Symbolic-manipulation constructions of Hilbert-space metrics in quantum mechanics

TL;DR

This paper demonstrates how computer-assisted symbolic algebra can be used to construct and analyze Hilbert-space metrics that make a given Hamiltonian self-adjoint in quantum mechanics, using a specific exactly solvable example.

Contribution

It introduces a method leveraging symbolic algebra and numerics to solve Dieudonne's operator equation for Hilbert-space metrics in quantum systems.

Findings

Successfully constructed the metric for the Gegenbauerian quantum-lattice oscillator

Demonstrated the use of MAPLE for symbolic and numerical analysis

Facilitated the determination of the metric's positivity domain

Abstract

The problem of the determination of the Hilbert-space metric which renders a given Hamiltonian $H$ self-adjoint is addressed from the point of view of applicability of computer-assisted algebraic manipulations. An exactly solvable example of the so called Gegenbauerian quantum-lattice oscillator is recalled for the purpose. Both the construction of suitable metric (basically, the solution of the Dieudonne's operator equation) and the determination of its domain of positivity are shown facilitated by the symbolic algebraic manipulations and by MAPLE-supported numerics and graphics.