Inverse resonance problem with partial information on the interval

Abstract

We consider the inverse resonance problem in scattering theory. In one-dimensional setting, the scattering matrix consists of $2\times 2$ entries of meromorphic functions. The resonances are defined as the poles of the meromorphic determinant. For the compactly supported perturbation, we are able to quantitatively estimate the zeros and poles of each meromorphic entry. The size of potential support is connected to the zero density of scattered wave field due to the form of Fourier transform. We will investigate certain properties of Fourier transforms in scattering theory and derive the inverse uniqueness on scattering source given certain knowledge on the perturbation and all the given resonances.

Keywords:

• Resonance
• wave scattering
• Schrödinger equation
• value distribution theory
• Faddeev theory

2010 Mathematics Subject Classifications:

• 34B24
• 35P25
• 35R30

Disclosure statement

No potential conflict of interest was reported by the author(s).