Critical Temperatures of Excited Holographic Superconductors in Born--Infeld Electrodynamics on a Reissner--Nordström--AdS Background

We study the ground state (GS) and the two lowest excited states (ES1, ES2) of a probe holographic superconductor with Born-Infeld (BI) electrodynamics on a charged Reissner-Nordstrom-AdS (RN-AdS) black hole background. Distinguishing these background geometries by their extended-thermodynamic pressure P=-Lambda/8pi requires the electric charge Q: the neutral Schwarzschild-AdS metric admits only one positive horizon for any M,L^2>0 and cannot support the SBH/LBH transition. Working on the correctly identified RN-AdS geometry, we set up the scalar/BI-gauge probe sector, formulate the linear eigenvalue problem for the onset of condensation, and solve it by a shooting method. We obtain T_c of GS, ES1, ES2 at three representative pressures (P>P_c, P=P_c, P