arXiv subjects
Publications and source records attributed to K. E. Hibberd.
Spectral equivalences of the quasi-exactly solvable sectors of two classes of Schrodinger operators are established, using Gaudin-type Bethe ansatz equations. In some instances the results can be extended leading to full isospectrality. In this manner we obtain equivalences between PT-symmetric problems and Hermitian problems. We also find equivalences between some classes of Hermitian operators.
In one-dimensional (1D) non-perturbative many-electron problems such as the 1D Hubbard model the electronic charge and spin degrees of freedom separate into exotic quantum objects. However, there are two different representations for such objects and associated scattering quantities whose relation is not well understood. Here we solve the problem by finding important information about the relation between the corresponding alternative choices for one-particle scattering states. Our study reveals why one of these representations, the {\it pseudofermion} representation, is the most suitable for the description of the unusual finite-energy spectral and dynamical properties of the model. This is a problem of physical importance, since the exotic independent charge and spin finite-energy spectral features observed by angle-resolved photoelectron spectroscopy in quasi-1D metals was found recently to correspond to the charge and spin quantum objects of the pseudofermion representation.
We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al., Phys. Rev. Lett. 93 (2004) 050403. Here we show that there is a second integrable manifold, established using the boundary Quantum Inverse Scattering Method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map in to a Hilbert space of PT-symmetric wavefunctions defined on a contour in the complex plane.
We investigate a static solution with spherical symmetry of a recently proposed field theory of gravitation. In this so-called NDL theory, matter interacts with gravity in accordance with the Weak Equivalence Principle, while gravitons have a nonlinear self-interaction. It is shown that the predictions of NDL agree with those of General Relativity in the three classic tests. However, there are potential differences in the strong-field limit, which we illustrate by proving that this theory does not allow the existence of static and spherically symmetric black holes.
We present an electronic model with long range interactions. Through the quantum inverse scattering method, integrability of the model is established using a one-parameter family of typical irreducible representations of gl(2|1). The eigenvalues of the conserved operators are derived in terms of the Bethe ansatz, from which the energy eigenvalues of the Hamiltonian are obtained.
We extend a recent construction for an integrable model describing Josephson tunneling between identical BCS systems to the case where the BCS systems have different single particle energy levels. The exact solution of the generalized model is obtained through the Bethe ansatz.
Two integrable quantum spin ladder systems will be introduced associated with the fundamental su(2|2) solution of the Yang-Baxter equation. The first model is a generalized quantum Ising system with Ising rung interactions. In the second model the addition of extra interactions allows us to impose Heisenberg rung interactions without violating integrability. The existence of a Bethe ansatz solution for both models allows us to investigate the elementary excitations for antiferromagnetic rung couplings. We find that the first model does not show a gap whilst in the second case there is a gap for all positive values of the rung coupling.
We analyze the stress-energy tensor necessary to generate a general stationary and axisymmetric spacetime. The constraints on the geometry arising from considering a perfect fluid as a source are derived. For a fluid with a nonzero stress tensor, we obtain two necessary conditions on the metric. As an example, we show that the rotating wormhole presented in the literature can not be described by either a perfect fluid or by a fluid with anisotropic stresses.
An integrable version of the supersymmetric U model with open boundary conditions and an impurity situated at one end of the chain is introduced. The model is solved through the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.
We investigate the propagation of electromagnetic waves through a static wormhole. It is shown that the problem can be reduced to a one-dimensional Schrödinger-like equation with a barrier-type potential. Using numerical methods, we calculate the transmission coefficient as a function of the energy. We also discuss the polarization of the outgoing radiation due to this gravitational scattering.
We present new exact cosmological inhomogeneous solutions for gravity coupled to a scalar field in a general framework specified by the parameter $λ$. The equations of motion (and consequently the solutions) in this framework correspond either to low-energy string theory or Weyl integrable spacetime according to the sign of $λ$. We show that different inflationary behaviours are possible, as suggested by the study of the violation of the strong energy condition. Finally, by the analysis of certain curvature scalars we found that some of the solutions may be nonsingular.