Fractional Borg-Levinson Problem with small non-negative potential of small growth

TL;DR

This paper proves the unique recovery of small, non-negative potentials in a fractional inverse spectral problem, specifically within the fractional Borg-Levinson framework for exponents between 0.5 and 1.

Contribution

It establishes the uniqueness of potential recovery in the fractional Borg-Levinson problem under smallness and growth conditions for the potential.

Findings

Potential can be uniquely determined from boundary spectral data.

The fractional exponent is restricted to (0.5, 1) for technical reasons.

Smallness of the potential depends only on domain and dimension.

Abstract

In this article, we investigate the fractional Borg-Levinson problem, an inverse spectral problem focused on recovering potentials from boundary spectral data. We demonstrate that the potential can, in fact, be uniquely determined by this data. However, for technical reasons, we restrict the fractional exponent to the interval (0.5, 1). Additionally, we assume that at least one of the potentials is small, non-negative, and exhibits mild growth. The smallness condition is made explicit in our calculations and depends only on the domain and the spatial dimension.