Improving the accuracy of WKB and Langer eigenvalues by higher-order variational method

Higher-order variational method is adapted for improving the accuracy of WKB and Langer eigenvalues by using the WKB and Langer eigenfunctions as trial functions. The results are compared with the first-order variational method using WKB and Langer functions as trial functions and with other approximate methods (finite element method, Langer, first-, third- and fifth-order WKB, first- and third-order WKB variational, first-order perturbation correction).