Quantization Condition of the Bound States in $n$th-order Schrödinger equations

Abstract: We prove a general approximate quantization rule $ \int_{L_{E}}^{R_{E}}k_0(x)$ $dx=(N+\frac{1}{2})\pi $ or $ \oint k_0(x)$ $dx=(2N+1)\pi $ (including both forward and backward processes) for the bound states in the potential well of the $n$th-order Schr\"{o}dinger equations $ e^{-i\pi n/2}{{}\frac{d^n\Psi(x)}{d x^n} } =[E-{} V(x)]\Psi(x) ,$ where ${} k_0(x)=(E-V(x) )^{1/n}$ with $N\in\mathbb{N}{0} $, $n$ is an even natural number, and $L{E}$ and $R_{E}$ the boundary points between the classically forbidden regions and the allowed region. The only hypothesis is that all exponentially growing components are negligible, which is appropriate for not narrow wells. Applications including the Schr\"{o}dinger equation and Bogoliubov-de Gennes equation will be discussed.