= Ground-state energy
The ground-state energy $E_0$ is the <energy eigenvalue> attained by a <ground state>. For a <self-adjoint operator> $H$ bounded below, distinguish this attained value from the always-defined spectral bottom $E_*=\inf\sigma(H)$. When a <ground state> exists, $E_0=E_*$. Some usage calls $E_*$ the <ground-state energy> even when it is unattained; here an unattained bottom is explicitly identified as an <infimum>. The <quantum variational principle> states
$$
E_*=\inf_{\substack{\psi\in D(H)\\\|\psi\|=1}}\langle\psi,H\psi\rangle.
$$
Approaching this <infimum> with trial <wavefunctions> does not ensure that a minimizing <wavefunction> exists. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf[Teschl, Mathematical Methods in Quantum Mechanics, Theorem 2.19]{ref}.
For the whole-line <free particle> with mass $m>0$,
$$
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2},\qquad \sigma(H)=[0,\infty).
$$
Its <spectrum> is continuous and contains no <energy eigenvalues> represented by <normalizable wavefunctions>. The <Fourier transform> makes $H$ multiplication by $\hbar^2 k^2/(2m)$; a fixed <energy> restricts $k$ to a set of zero length, on which a nonzero <square-integrable function> cannot be supported. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf[Teschl, Mathematical Methods in Quantum Mechanics, section 7.2, Theorem 7.8]{ref}. An explicit family of normalized <Gaussian wave packets> demonstrates how the bottom is approached:
$$
\psi_s(x)=(2\pi s^2)^{-1/4}e^{-x^2/(4s^2)},\qquad \langle H\rangle_s=\frac{\hbar^2}{8ms^2}\longrightarrow0\quad(s\longrightarrow\infty).
$$
The formula follows from $\psi_s'=-x\psi_s/(2s^2)$, <integration by parts>, and $\langle x^2\rangle_s=s^2$. Every finite width gives positive <kinetic energy>. At zero <energy>, $H\psi=0$ would require an affine function on the whole line, which is never a nonzero <normalizable wavefunction>. Spreading a <Gaussian wave packet> indefinitely produces an unattained <infimum>, rather than a zero-energy <ground state>.
An additive choice of <energy> origin changes the numerical <ground-state energy> without changing the <ground-state subspace>. For $H'=H+C I$, every <energy eigenvalue> becomes $E'=E+C$, while all differences $E_j-E_i$ stay fixed. Time evolution gains only the <global phase> $e^{-iCt/\hbar}$. This makes excitation costs and spectroscopic differences more intrinsic than the isolated number $E_0$.
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