Reconstruction of the potential from I-function

TL;DR

This paper establishes a one-to-one correspondence between the I-function, scattering data, and spectral function for Sturm-Liouville operators, providing analytical methods to reconstruct the potential from the I-function.

Contribution

It introduces a novel approach to reconstruct the potential of Sturm-Liouville operators using the I-function, linking it with scattering data and spectral functions.

Findings

I(k) uniquely determines the scattering triple and spectral function.

Analytical methods are developed for converting between I(k), scattering data, and spectral functions.

These methods enable potential reconstruction from the I-function.

Abstract

If $f(x,k)$ is the Jost solution and $f(x) = f(0,k)$, then the $I$-function is $I(k) := \frac{f^\prime(0,k)}{f(k)}$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple ${\mathcal S} :=\{S(k), k_j, s_j, \quad 1 \leq j \leq J\}$ and with the spectral function $\rho(\lambda)$ of the Sturm-Liouville operator $l= -\frac{d^2}{dx^2} + q(x)$ on $(0, \infty)$ with the Dirichlet condition at $x=0$ and $q(x) \in L_{1,1} := \{q: q= \bar q, \int^\infty_0 (1+x) |q(x) dx < \infty\}$. Analytical methods are given for finding $\mathcal S$ from $I(k)$ and $I(k)$ from $\mathcal S$, and $\rho(\lambda)$ from $I(k)$ and $I(k)$ from $\rho(\lambda)$. Since the methods for finding $q(x)$ from $\mathcal S$ or from $\rho(\lambda)$ are known, this yields the methods for finding $q(x)$ from $I(k)$.