A ground state of a quantum Hamiltonian 
is a quantum
state supported on its lowest eigenspace. For a Hamiltonian
bounded below, put 
,
where 
is its operator spectrum. A pure ground state
is represented by a unit vector 
satisfying 
. The infimum need
not be an eigenvalue, in which case no normalizable
ground state exists. A ground state is unique when this eigenspace has dimension
one.
For a unique ground vector, a spectral gap of at least 
means
 |
This operator inequality tests every vector orthogonal
to 
, rather than one trial excitation.
