On the degeneracy in the ground state of the N=2 Wess–Zumino supersymmetric quantum mechanics

It is known that the N=2 Wess–Zumino supersymmetric quantum mechanical model has p–1 degenerate zero-energy ground states consisting of only bosonic states, where p>3 is the degree of the polynomial superpotential V(z)(z∈C) of the model [Jaffe et al. Ann. Phys. (NY) 178, 313 (1987)]. In this paper, the mathematical structure of the degenerate ground states is analyzed in the special case V(z)=λz^p (λ>0). The following facts are discovered: (i) there exists a strongly continuous one parameter unitary group acting as a symmetry group in the quantum system under consideration; (ii) the generator of the symmetry group has infinitely many eigenspaces Hm, m∈Z, and the bosonic part H+ of the Hamiltonian of the model is reduced by each of them; and (iii) there exist exactly p–1 Hm's in each of which the reduced part of H+ has a unique zero-energy ground state. It is noted also that H+ has infinitely many generalized eigenfunctions with eigenvalue zero. Moreover, a family of operators interrelating the zero-energy ground states is constructed. The coupling constant dependence of the nonzero eigenvalues of H+ is exactly found