Tests of the Envelope Theory in One Dimension

Abstract

The envelope theory is a simple technique to obtain approximate, but reliable, solutions of manybody systems with identical particles. The accuracy of this method is tested here for two systems in one dimension with pairwise forces. The first one is the fermionic ground state of the analytical Calogero model with linear forces supplemented by inverse-cube forces. The second one is the ground state of up to 100 bosons interacting via a Gaussian potential. Good bounds can be obtained depending on values of the model parameters. *

Improvement of the Envelope Theory for Systems with Different Particles

“…The envelope theory (ET) is a remarkably easy to use method to obtain approximate but reliable eigensolutions of some quantum many-body systems [1][2][3][4][5][6][7][8][9][10]. It is particularly useful when analytical or numerical results with only a reasonable accuracy are asked for [23], as simple tests for numerical codes [14,33], or when a systematic study as a function of the number of particles is necessary as in the Large-N approach of QCD [21,22].…”

“…The problem could arise from the fermionic nature of the electrons, but results are not better when the two only electrons are treated as bosons. Moreover, good results are obtained for the fermionic ground state of the one-dimensional Calogero model with linear forces supplemented by inverse-cube forces [10]. We think that the origin of the poor quality of the ET approximation is the mixing of attractive and repulsive parts in the potentials, forbidding the existence of a definite variational character for the solutions.…”

“…For systems with identical particles, it was possible to test the ET (D = 1 and 3) and the IET (D = 3) [10,14,18] thanks to the existence of very accurate results on simple many-body systems in the literature [15,27,28]. Accurate results for simple systems with one particle different from all others are more scarce.…”

“…It is quite easy to implement if all the particles are identical [6] and becomes a little trickier if the number of different types of particles increases [7]. Quite general type of Hamiltonians in D dimensions can be considered [8][9][10] for which the ET can yield analytical bounds for the spectra in favourable situations. If this is not the case, numerical approximations with or without variational character can always be computed.…”

Compact equations for the envelope theory

“…The envelope theory (ET) [1][2][3], also known as the the auxiliary field method, is a technique to compute approximate eigenvalues and eigenvectors of N -body systems in D dimensions. The method, first developed for systems with identical particles [4,5], has been recently generalized for systems with different particles [6]. The big advantage of this method is that the computation cost is independent from the number of particles.…”