The Gram matrix of a PT-symmetric quantum system ∗)

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. PACS: 03.67.–w Key words: Gram Matrix, PT symmetry, bi-orthonormal basis The spectrum of a non-hermitean Hamiltonian H ̂ is real if the Hamiltonian is invariant under the combined action of self-adjoint parity P and time reversal T, [Ĥ,PT] = 0, (1) and if the energy eigenstates are invariant under the operator PT [1]. Pairs of com-plex conjugate eigenvalues are also compatible with PT symmetry but the eigen-states of H ̂ are no longer invariant under PT. Wigner’s representation theory of anti-linear operators [2], when applied to the operator PT [3], explains these ob-servations in a group-theoretical framework. Alternatively, they follow from the properties of pseudo-Hermitean operators [4] satisfying ηĤη−1 = Ĥ † equivalent to Eq. (1) if η = P. Consider a (diagonalizable) non-Hermitean Hamiltonian H ̂ with a discrete spec-trum [5]. The operators H ̂ and and its adjoint Ĥ † have complete sets of eigenstates: Ĥ|En 〉 = En|En 〉 , Ĥ†|En 〉 = En|En 〉 , n = 1, 2,..., (2) with, in general, complex conjugate eigenvalues, En = E∗n. The eigenstates consti-tute bi-orthonormal bases in H with two resolutions of unity,∑ n |En〉〈En |