Abstract
We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.
Highlights
At first sight it is suprising that a subset of eigenvalues of a complex-valued non-hermitian operator Hcan be real
We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it
The procedure followed by Bender et al [2] for the construction of the suitable basis set is reminiscent of the one used by Porter [3] in the study of matrix representations of Hermitian operators
Summary
At first sight it is suprising that a subset of eigenvalues of a complex-valued non-hermitian operator Hcan be real (see [1] and references therein). In order to provide a simple and general explanation of this fact Bender et al [2] showed that it is possible to construct a basis set of vectors so that the matrix representation of such an operator is real. We illustrate this point by means of the well known harmonic-oscillator basis set and show how to overcome that shortcoming.
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