The Method of Particular Solutions Using Trigonometric Basis Functions

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method

Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schr\"odinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schr\"odinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with pairwise delta-function interactions is indicated.Comment: 23 pages; Version 2: expanded introduction and misprints correcte

SU(3) Richardson-Gaudin models: three level systems

We present the exact solution of the Richardson-Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis-Cummings model for three level atoms interacting with two modes of the quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo

Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables

Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T-Q equations of Baxter's type, hence proving the completeness of the solutions to the associated systems of Bethe-type equations. More precisely, we consider here two such reformulations. The first one is given in terms of Q-solutions, in the form of trigonometric polynomials of a given degree $N_s$, of a one-parameter family of T-Q functional equations with an extra inhomogeneous term. The second one is given in terms of Q-solutions, again in the form of trigonometric polynomials of degree $N_s$ but with double period, of Baxter's usual (i.e. without extra term) T-Q functional equation. In both cases, we prove the precise equivalence of the discrete SOV characterization of the transfer matrix spectrum with the characterization following from the consideration of the particular class of Q-solutions of the functional T-Q equation: to each transfer matrix eigenvalue corresponds exactly one such Q-solution and vice versa, and this Q-solution can be used to construct the corresponding eigenstate.Comment: 38 page

A new perturbative expansion of the time evolution operator associated with a quantum system

A novel expansion of the evolution operator associated with a -- in general, time-dependent -- perturbed quantum Hamiltonian is presented. It is shown that it has a wide range of possible realizations that can be fitted according to computational convenience or to satisfy specific requirements. As a remarkable example, the quantum Hamiltonian describing a laser-driven trapped ion is studied in detail.Comment: 32 pages; modified version with examples of my previous paper quant-ph/0404056; to appear on the J. of Optics B: Quantum and Semiclassical Optics, Special Issue on 'Optics and Squeeze Transformations after Einstein