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

[摘要] In this paper, we present a semiclassical eigenenergy expansion for the potential |𝑥|𝛼 when 𝛼 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 𝑥𝑁 (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|𝛼 when 𝛼 is any positive real number.

[学科分类] 物理（综合）

[关键词] Eigenenergies;asymptotic;WKB;polynomial potentials.