Inverse Problem For Periodic Potential With The Transmission Condition On The Half-line And The Property Of The First Eigenvalue Of The Spherecial Laplacian Operator

We have two topics in this paper.In chapter 2,we discuss the first topic,which is about the inverse problem of Schrodinger operators with periodic potential and the trans-mission condition on the half-line.First,we discuss the properties of the Weyl function and the eigenvalues of this operator.The eigenvalues obtain at {vn}n=0?,{?n}n=-??,which are the zeros of the denominator of the Weyl function.Second,we give the asymptotic formulae of{vn}n=0?,{?n}n=-??.Third,we give the trace formula of {vn}n=0?,{?n}n=-?? by residue theorem.Finally,we reconstruct the potential and the transmission condition from spectral data and other information by E.Trubowitz's method,which is also known as the shift of potential method.It is different to the situation that there is no transmission condition,we need to shift not only the potential but also the transmission condition.We find the transmission condition is determined by {?n}n=-??,then we reconstruct the potential by the trace formula.In chap-ter 3,we discuss the second topic,which is about the properties of the first eigenvalue of Laplace operator on a sphere.If the shape is under certain constraints,we give the condition of the shape for which the first eigenvalue gets its maximum...