Quantum states of a sextic potential: hidden symmetry and quantum monodromy

Abstract

Abstract. A known terminating polynomial ansatz for a finite sub-set of states of the sextic central potential V (a, b, c; r) = ar2 + br4 + cr6, c> 0, subject to integer related parameter constraints, is confirmed to apply in two as well as three dimensions. A hidden symmetry relating the eigenstates of V (a, b, c; r) to those of V (a,−b, c; r) is used to establish the boundary between the ansatz sub-set and the infinite set of higher states. Connections between the radial wavefunctionsRn(a, b, c; r) and Rn ′ (a,−b, c; r) also provide semiclassical insight into the origin of the parameter constraint. Finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy. 1.