The Inverse Spectral Problem For Sturm-Liouville Operator With Eigenparameter In The Boundary Condition And The Trace Formula For A Class Of Impulsive Sturm-Liouville Operators

The inverse spectral problems for Sturm-Liouville operator are to reconstruct the potential function from the observable spectral data(such as a set of eigenvalues and normalization constants called spectral data),consequently,to obtain the differential operator,the problems involve the existence,uniqueness,stability of the solution and the restructing algorithm.In this paper,we consider a class of Sturm-Liouville operator with eigenparameter in the boundary condition,we obtain the uniqueness of the solution by the completeness of cosine functions,and get the local solvability and stability by the method of Borg equation.As the trace formula,for the finite dimensional linear operators,the trace is the sum of eigenvalues.But the summation of eigenvalues for the unbounded differential operator is divergent,one tries to regularize by substracting some divergent,which is the so-called trace of Gelfand-Levitan type.It shows the impact to the spectral.In this paper,we consider a class of the impulsive Sturm-Liouville operator and obtain the asymptotic formula of the eigenvalues and trace formula of this operator mainly by the residue theorem.