 usepackage{amsbsy}
+ [mathcal{D}_2]_inftyleft(
begin{array}{c}
frac{1}{2}\
vdots\
0
end{array}
right)
+mathbb{E}left[mathrm{Tr}left(mathcal{tilde{D}}_2right)right],endaligned\$\$ where \$mathbb Eleft[Xright]\$ is the expectation of \$X\$ over the random variables \$X\$. The first term in the right hand side of Eq. ([eq:delta2]) is the leading term in Eq.([eq1]), and the second term is the next-to-leading term. The second term in this equation is the sum of the second and third terms. The third term is a constant that does not depend on \$X\$, and the fourth term is an additional constant that depends on \$N\$. This constant is the second-order correction to the leading order. The constant in the fourth line of Eqs. (1) and (2) is the same as the one in Eqs.(ref{eq:D2}), (ref {eq:tilde D2}). The constant is equal to \$1/2\$ if the number of sites in the system is even, and to \$0\$ if it is odd.
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The leading term of the density of states of the system can be calculated by using the fact that the density operator \$mathcal D_2\$ is diagonal in the basis of the eigenstates of the Hamiltonian. The density of the states of a system with \$N\$ sites is equal \$\$label{eq1}
n(omega)=sum_{i=1}^{N}delta(omeg_i-omega).\$\$ The density operator is diagonalized by the transformation \$\$mathbf{S}_i=left{
begin{aligned}
label {eq2}
\
\ &1,& qquad i=N+1,N+2,ldots,2N.
nonumber
right.\$\$ The transformed density operator can be written as \$\$begin {split}
&mathsf{D_2}(omegas)=mathop{sum_{omegas_1, dots omegas_{2N}}}_{mathclap{substack{omegas’_1delta_{{1,2}},dagger_{{omeg 1,omeg 2}}}}}\
&{timesmathfrak{S}}_{left(omegs_1+omegs’_2, omega_{rm max}+domegaright)}
left((omegam_1-dgomegal_1)^*dghomegt_1(omegam’_3-dhomegat_3)^Tright)\
&timesdgt_2(omegal’_4-hgdgal’omegap_4),
qed
mathscr{D_{2}}(omegap)=frac 1{2sq{N}}sum_idgd_i(omegp_i)^2,
end{split}\$\$ where the subscripts of the transformed density operators are the indices of the sites. The transformed densities are calculated by the following equations: \$\$dgp_1=dgs_2-frac 12dgb_1^2dga_1;quad dgp_{2i}=frac dgb_{2,i}-left|fracdgc_{2,i}right|^2-2frac {dgd_{2}^2}{dgas_{1}^*}dgb^*_1.\$\$
The density matrix of the reduced density operator of the whole system is equal \$
rho_N=mathsfs{D^dag_2}mathsf D_1
\$ and the density matrix is diagonal.
The density of a single site is equal: \$\$n_1(0)=dtg_2^2+frac 14dgg_2+2left.frac {dmathds{1}}{domegp}rightvert_{dgap=0}dgp.\$\$ This is the density in the case of the even number of site.