The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

Abstract

We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supplied by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

14 pages

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem

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