奇异位的几何谱反演
Geometric spectral inversion for singular potentials
摘要：
函数E = F（v）表示薛定谔哈密顿量H = -Δ+ vf（r）的离散本征值E对耦合参数v的依赖性。我们使用包络理论生成函数序列\{f^{[k]}（r）\}以从种子势f^{[0]}（r）开始从F（v）重构f（r）。在幂律或对数情况下，反演可以通过分析来实现，并且只需两个步骤即可完成。在其他情况下，收敛观察数值。为了提供反演方法的具体说明，首先将其应用于Hulthén势，然后将其用于反演由形状为f（r）= -a/r + b \sgn（q）r^q和f（r）= -a/r + b \ln（r），a，b > 0的奇异势生成的谱数据。对于形状为f（r）= g（r）/r，g（0）< 0，g '（r）\ge 0的吸引中心势类，我们证明了基态能量曲线F（v）唯一地决定f（r）.
英文摘要：
The function E = F(v) expresses the dependence of a discrete eigenvalue E of the Schroedinger Hamiltonian H = -Δ+ vf(r) on the coupling parameter v. We use envelope theory to generate a functional sequence \{f^{[k]}(r)\} to reconstruct f(r) from F(v) starting from a seed potential f^{[0]}(r). In the power-law or log cases the inversion can be effected analytically and is complete in just two steps. In other cases convergence is observed numerically. To provide concrete illustrations of the inversion method it is first applied to the Hulthén potential, and it is then used to invert spectral data generated by singular potentials with shapes of the form f(r) = -a/r + b\sgn(q)r^q and f(r) = -a/r + b\ln(r), a, b > 0. For the class of attractive central potentials with shapes f(r) = g(r)/r, with g(0)< 0 and g'(r)\ge 0, we prove that the ground-state energy curve F(v) determines f(r) uniquely.
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