Still a WIP.
@David_Tschumperle brings forth eigenvalues only on command line -meigen and its underlying math expression equivalent meig(A,m,n). As I was rummaging through his implementation in ,gmic_stdlib.gmic, became increasingly disgusted with myself on the “plugging the numbers, turning the crank” approach. Don’t like it when I’m finding myself trying to find eigenvalues and eigenvectors of giganormous matrices. Feels like the dumb näif approach, numerically speaking. Nice when you have a workstation with 128GB RAM, What about people with 8GB laptops? So I took a step back to ask myself 'What am I doing?"
I believe what I’m doing is depicted here, solving for the ψi (the wave functions) in this “Shrodinger’s Equation” for energy levels, where the Hamiltonian operator (Ho) is in play. The solution is a numerical one (not analytic), so that the solution looks like a square data set of particle probabilities p at discrete <x>,<y> locations. This being quantum mechanics, Shrodinger’s Equation is true for only discrete energy levels. Finding (some) of those energy levels involves extracting eigenvectors (a.k.a eigenfunctions), the ψi, and eigenvalues, the Ei. These ψi are the basis of the pretty pictures.
I don’t understand the analytic justification for the mechanics behind this construction of Ho, unrolling the well (seed) image into off-diagonal coefficients, the scaling and spacing of the off-diagonals and the main. I want to grasp that basis - connect those dots - and, from that, grasp if there is any way better than this näive “follow-your-nose” approach that we seem to be doing. So I’ve made progress, I think, in coming to know what I don’t know. `Tis the negative that shapes the positive, and all that. I feel halfway up to writing my own version of -meigen that solves for the ψi in a not-so-memory sprawling way. Another weekend, maybe.