On the ℏ2correction terms in quantum integrability

Abstract

The authors study the question of quantum integrability for two-dimensional Hamiltonian systems with special attention on the 2 correction terms in the potential. A class of Hamiltonians of type H=1/2(px2 + py2)+ v4y4 + y2(f2(x) + α 6) + f0(x) with a second invariant of type I = px4 + A(x,y)px2 + B(x,y)pxpy + C(x,y) py2 + D(x,y) is considered. The general solution for f2 involves elliptic integrals. For quantum integrability the potential must be modified with 2-dependent terms. They construct a point transformation which, after coupling constant metamorphosis, takes the Hamiltonian to a new quantum integrable Hamiltonian for which no correction terms are necessary. The new system does not in general have a flat space kinetic part.