Coupled supersymmetry and ladder structures beyond the harmonic oscillator

Abstract

The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general-except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalize the Heisenberg uncertainty principle. Coupled SUSY is explicitly realized by an infinite family of differential operators.

Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

“…As for the new MUB, these all may be implemented in the coordinate representation with the correct measure guaranteed by the canonical transformation. We note that the Heisenberg-Weyl Lie algebra governs the W-oscillator, in contrast to the work of Williams, et al [34]. In the coordinate representation, this is…”

Point Transformations and Relationships among Linear Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

“…Because of the monotonic increasing character of the PT, d d W x is never negative and the ground state is always positive semi-definite. This is new and it reflects the fact that our formalism is closely related to the "Coupled SUSY" formalism introduced elsewhere [42] [43].…”

Segal-Bargmann Transforms Associated to a Family of Coupled Supersymmetries

“…By way of the SUSY structure, these eigenfunctions can be mapped to eigenfunctions of the partner Hamiltonian by applying a and a * (or b and b * ) accordingly. By reasoning presented in [31], these are all of the L 2 eigenfunctions of the coupled SUSY Hamiltonians. The ladder structure is summarized in the diagrams below.…”

“…z l under the Segal-Bargmann transform. In [31], the notion of a coupled supersymmetry (coupled SUSY) was introduced as a generalization of the quantum harmonic oscillator with a built-in supersymmetric nature, informed by supersymmetric quantum mechanics (SUSY QM) [9,10]. Whereas SUSY QM only has a Lie superalgebra structure that is not strong enough to fully identify the eigenstructure of the associated quantum systems, coupled SUSYs have associated to them additional su(1, 1) Lie algebra structures which allow for the exact solvability of the eigenstructure.…”

“…The remainder of the paper is organized as follows. In Section 2, we will briefly discuss the notion of a coupled supersymmetry, particularly focusing on a family of coupled supersymmetries on R first introduced in [31] of which the quantum harmonic oscillator is a special case. In Section 3, we will develop generalizations of the Segal-Bargmann space on which holomorphic representations of the coupled SUSYs live and remark on the reproducing kernel Hilbert space nature thereof.…”