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A physically meaningful limit of an infinite particle number can be obtained as the limit d/Δ0→0, where d is the mean single-particle level spacing d=j+1-j and Δ0 is the ground state average Δ0=gb. The quantity Δ0 remains finite in the thermodynamic limit and plays the role of the BCS gap. The interaction strength g should be scaled as g=λd, where λ is a dimensional constant, so that the total energy of the system is an extensive quantity.
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The coordinate, x and the momentum, p of the oscillator are related to the fields b and b̄ as x=2bx=(b+b̄)/2 and p=2by=(b̄-b)/(2i). Then, we have the usual coordinate-momentum (canonical) brackets {x,p}=1, while the second term in the Hamiltonian 2 3 reads b̄b=p2/2+x2/2.
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Note that if the root E0 is real, it has to be degenerate, since according to Eq. 5 9 Q2n+2(u)≥0 for real u. In this case L2(E0)=0 implies Lz(E0)=L-(E0) and, since zeros of L-(u) are separation variables, one of them has to be frozen at E0.
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Note that if the root E0 is real, it has to be degenerate, since according to Eq. 5 9 Q2n+2(u)≥0 for real u. In this case L2(E0)=0 implies Lz(E0)=L-(E0) and, since zeros of L-(u) are separation variables, one of them has to be frozen at E0.
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