Julia code to generate the plots used in the counter examples sections of our paper on the solutions to Schrödinger equations with analytic potentials, available on arXiv.

Dependencies

Julia 1.9.3 with the following libraries:

• DFTK.jl 0.6.11
• PyPlot, LineSearches, LinearAlgebra, Optim
• For the linear cases : DoubleFloats and GenericLinearAlgebra

Usage

To perform the computations, first open the Julia shell with julia --project from the location of this repository and then run

using Pkg
Pkg.instantiate()

to install the required dependencies.

Linear case with V only analytic on a finite band

To generate the plots from Section 3.4.2 - Figure 1 of the paper, just run within the Julia shell:

include("linear_egval.jl")

Plots are then saved in test_decay_linear_egval.png.

Exponential convergence of the plane wave discretization

To generate the plots from Section 3.4.2 - Figure 2 of the paper, just run within the Julia shell:

include("pw_discretization.jl")

Plots are then saved in test_pw_discretization.png.

Nonlinear Gross-Pitaevskii equation

To generate the plots from Section 4.2.1 of the paper, just run within the Julia shell:

include("gross_pitaevskii_egval.jl")

Plots are then saved in test_decay_gp.png and u0_gp.png.

Nonlinear elliptic equation

To generate the plots from Section 4.2.2 of the paper, just run within the Julia shell:

include("gross_pitaevskii_source_term.jl")

Plots are then saved in test_decay.png and u0.png.

Contact

This is research code, not necessarily user-friendly, actively maintened or extremely robust. If you have questions or encounter problems, get in touch!