Exact solutions for semirelativistic problems with non-local potentials

TL;DR

This paper demonstrates how to find exact solutions for the energy eigenvalues of semirelativistic Hamiltonians with non-local, separable potentials, providing explicit examples and bounds for many-body problems.

Contribution

It introduces a method to obtain exact solutions for a class of semirelativistic Hamiltonians with non-local potentials, including explicit examples and bounds for N-boson systems.

Findings

Exact solutions for specific non-local potentials in 1D and 3D

Lower bounds for N-boson energy using these solutions

Upper bounds via Gaussian trial functions

Abstract

It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form H = sqrt{m^2+p^2} + hat{V}, where hat{V} is a non-local potential with a separable kernel of the form V(r,r') = - sum_{i=1}^n v_i f_i(r)g_i(r'). Explicit examples in one and three dimensions are discussed, including the Yamaguchi and Gauss potentials. The results are used to obtain lower bounds for the energy of the corresponding N-boson problem, with upper bounds provided by the use of a Gaussian trial function.