A note on the eigenfunction in a relativistic closed Toda chain

Recently, the eigenfunction for an N-particle relativistic closed Toda chain has been calculated by association with gauge theory. The results involve only the behavior of the eigenfunction of the energy level but do not include nonphysical energy. A conjecture was thus proposed to determine the eigenfunction of the nonphysical energy and a numerical calculation was conducted to prove that it is impossible to get an entire eigenfunction of the nonphysical energy. Because of the pole cancellation condition, the relation between the open and closed Bogomol’nyi-Prasad-Sommerfield (BPS) invariants could be derived.