Simultaneous eigenstates of the number-difference operator and a bilinear interaction Hamiltonian derived by solving a complex differential equation

As a continuum work of Bhaumik et al.(3) who derived the common eigenvector of the number-difference operator Q (a(dagger)a-b(dagger)b) and pair-annihilation operator ab, we search for the simultaneous eigenvector of Q and (ab-a(dagger)b(dagger)) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Ref. 5 in which the common eigenkets of Q and pair creators a(dagger)b(dagger) are discussed.