Envelope Theory for Systems with Different Particles
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Claude Semay
· Lorenzo Cimino · Cintia Willemyns
Envelope Theory for Systems with Different Particles
Received: 22 April 2020 / Accepted: 5 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The eigensolutions of many-body quantum systems are always difficult to compute. The envelope theory is a method to easily obtain approximate, but reliable, solutions in the case of identical particles. It is extended here to treat systems with different particles (bosons or fermions). The accuracy is tested for several systems composed of identical particles plus a different one. 1 Introduction The determination with a high accuracy of eigenvalues and eigenvectors of many-body quantum systems is always a hard problem requiring generally heavy computations. Several methods have been developed to tackle this issue. Among the most efficient, one can note the expansions in large oscillator basis [1] or in correlated Gaussian basis [2], the Lagrange-mesh method [3], etc. The envelope theory (ET) [4–6], also known as the auxiliary field method, is a simple technique to compute approximate eigenvalues and eigenvectors for N -body Hamiltonians. The method has been extended to treat arbitrary kinematics in D dimensions with various potentials, but only for identical particles [7–10]. Numerical approximations can always be easily computed. Moreover, in the most favourable cases, analytical lower or upper bounds can be obtained. Besides two-body interactions [7,8], a special type of many-body forces [9] can be handled. The accuracy of the method has been checked for D = 1 [10] and D = 3 [11,12] for various potentials. In particular, two-body interactions with a repulsive short range part are considered for D = 1 with the Calogero model [10] and for D = 3 with confined bosons [11]. Many tests have been performed with the ground state, but the spectra can be calculated as well. An example is given in [11] for a three-quark system for which an accuracy of a few percent has been reached for the 16 lowest levels. Not only approximations of the eigenvalues can be obtained with the ET but also approximations of the eigenvectors. In [11], mean values for the interparticle distance and values of the pair correlation function at the origin are computed for various systems with a reasonable accuracy. The key element of the ET is the fact that the complete solution of a N -body harmonic oscillator Hamiltonian with one-body and two-body forces, says Hho , can be obtained by the diagonalisation of a matrix of order (N −1) [13]. This diagonalisation can be performed analytically in many situations. If H is the general N -body C. Semay (B) · L. Cimino · C. Willemyns Service de Physique Nucléaire et Subnucléaire, UMONS Research Institute for Complex Systems, Université de Mons, Place du Parc 20, 7000 Mons, Belgium E-mail: [email protected] L. Cimino E-mail: [email protected] C. Willemyns E-mail: [email protected]
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