Existence of infinitely many zero-energy states in a model of supersymmetric quantum mechanics

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Abstract
The general framework of the N=2 Wess–Zumino holomorphic supersymmetric quantum mechanics with polynomial superpotentials is extended to the case of nonpolynomial superpotentials V(z) (z∈C) in a mathematically rigorous way. It is also proved that there exist no fermionic zero-energy states. Under some conditions for V, the operator domain of the supercharges and the supersymmetric Hamiltonian are identified. As an example, the model with V(z)=λeαz (λ∈C■{0}, α>0) is analyzed in view of index theory. The following remarkable result is proved: There exist infinitely many bosonic zero-energy states which are localized in the momentum space dual to the Im z direction. The results are applied to two models in atomic and nuclear physics.

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