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Condensed Matter > Strongly Correlated Electrons
Title: Quantization Condition of the Bound States in $n$th-order Schrödinger equations
(Submitted on 3 Apr 2023 (v1), last revised 25 Apr 2023 (this version, v2))
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.
Submission history
From: Xiong Fan [view email][v1] Mon, 3 Apr 2023 12:07:34 GMT (208kb,D)
[v2] Tue, 25 Apr 2023 04:31:31 GMT (209kb,D)
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