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Journal of Physics A Mathematical and General

On the nonlinear Fock description of quantum systems with quadratic spectra

V. M. Eleonsky, V. G. Korolev

Published May 21, 1996• 11 citations
Publisher

Abstract

The nonlinear version of the Fock approach, developed previously for systems with (extended) equidistant spectra (Eleonsky V M and Korolev V G 1995 J. Phys. A: Math. Gen. 28 4973), is also shown to describe systems with quadratic spectra. Some general properties of shift operators for such spectra are studied. It is found, in particular, that families of shift operators related to the generators of the algebra su(1,1) as well as supersymmetry ladder operators arise naturally in that approach in a unified way.

DOI
10.1088/0305-4470/29/10/004

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The nonlinear version of the Fock approach, developed previously for systems with (extended) equidistant spectra (Eleonsky V M and Korolev V G 1995 J. Phys. A: Math. Gen. 28 4973), is also shown to describe systems with quadratic spectra. Some general properties of shift operators for such spectra are studied. It is found, in particular, that families of shift operators related to the generators of the algebra su(1,1) as well as supersymmetry ladder operators arise naturally in that approach in a unified way.
A simple way to construct exactly solvable Hamiltonians whose spectra contain two equidistant ladders, one finite and another infinite, appears when applying supersymmetric quantum mechanics to the harmonic oscillator. Some of those supersymmetric partners have third order differential ladder operators, although the order of the transformation is higher than one. In this work the linearized coherent states for these specific Hamiltonians are studied. To each SUSY partner Hamiltonian corresponds two families of linearized coherent states: one inside the subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states inserted through the SUSY technique.
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 (QMHO) 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 generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m -state quantum spin Hamiltonian with nearest-neighbour interaction and periodic boundary condition is considered. For this representation we use the quadratic algebra defined in Krebs and Sandow ( 1997 J. Phys. A: Math. Gen. 30 3165 ), which is generated by 2 m operators which fulfil m 2 quadratic relations, endowed with a trace-like function reflecting periodic boundary conditions. It is shown that not every eigenstate with energy and momentum zero can be written in this way. An explicit counter-example is given. This is in contrast to the case of open boundary conditions where every zero-energy eigenstate can be written as a matrix product state using a Fock-like representation of the same quadratic algebra.
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.