General comparison theorem for eigenvalues of a certain class of Hamiltonians

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form (T + V)|psi> = E|psi>, where T is a kinetic part which depends only on momenta and V is a potential which depends only on positions. We assume that H^(1) = T + V^(1) and H^(2) = T + V^(2) (H^(1) = T^(1) + V and H^(2) = T^(2) + V) support both discrete eigenvalues E^(1)_{a} and E^(2)_{a}, where {a} represents a set of quantum numbers. We prove that, if V^(1) <= V^(2) (T^(1) <= T^(2)) for all position (momentum) variables, then the corresponding eigenvalues are ordered as E^(1)_{a} <= E^(2)_{a}. Some analytical applications are given.