Fourier spectral method on sparse grids for computing ground state of many-particle fractional Schrödinger equations

ABSTRACT

In this paper, we consider the Fourier spectral method on the sparse grids for computing the ground state of the many-particle fractional Schrödinger equations. The appropriate sparse grids for many-particle fractional Schrödinger equations are given, and the estimation for the number of grid points is obtained. Then, the iterative scheme of the inverse power method is presented to compute the ground state. In the numerical experiments, we consider two kinds of fractional Schrödinger equations, i.e. the fractional Schrödinger equation with harmonic potential and the electronic fractional Schrödinger equation. For the fractional Schrödinger equation with harmonic potential, when the scaled Planck constant ℏ is equal to 1, the sparse grid method has the obvious advantages in the high-resolution approximation both for the integer-order problem and the fractional-order problem, but when ℏ is equal to 1/4, the advantage of sparse grid method exists only for the fractional-order problem. We also show that for the electronic fractional Schrödinger equation, the sparse grid method is much better than the full grid method when the number of electrons is relatively large.

Keywords:

• Sparse grid
• Fourier pseudospectral method
• ground state
• Fractional Schrödinger equations
• many-particle systems

AMS subject classifications 2010:

• 35R11
• 65F15
• 65M70
• 65T50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China (11601461,11671343), Natural Science Foundation of Hunan Province (2016JJ3115) and Education Department of Hunan Province (16C1541).