Bound-state equivalent potentials with the Lagrange mesh method

Abstract

The Lagrange mesh method is a very simple procedure to accurately solve eigenvalue problems starting from a given nonrelativistic or semirelativistic two-body Hamiltonian with local or nonlocal potential. We show in this work that it can be applied to solve the inverse problem, namely, to find the equivalent local potential starting from a particular bound state wave function and the corresponding energy. In order to check the method, we apply it to several cases which are analytically solvable: the nonrelativistic harmonic oscillator and Coulomb potential, the nonlocal Yamaguchi potential and the semirelativistic harmonic oscillator. The potential is accurately computed in each case. In particular, our procedure deals efficiently with both nonrelativistic and semirelativistic kinematics.

Effective potential between two transverse gluons from lattice QCD

“…It is readily checked that this nice feature allows to rewrite Eq. (3) as Vðhx i Þ ¼ f½T ij ; m 0 þþ ; Rðhx i Þ, thus to solve the inverse problem by computing the potential at the mesh points [22]. In our case, both m 0 þþ and Rðhx i Þ are known thanks to the lattice results (1) and (2).…”

Lagrange-mesh calculations and Fourier transform

“…We will focus on the quality of wave functions and observables in the momentum space since the efficiency of the method in the position space has already been demonstrated elsewhere [1][2][3][4][5][6][7][8][9]. In order to estimate the quality of the Fourier transform (22) more precisely, we define a quality factor Q(p * ),…”

“…The Lagrange-mesh method is a very accurate procedure for computing eigenvalues and eigenfunctions of a two-body Schrödinger equation [1][2][3][4][5] as well as a semirelativistic Hamiltonian [6][7][8][9]. The trial eigenstates are developed in a basis of well chosen functions, the Lagrange functions.…”

Lagrange-mesh calculations in momentum space

“…Among these techniques, the Lagrange-mesh method (LMM), which is especially easy to implement, can produce very accurate results. First created to compute eigenvalues and eigenfunctions of a two-body Schrödinger equation [2][3][4][5][6][7], it has been extended to treat semirelativistic Hamiltonian [8][9][10][11]. The trial eigenstates are developed in a basis of particular functions, the Lagrange functions, which vanish at all mesh points, except one.…”