Proof of non-extremal nature of excited-state energies in   nonrelativistic quantum mechanics with the use of constrained derivatives
  With the use of a second derivative test based on constrained second derivatives, a proof is given that excited states in nonrelativistic quantum mechanics are saddle points of the energy expectation value, and is shown, further, how to determine their Morse index. 
Tamas Gal
math-ph math.MP quant-ph