Constraining self-interacting scalar field dark matter with strong gravitational lensing in cluster scale?

We present a method to investigate the properties of solitonic cores in the Thomas-Fermi regime under the self-interacting scalar field dark matter framework. Using semi-analytical techniques, we characterize soliton signatures through their density profiles, gravitational lensing deflection angles, and surface mass density excess in the context of strong lensing by galaxy clusters. Focusing on halos spanning two mass scales -- $M_{200}= 2 \cdot 10^{15}\rm M_\odot$ and $2 \cdot 10^{14} \rm M_\odot$ -- we compute lensing observables to assess the viability of the SFDM model. Our analysis establishes constraints on the soliton core mass, directly probing the self-interaction parameter space of scalar field dark matter. This work bridges semi-analytical predictions with astrophysical observations, offering a lensing-based framework to test ultralight dark matter scenarios in galaxy cluster environments.