AI 中文总结
该研究分析了由变形超球支撑的秩一δ相互作用的束缚态，推导了未变形超球的束缚态方程，得出一阶近似下束缚态能量仅依赖平均法向位移的核心结论。
AI 中文摘要
我们研究由n维超球S_R^n⊂ℝ^{n+1}的小法向变形所支撑的秩一δ相互作用，该相互作用由支撑上的归一化表面测度定义，对应于超球壳问题的旋转不变扇区。对于未变形的超球，我们根据修正贝塞尔乘积I_{(n-1)/2}(νR)K_{(n-1)/2}(νR)推导出E=-ν²的束缚态方程。随后考虑内法向变形X_ε(ω)=(R-εh(ω))ω，其中ω∈S^n，0<ε≪1。我们的主要结果是，在ε的一阶近似下，束缚态能量仅依赖于平均法向位移⟨h⟩=1/|S^n|∫_{S^n}h(ω)dΩ_n(ω)。等价地，变形超球在O(ε²)精度内谱等价于有效半径为R-ε⟨h⟩的圆超球，特别地，均值为零的变形不会在一阶近似下改变秩一束缚态能量。
英文摘要
We study rank-one delta interactions supported by small normal deformations of an \(n\)-dimensional hypersphere \(S_R^n\subset\mathbb{R}^{n+1}\). The interaction is defined by the normalized surface measure on the support and therefore corresponds to the rotationally invariant sector of the hyperspherical shell problem. For the undeformed hypersphere, we derive the bound-state equation for \(E=-ν^2\) in terms of the modified Bessel product \[ I_{\frac{n-1}{2}}(νR)K_{\frac{n-1}{2}}(νR). \] We then consider an inward normal deformation \[ X_\varepsilon(ω)=(R-\varepsilon h(ω))ω, \qquad ω\in S^n, \qquad 0<\varepsilon\ll1. \] Our main result is that, to first order in \(\varepsilon\), the bound-state energy depends only on the average normal displacement \[ \langle h\rangle = \frac1{|S^n|}\int_{S^n}h(ω)\,dΩ_n(ω). \] Equivalently, the deformed hypersphere is spectrally equivalent, up to \(O(\varepsilon^2)\), to a round hypersphere with effective radius \(R-\varepsilon\langle h\rangle\). In particular, mean-zero deformations do not change the rank-one bound-state energy at first order.
Comments17 pages
