Solving one or few particles schrodinger equation

[benoitseron]

I am starting to work on a new project, based on this PhD thesis. The relevant sections for this discussions are Chapter 3, in particular 3.7.2.

I am looking to solve the Schrodinger equation through finite element methods, as done in the PhD thesis. It's an eigenvalue problem, and from the little I understand, might be slightly different from the typical problems treated by Gridap.jl.

1. I would like to know if it would be easy to use/modify the package to solve this type of equation? I already wrote some code for a 1D schrodinger equation, with one particle, on my side, this was not too difficult. I assume, however, that Gridap is highly optimised, especially for higher dimensions, and I see no point in reinventing the wheel.

2. One of the difficulties encounter in the thesis linked above, is the proper choice of discretizations. For instance, see Fig. 3.7., where physical constraints impose a specific type of elements due to the coulomb potential, symmetries, etc. Would it be easy, and how could I modify the various elements/spaces to account for these symmetry considerations?

3. In the paragraph "Limitations and Further Comments on N ≥ 4 Particles", it seems that complications arise. Is there some intuition on whether Gridap is well suited for these higher dimensional spaces?

These questions are a preliminary investigation for me, and I am not asking directly for complete solutions. My goal is to see whether it makes sense to pursue what my advisor would strongly like me to do: converting the (working) mathematica code of this PhD thesis, into a more performant code (Julia being my language of predilection), in particular to account for more particles, and in more dimensions. My feeling is that his intuition might be misguided, given the limitations stated in point number 3.

Even if many particle states cannot be efficiently simulated, being able to accurately form basis states for a single particle using Gridap could still be very valuable.

Many thanks in advance!