论文标题
分形的几何形状和埃菲莫夫态映射到bloch状态
The fractal geometry and the mapping of Efimov states to Bloch states
论文作者
论文摘要
已知Efimov州具有离散的实际空间尺度不变性，在动量空间中工作，我们确定了定义其WeierStrass功能的散射幅度的相关离散量表不变性。通过将数学形式主义用于散射幅度的离散尺度不变性，我们从相应的Zeta函数的极点结构中标识了缩放参数，它的Zeroth Order Pole由Efimov物理固定。 Efimov物理在动量空间中的相应几何分形结构被识别为对数螺旋的射线。这种几何结构也出现在将其与Efimov物理学联系起来的相对论政权的原子崩溃物理学中。在动量空间中转换为对数变量，我们将三体散射幅度映射到Bloch状态，而Efimov状态的能量梯子仅获得了Bohr-Sommerfeld量化规则的Interms。因此，通过映射，三体短距离相互作用的复杂问题被转化为离散晶格中非相互作用的单个粒子的复杂问题。
Efimov states are known to have a discrete real space scale invariance, working in momentum space we identify the relevant discrete scale invariance for the scattering amplitude defining its Weierstrass function as well. Through the use of the mathematical formalism for discrete scale invariance for the scattering amplitude we identify the scaling parameters from the pole structure of the corresponding zeta function, it's zeroth order pole is fixed by the Efimov physics. The corresponding geometrical fractal structure for Efimov physics in momentum space is identified as a ray across a logarithmic spiral. This geometrical structure also appears in the physics of atomic collapse in the relativistic regime connecting it to Efimov physics. Transforming to logarithmic variables in momentum space we map the three-body scattering amplitude into Bloch states and the ladder of energies of the Efimov states are simply obtained interms of the Bohr-Sommerfeld quantization rule. Thus through the mapping the complex problem of three-body short range interaction is transformed to that of a non-interacting single particle in a discrete lattice.
