The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

We propose a unique way 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 minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer 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 supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.