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Test of a modified BCS theory performance in the Picket Fence Model

Analyses of a modified BCS (MBCS) theory performance at finite temperatures in the Picket Fence Model (PFM) for light and heavy systems is presented. Both symmetric, $Ω=N$ ($N$ particles on $Ω$ twice-degenerate levels), and asymmetric, $Ω\neq N$, versions of the PFM are considered. The quantities known exactly from particle-hole symmetry of the $Ω=N$ PFM are calculated. Starting from very low temperatures, these quantities are found in dramatic deviation from the exact values in MBCS results. Consequences of the MBCS prediction that heating generates a thermal constituent of the pairing gap, are discussed. Thermodynamical inconsistency of the MBCS is also addressed.