Related papers: Quantization Condition of the Bound States in $n$th-order Schr\"{o}dinger equations

Quantization Condition of the Bound States in $n$th-order Schr\"{o}dinger equations

URL: http://arxiv.org/abs/2304.00914v2
Date: Tue, 25 Apr 2023 04:31:31 GMT
Title: Quantization Condition of the Bound States in $n$th-order Schr\"{o}dinger equations
Authors: Xiong Fan
Abstract summary: We will prove a general approximate quantization rule $% int_L_ER_Ek. The only hypothesis is that all exponentially growing components are negligible, which is appropriate for not narrow wells.
Score: 2.5822051639377137
License: http://creativecommons.org/licenses/by/4.0/
Abstract: We will prove a general approximate quantization rule $% \int_{L_{E}}^{R_{E}}k_0$ $dx=(N+\frac{1}{2})\pi $ for the bound states in the potential well of the equations $e^{-i\pi n/2}\nabla_x ^{^{n}}\Psi =[E-\Delta (x)]\Psi ,$ where $k_0=(E-\Delta )^{1/n}$ 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.

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