Geometric Spectral Inversion

We consider a Schrödinger Hamiltonian operator H = -Delta + vf(x), where f(x) is the shape of an attractive potential, and v > 0 is the coupling parameter. We suppose that the dependence of a discrete eigenvalue E on v is given by the trajectory function E = F(v). Geometric spectral inversion is the reconstruction of f from F. Various approaches to this problem have been studied.

For example, if f(x) is negative, bounded, symmetric, monotone increasing to zero for x > 0, and has area, then the given F(v) can be normalized by scale changes so that, without loss of generality, we can assume f(0) = -1, and the area under f(x) is -2. If F(v) is the ground-state trajectory for this problem, then, for small v, F(v) ~ -v², and, for large v, F(v)/v ~ -1. Hence, superficially, the whole class of energy trajcetories look rather similar. Coded in the fine details of each energy curve is all the information needed to reconstruct the corresponding f(x), which might be as different as the square well from, say, f(x) = -sech²(x). The higher trajectories F_n(v) (or even the bottom trajectory for dimension > 1), only exist for v sufficiently large. This might lead one to suspect that these trajectories do not contain enough information for the reconstruction of f(x). However, WKB studies strongly suggest that full reconstruction is not only possible but, in a sense, becomes more efficient, as n increases and the problem becomes more classical in nature.

Specific topics and the locations of their treatment in this work:

Concentration lemma: as v increases, the wave function concentrates towards x = 0  [59]

Potentials with flat bottoms: reconstructing the size of the flat piece  [59]

Refinements to the comparison theorem of QM: intersecting potentials  [52][68]

Normalization of potentials, uniqueness theorem  [58]

Square-well trajectories are extremal and bound the domain of the inverse  [58]

Variational inverse  [58]

Inverse for separable potentials  [58]

Inversion from higher trajectories by WKB  [61]

Constructive inversion algorithm  [71]

An inversion inequality if f(x) = g(x²)  [73]