Finite-Dimensional 𝒫𝒯-Symmetric Hamiltonians*

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This paper investigates finite-dimensional 𝒫𝒯-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex 𝒫𝒯-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the 𝒫𝒯 symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D > 2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional 𝒫𝒯-symmetric matrix Hamiltonian.

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