The Intersection between Dual Potential and SL(2) Algebraic Spectral Problems

TL;DR

This paper explores how expressing dual Hamiltonians in terms of SL(2) algebra simplifies spectral problems in quantum mechanics, revealing restrictions on potentials and suggesting methods to construct spectra via algebraic and partner transformations.

Contribution

It demonstrates that representing dual Hamiltonians with SL(2) algebra limits potentials to Coulomb and harmonic oscillator types, and proposes spectrum construction using partner potentials and quasi-solvability.

Findings

SL(2) algebra restricts allowable potentials to Coulomb and harmonic oscillator.

Expressing Hamiltonians via SL(2) relates to known solutions for these potentials.

Partner potential transformations can aid in spectrum construction through quasi-solvability.

Abstract

The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation $x{\rightarrow}\bar{x}^{\bar{\alpha}}$ has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of SL(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only non-trivial potentials allowed being the Coulomb $\frac{1}{r}$ potential and the Harmonic Oscillator $r^2$ potential, for both of which the SL(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian's spectrum from the quasi-solvability of its partner Hamiltonian.