= Solution
Put $d_k=\sqrt{(g-\cos k)^2+\sin^2k}$. The matrix in the <Ising-chain Nambu spinor> has eigenvalues $\pm d_k$. A canonical <Bogoliubov transformation>, with paired angles satisfying $\cos\vartheta_k=(g-\cos k)/d_k$ and $\sin\vartheta_k=-\sin k/d_k$, diagonalizes it. Accounting for <full-zone fermionic Nambu energy counting> gives
$$
\boxed{H=\sum_k\epsilon_k\left(\gamma_k^\dagger\gamma_k-\frac12\right),\qquad
\epsilon_k=2Jd_k.}
$$
Thus the bulk <Ising-chain ground-state energy> is
$$
\boxed{E_0=-\frac12\sum_k\epsilon_k,\qquad
\frac{E_0}{N}\longrightarrow-\frac{J}{2\pi}\int_{-\pi}^{\pi}\sqrt{1+g^2-2g\cos k}\,dk.}
$$
Exact finite-chain states additionally obey their <fermion parity> sector.
The <Ising-chain quasiparticle dispersion> has bulk gap $2J|1-g|$, closing at the <Ising-chain quantum critical point>. Finite-system parity splittings and the bulk gap are distinct scales. The three requested limits are illustrated below.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2012/paper-1-qcm2012-ising-bands.png]
{title=Ising quasiparticle bands}
{source=paper-1-qcm2012-ising-bands.py}
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