Inverse resonance problem on the line for perturbations of P\"oschl-Teller potentials

We study an inverse resonance problem on the line in which we aim at determining a compactly supported and integrable perturbation of a fixed P\"oschl-Teller potential. We define the resonances as the poles of the reflection coefficients with a negative imaginary part. Given the zeros and the poles of one of the reflection coefficients, we are able to determine uniquely the perturbation of a P\"oschl-Teller potential when its support is to the left or to the right of zero on the line and also in the remaining cases under the addition of other hypothesis or extra spectral data. We also give asymptotics of the resonances and show that they are asymptotically located on two logarithmic branches.