The Gram Matrix of a 𝒫𝒯-Symmetric Quantum System*

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The eigenstates of a diagonalizable 𝒫𝒯-symmetric Hamiltonian satisfy unconventional completeness and orthonormality relations. These relations reflect the properties of a pair of bi-orthonormal bases associated with non-hermitean diagonalizable operators. In a similar vein, such a dual pair of bases is shown to possess, in the presence of 𝒫𝒯 symmetry, a Gram matrix of a particular structure: its inverse is obtained by simply swapping the signs of some its matrix elements.

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