Higher-order semiclassical energy expansions for potentials with non-integer powers

In this paper, we present a semiclassical eigenenergy expansion for the potential \x\(alpha) when a is a positive rational number of the form 2n/m (n is a positive integer and m is an odd positive integer). Remarkably, this expansion is found to be identical to the WKB expansion obtained for the potential x(N) (N-even), if 2n/m is replaced by N. Taking the limit m --> 2 of the above expansion, we obtain an explicit asymptotic energy expansion of symmetric odd power potentials \x\(2j+1) (j-positive integer). We then show how to develop approximate semiclassical expansions for potentials \x\(alpha) when alpha is any positive real number.