Meromorphic N=2 Wess–Zumino supersymmetric quantum mechanics

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Abstract
The ordinary (holomorphic) N=2 Wess–Zumino model in supersymmetric quantum mechanics is extended to the case where the superpotential V(z) is a meromorphic function on C■{∞}. The extended model is analyzed in a mathematically rigorous way. Self-adjoint extensions and the essential self-adjointness of the supercharges are discussed. The supersymmetric Hamiltonian defined by one of the self-adjoint extensions of the supercharges has no fermionic zero-energy states (‘‘vanishing theorem’’). It is proven that if V(z) has only one pole at z=0 in C, then the supercharges are essentially self-adjoint on C∞0(R2■{0};C4). The special case where V(z)=λz−p(p∈N,λ∈C■{0}) is analyzed in detail to prove the following facts: (i) the number of the bosonic zero-energy ground state(s) is equal to p−1; (ii) the supercharges are not Fredholm.