchmarsh function M±(z,x),based on the matrix-valued function M±(z,x)defines the Green function g(z,x).Next,the Schrodinger operators H is introduced,using the matrix-valued function M±(z,x)to give the Riccati equation of the operators H,M±(z,x)highenergy asymptotic,the high-energy asymptotic formula of the difference norm of the Weyl-Titchmarsh function corresponding to the two Schrodinger operators under the condition of continuous and locally smooth combined potential function of 2k order.Finally,by fixing x0,it is concluded that g0(z)and g'0(z)can uniquely determine m × m in the case of partial smoothness,order matrix valued potential function Q(x).Chapter 3:First,the Dirac operators D is introduced,using the newly defined matrix-valued function M±(z,x),the Riccati equation of the operators D,M±(z,x)high-energy asymptotic,the high-energy asymptotic formula of the difference norm of the Weyl-Titchmarsh function corresponding to the two Dirac operators under the condition of continuous and local smoothness of order k.Finally,by fixing x0,it is concluded that g0(z)and g'0(z)can uniquely determine 2m × 2m in the case of partial smoothness,order matrix valued potential function B(x).