Geometric spectral inversion for singular potentials

TL;DR

This paper introduces a spectral inversion method using envelope theory to reconstruct potential functions from eigenvalue data, with analytical solutions for specific cases and numerical convergence for others, demonstrated on singular potentials.

Contribution

It develops a novel inversion technique for Schrödinger potentials from spectral data, including analytical solutions for power-law and log cases, and proves uniqueness for a class of potentials.

Findings

Analytical inversion for power-law and logarithmic potentials in two steps.

Numerical convergence of the inversion method for other potential shapes.

Uniqueness of potential reconstruction from ground-state energy curves for certain attractive potentials.

Abstract

The function E = F(v) expresses the dependence of a discrete eigenvalue E of the Schroedinger Hamiltonian H = -\Delta + 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 Hulth\'en 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.