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THE SEMI-INFINITE q-BOSON SYSTEM WITH BOUNDARY INTERACTION

J. F. Van, Diejen, Erdal Emsiz

Published Aug 17, 2016• 6 citations
OpenAlex

Abstract

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald’s hyperoctahedral Hall–Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.

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Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald’s hyperoctahedral Hall–Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.