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

TL;DR

This work extends semiclassical quantization to nth-order Schrödinger-type equations with even order $n$, deriving a general bound-state condition via turning-point analysis, Frobenius expansions, and a Bessel-function representation. By enforcing continuity and symmetry constraints on the local solutions, the authors obtain the universal rule $\int_{L_E}^{R_E} k_0 dx=(N+\tfrac{1}{2})\pi$ (or $\oint k_0 dx=(2N+1)\pi$) with $k_0=(E-V)^{1/n}$, valid across both parity families $n=2\, (2n_c)$ and $n=2\, (2n_c+1)$. The framework unifies Schrödinger and Bogoliubov-de Gennes equations, and extends to multipartite BdG systems, offering a practical semiclassical tool for estimating bound-state energies in wide, smoothly varying wells. The approach relies on controlling exponentially growing components and leverages phase matching of free-particle-like components to yield robust quantization relations that generalize the classic WKB result to higher-order differential equations.

Abstract

We prove a general approximate quantization rule $ \int_{L_{E}}^{R_{E}}k_0(x)$ $dx=(N+\frac{1}{2})π$ or $ \oint k_0(x)$ $dx=(2N+1)π$ (including both forward and backward processes) for the bound states in the potential well of the $n$th-order Schrödinger equations $ e^{-iπn/2}{{}\frac{d^nΨ(x)}{d x^n} } =[E-{} V(x)]Ψ(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.

Figures (4)

• Figure Appendix Figure 1: Illustration of a bound state $\Psi(x)$ in a potential well ${{} V}(x)$. The bound state wave function $\Psi(x)$ dissipates gradually within the barrier ${{} V(x) }$.
• Figure Appendix Figure 2: The ranges of exponentially growing components (purple), free particle states, and dissipating components (indigo).
• Figure Appendix Figure 1: The first four bound states for $n=4$. Note that there are tails or small oscillations when the wave functions are damped to zero, which are the high-order effect and absent in the Schrödinger equation as expected from our asymptotic analysis.
• Figure Appendix Figure 2: (a) The double-well potential for ${{} V(x) }=x^4-5x^2$; (b) the intersection points between $\tan \phi_1$ and $\pm2e^{\phi_2}$ are the bound states in (a). $x_1$ and $x_2$ are the intersection points of $E$ and ${{} V(x) }$ on the right-side well. (c) The double-well potential for ${{} V(x) }=x^4-10x^2$. (d) The intersection points between $\tan \phi_1$ and $\pm2e^{\phi_2}$ are the bound states in (c). n=4 for all plots.

Theorems & Definitions (8)

• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Remark 4.5
• Remark 4.6
• Remark 4.7
• Remark 4.8