On a special property of tridiagonal matrices. Application to dual quasi-exactly solvable sextic potentials in quantum mechanics

TL;DR

This paper presents a theorem about the eigenvalues of tridiagonal matrices and applies it to demonstrate a duality property in the spectra of certain quantum mechanical sextic potentials, providing both algebraic and quantum-mechanical proofs.

Contribution

The paper introduces a simple theorem on eigenvalue sign changes in tridiagonal matrices and applies it to establish a duality property in quasi-exactly solvable sextic potentials in quantum mechanics.

Findings

Eigenvalues of tridiagonal matrices change sign when diagonal signs are flipped.

Dual potentials in quantum mechanics have spectra with opposite signs.

The theorem offers a concise proof of spectral duality in sextic potentials.

Abstract

We put forward and prove a simple theorem stating that the eigenvalues of a tridiagonal matrix change their sign (as a set), once the signs of the diagonal elements of the matrix are changed. We also provide an example of application of this theorem in quantum physics. Specifically, we introduce the notion of duality and self-duality for a sextic-polynomial quasi-exactly-solvable potential, and demonstrate that the algebraic parts of the spectrum of the dual potentials have signs opposite to one another (as sets). Our Theorem furnishes an elegant one-line proof of this statement. In addition, we also prove it by purely quantum-mechanical means - a far less straightforward method that requires some effort.