= Ground-state degeneracy
When a <ground state> exists, its ground-state degeneracy is the <dimension> of the <ground-state subspace>:
$$
g_0=\dim\ker(H-E_0I),\qquad E_0=\inf\sigma(H).
$$
Here $I$ is the <identity operator>, and the <kernel of a linear map> is taken within the <operator domain> of the <Hamiltonian operator> $H$. The count concerns <linearly independent> <energy eigenstates>, rather than every normalized <quantum superposition> or every choice of <global phase>. Even a single <energy eigenspace> contains infinitely many representing vectors; its <dimension> measures independent freedom. <Ground-state degeneracy> is the lowest-energy instance of <degenerate energy levels>.
For an exact finite model with <energy> unit $\varepsilon>0$, let the <Hamiltonian operator> be the <diagonal matrix>
$$
H_{\mathrm{toy}}=\varepsilon\operatorname{diag}(0,0,2,5).
$$
Its <ground-state energy> is zero and $g_0=2$: the first two coordinate <unit vectors> $e_1,e_2$ span the <ground-state subspace>. Every normalized $a e_1+b e_2$, with $|a|^2+|b|^2=1$, is a <ground state>, including $(e_1+e_2)/\sqrt2$. Its <energy measurement> is certain to return zero, even though a measurement distinguishing $e_1$ from $e_2$ need not have a definite outcome. Under the unchanged <Hamiltonian operator>, this <quantum superposition> is a <stationary state> because both components acquire the same <global phase>.
\Image[/ground-excited-states-spectral-bottom.png]
{title=Attained ground energies, ground-state degeneracy, and an unattained spectral bottom}
{description=Exact analytic models, not measured data. The <quantum harmonic oscillator> has an attained <ground-state energy>. The finite <diagonal matrix> has two independent <ground states> at one <energy eigenvalue>. The whole-line <free particle> has <continuous spectrum> beginning at zero and no normalizable zero-energy <energy eigenstate>; the shaded interval denotes spectral values, not a collection of normalized <energy eigenstates>. Each panel uses its own stated <energy> unit.}
<Ground-state degeneracy> matters when an interaction selects or splits previously equal <energy eigenvalues>. In the finite model, adding $\varepsilon\operatorname{diag}(\delta,-\delta,0,0)$ with $0<\delta<2$ gives the formerly lowest pair the distinct energies $\pm\varepsilon\delta$, leaving a unique <ground state>. This direct calculation illustrates why <degenerate perturbation theory> first examines the perturbation inside the whole <ground-state subspace>.
A classical continuous-symmetry model can instead have a <manifold> of minimizing configurations, such as the sphere of orientations of a <Néel state> in an isotropic <Heisenberg antiferromagnet>. That geometric description does not by itself count the <dimension> of a finite quantum <ground-state subspace>.
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