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Physical Review Letters

Exact solution for quantum Davey-Stewartson I system

Gen-Di Pang, Fu‐Cho Pu, Bao‐Heng Zhao

Published Dec 24, 1990• 34 citations
Publisher

Abstract

The (2+1)-dimensional Davey-Stewartson I system is quantized, and the exact eigenvalues and eigenfunctions of the Hamiltonian are obtained. Thermodynamic quantities are also given. Although it is quantized by using the commutation relations for bosons, the energy spectrum and thus the thermodynamic quantities reveal the characteristics of a system with two species of independent fermions.

DOI
10.1103/physrevlett.65.3227

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The (2+1)-dimensional Davey-Stewartson I system is quantized, and the exact eigenvalues and eigenfunctions of the Hamiltonian are obtained. Thermodynamic quantities are also given. Although it is quantized by using the commutation relations for bosons, the energy spectrum and thus the thermodynamic quantities reveal the characteristics of a system with two species of independent fermions.
The quantum 2-component DS1 system in two dimensions has been reduced to two 2-component many-body problems both in one dimension by an analogue of the method of separation of variables. The solutions to 2 dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For latters with the delta\\deltadelta-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
The quantum 2-component DS1 system was reduced to two 1D many-body problems with δ−δ-δ−function interactions, which were solved by Bethe ansatz. Using the ansatz in ref.[1] and introducing symmetric and antisymmetric Young operators of the permutation group, we obtain the exact solutions for the system.
The Davey-Stewartson 1 (DS1) system [Proc. R. Soc. London, Ser. A 338, 101 (1974)] is an integrable model in two dimensions. A quantum DS1 system with two color components in two dimensions has been reformulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with two color components. The solutions to the two-dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For the latter problem with delta-function interactions and solution by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.
The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.
A simple one-dimensional model with scalar variables −1≤sn≤+1, coupled according to a Hamiltonian, H=−J∑n‖sn−sn+1‖, is presented. This Ising-like model is an interesting example of a graduate-level statistical mechanics problem where the eigenfunctions and eigenvalues of an integral operator, the transfer integral, can be determined exactly. This can be contrasted with the determination of eigenfunctions and eigenvalues of differential operators, such as the quantum mechanical finite-depth square well. In this problem transcendental equations leading to the eigenvalues and eigenfunctions are also obtained. The largest eigenvalue is identified and the free energy is determined. The thermodynamic properties depend on the sign of J, unlike classical Ising or Heisenberg models.