Bogged

The singularity of the map S²→R² makes any straightforward simulation impossible. A possible turnaround: rotate the axis every time θ goes close to 0 or π.

Another idea: what if the surface is a, say, tetrahedron? Keep in mind the relationship between the kicked rotator and tight binding. Both manifests the lack of a continuous symmetry, but both are solvable.

SAR suggests investigating the dynamics with a perturbed potential surface. Stöckmann points out the kinship between the Anderson (spatial) localization and dynamical localization. Heller's paper provides a good ground for studying the semiclassical aspects.

It is likely that the localization effect is absent in the classical limit (strong diffusion). If S¹ can be taken as an analog of the kicked rotator (loosely), what about S²?

This can relate the current project to both previous projects. Also valuable would be to re-evaluate the results with a different separable potential profile (can be simply ½k(r-r₂)² + V(r=r₀,θ,φ)) against the Morse potential.

Edit:

A possible approach:
Consider the Hamiltonian of the form:
H_ε = H₀ + ε*V(θ,φ)
H₀ is integrable, then Kolmogorov–Arnold–Moser theorem can be applied.

Binary choice experiment and so on

So I made 4 bets on NVDA in just 7 days: 3 shorts & 1 long. Fortunately I got all of them worked out with positive returns (one as small as $20), but in retrospect, I blame myself for an extreme irrationality when making the choices. Let me explain.

I recently read about this from a book (Kuenne, Readings in Applied Microeconomic Theory). It is not really anything “economics”, but rather some psychology experiment helping to understand why people do not always maximize their expected returns or even minimax the losses.

The subject is given two choices A, B in each round and then asked to pick one. If he picks the right one, he gets a reward, and if he picks the wrong one, he gets nothing. The experimenter sets A or B to be correct on a completely random basis, for example, for 2/3 of the times A is the correct answer and for the rest 1/3 B is. Interestingly enough, the subject can soon realize this distribution, but he would not always bet on A, which is the optimal strategy that rewards 2/3 of the time. Instead, he bets on A 2/3 of the times and on B 1/3 of the times, randomly. This strategy rewards 5/9 of the time, therefore less optimal than the aforementioned one.

So why? Actually nobody knows for sure why subjects would choose to be both random and pattern-following. It is plausible that men have a tendency to be competitive in situations, against “nature” in this experiment. Men want to “outguess” his opponents, and in this case, he assumes the nature is playing against him. It is like poker: if you know your opponent is bluffing 2/3 of the times, you don't want to call always, because that would change his strategy - so you probably want to exploit this to the full and consequently randomize your responses.

Now you see why the hidden rationale behind my bets is so irrational. I predicted the market goes down 75% of time and up 25% of time. So I made 3 bets of it going down and 1 bet of it going up. Exactly the same mistake made by the subjects in the experiment. I was trying to outguess the market (who doesn't even care what I have guessed).

I can not help linking this to the philosophical ideas of Lao-Zi. “In harmony with the natural laws, not against them.” I also think now the news in which SAC traders were forced to take female hormones to enhance their performance, makes perfect sense. Spirit of competition (or in my favorite Nietzsche's words, The Will to Power) is probably innate to the male aggression. And it is a good thing in general - just that we have to put a check to it so it doesn't act foolish.

N-well tunneling

Basically t~2Zγ=ZE_tun. General to arbitrary dimensions or symmetries (rotational/translational doesn't matter). See my write-up.