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.
