Of Sturm-liouille Operator Eigenvalues ​​and The Characteristic Function Of The Exact Solution And The Trace Formula Calculation

In this paper, we discuss the asymptotic formulae for the eigenvalue and eigenfunction of Sturm-Liouville operator with all kinds of separated self-adjoint boundary condition. For the potential q∈Cm[0,π] m=2,3,we calculate the coefficients in asymptotic formulae.And more, we introduce a new concept of trace of the Sturm-Liouville operator and give expres-sions of the traces of the Sturm-Liouville operator with smooth potential.