Sturm-Schroedinger equations: formula for metric

TL;DR

This paper presents a new, simplified formula for the metric operator in Sturm-Schroedinger equations, improving the computational approach by replacing a complex double-series with a single-series expression.

Contribution

It introduces an amended, single-series definition of the metric operator, addressing a key numerical challenge in the theory of non-Hermitian Sturm-Schroedinger equations.

Findings

Simplified the metric operator calculation from double-series to single-series.

Enhanced the numerical stability and efficiency of solving Sturm-Schroedinger equations.

Provided a theoretical foundation for more practical applications of non-Hermitian quantum mechanics.

Abstract

Sturm-Schroedinger equations $H\psi=E W\psi$ with $H\neq H^\dagger$ and $W\neq W^\dagger \neq I$ are considered, with a weak point of the theory lying in the purely numerical matrix-inversion form of the double-series definition of the necessary metric operator $\Theta$ in the physical Hilbert space of states [M. Znojil, J. Phys. A: Math. Theor. 41 (2008) 215304]. This shortcoming is removed here via an amended, single-series definition of $\Theta$.