Entropy Maximization, Cutoff Distribution, and Finite Stellar Masses

Table 1

Properties of one-body semiclassical spectrum labeled by the integer j for four solvable potentials 𝑊(𝑟) in D dimensions. The well-depth 𝑊₀≡−|𝑊(0)| and the range R of the rectangular, linear, and oscillator wells are made finite using the unit step function. The Coulomb well is left unrestricted over all r. For other notations, see Appendix A. (We do not tabulate the rigid box potential model which has the upper end of the energy spectrum at 𝜀_𝐽=𝐸−(𝑁−1)𝑊₀≈+∞. Such a model is of little interest in cosmology In usual physical applications, one puts 𝐷=3.)

Potential	Rectangular	Linear	Oscillator	Coulomb
Form 𝑊(𝑟)=	𝑊0step(𝑅−𝑟)	𝑊0𝑟{1−𝑅}⋅step(𝑅−𝑟)	𝑊0𝑟{1−^2𝑅^2}⋅step(𝑅−𝑟)	𝐶𝑟=𝜀1𝑟1𝑟:0≤𝑟≤∞
Highest label 𝐽=	(8𝑚|𝑊0|)^1/2𝑅𝜋ℏ	43⋅(2𝑚|𝑊0|)^1/2𝑅𝜋ℏ	(𝑚|𝑊0|2)^1/2𝑅ℏ	∞
Ionization threshold 𝜀𝐽=	0	0	0	0=𝜀∞
Spectrum 𝜀𝑗=	𝑊0𝑗{1−(𝐽)^2}	𝑊0𝑗{1−(𝐽)^2/3}	𝑊0𝑗{1−(𝐽)}	−𝑚𝐶^22ℏ^2𝑗^2=𝜀1𝑗^2
Turning point 𝑟𝑗=	𝑅	(𝑗𝐽)^2/3𝑅	(𝑗𝐽)^1/2𝑅	2ℏ^2𝑗^2𝑚|𝐶|=𝑟1𝑗^2
Separation Δ=	2|𝑊0|𝑗𝐽^2	23⋅|𝑊0|𝑗^−1/3𝐽^2/3	|𝑊0|𝐽	2|𝜀1|𝑗^3
Geometrical coefficient𝐵=	𝜋^𝐷2^2𝐷−2𝐷^2Γ^2(𝐷/2)	(3𝜋/4)^𝐷Γ(𝐷)2^𝐷−1Γ(𝐷/2)Γ(3𝐷/2+1)	1Γ(𝐷+1)	1𝐷Γ(𝐷/2+1/2)
Accumulated states 𝐼𝑗	𝐵𝑗^𝐷	𝐵𝑗^𝐷	𝐵𝑗^𝐷	𝐵𝑗^𝐷
Level degeneracy 𝑔𝑗=	𝐷𝐵𝑗^𝐷−1	𝐷𝐵𝑗^𝐷−1	𝐷𝐵𝑗^𝐷−1	𝐷𝐵𝑗^𝐷−1