The Gram Matrix of a PT-symmetric Quantum System

Abstract

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.

arXiv comment3 pages, no figures

References in corpus (5)

• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Space of State Vectors in PT Symmetrical Quantum Mechanics
• Completeness and Orthonormality in PT-symmetric Quantum Systems
• Solvable simulation of a double-well problem in PT symmetric quantum mechanics

Cited by in corpus (6)

• A return to observability near exceptional points in a schematic PT-symmetric model
• Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
• Experiments in PT-symmetric quantum mechanics
• Linear representation of energy-dependent Hamiltonians
• Quantumness of pure-state ensembles via coherence of Gram matrix based on generalized $\alpha$-$z$-relative Rényi entropy
• PT-symmetric quantum models living in an auxiliary Pontryagin space