Intertwined Hamiltonians in Two Dimensional Curved Spaces

TL;DR

This paper explores intertwined Hamiltonians in various two-dimensional curved spaces, revealing their connection to Killing vectors and isometry groups, and applies the formalism to specific spectral problems.

Contribution

It provides explicit results for different curved spaces and links intertwining operators to geometric symmetries, advancing the understanding of Hamiltonians in curved geometries.

Findings

Intertwining operators relate to Killing vector fields.

Intertwined potentials connect to integral curves of Killing vectors.

Applications include Hamiltonians with equispaced and free-particle-like spectra.

Abstract

The problem of intertwined Hamiltonians in two dimensional curved spaces is investigated. Explicit results are obtained for Euclidean plane,Minkowski plane, Poincar{\' e} half plane ($AdS_2$), de Sitter Plane ($dS_2$), sphere, and torus. It is shown that the intertwining operator is related to the Killing vector fields and the isometry group of corresponding space. It is shown that the intertwined potentials are closely connected to the integral curves of the Killing vector fields. Two problems of considered as applications of the formalism presented in the paper. The first one is the problem of Hamiltonians with equispaced energy levels and the second one is the problem of Hamiltonians whose spectrum are like the spectrum of a free particle.