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 Schrödinger 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 Schrödinger equation and Bogoliubov-de Gennes equation will be discussed.