Does Uncertainty Principles apply to solutions of finite duration (so, to something it stops moving in finite time)?
I have incorporated the calculations I would like to reproduce for Gabor's Uncertainty Principle at the end. I got stuck on the Fourier Transform, since there is a integration limit dependence I was not able to resolve. Also, I am conceptually confused about if I have to take the Bidimensional Fourier Transform, or instead only a Fourier Transform in the time variable... How it is done Quantum Mechanics?
Intro____________
In this another question I think I found an ansatz for solving the PDE $$ \nabla |u|^{\frac12} = 0$$ through a particular singular solution given by $$u(x,t) = \frac{\text{sgn}(u(0,0))}{4}\left(2\sqrt{|u(0,0)|}-(x+t)\right)^2\cdot\theta\!\left(2\sqrt{|u(0,0)|}-(x+t)\right)$$ with $\theta(t)$ the Heaviside step function. I am still not sure if it is completely right, since boundary conditions are fully determined by just the constant $u(0,0)$ so it lost many degrees of freedom, but it is interesting the solution shows finite duration so after a time $T=2\sqrt{|u(0,0)|}<\infty$ the system will achieve a rest position.
I don't know yet if the system has any physical interpretation (I am asking this right now here), but please try to follow the following idea as it does since it would be easier to explain my question in this way: imagine this system is describing some particle properties, then I expect to have some Uncertainty Principle relation among their variables:
- I don't think it could be some related quantum mechanics analogue, since it would require some very weird potentials (to begin with some element cancelling the imaginary part losing all it's sense - but maybe you could think of one by some change of variable $\sqrt{\varphi}=\psi$ and doing something similar to this question, but I doubt it would make any sense anyway), but please don't fix in the physical interpretation, but instead in the math of having a solution that achieve rest in finite time. At least the basic Shrödinger Equation is linear so it would never show something like the answer (it solved through power series), but imagine you made some non-linear version that could have a potential like $u(x,t)$ for this question.
- Maybe the Gabor's Uncertainty relation among the time and frequency descriptions (here I will admit I tried to take the bidimensional Fourier Transform but got lost on the variable integration limits - my apologies, any help will be welcomed). This one I think it would be valid for a classic system also.
- I don't know but maybe an Uncertainty relation for the Entropy could be defined.
But what makes this question, if that if I think on this particle position and momentum (using them as examples/analogues for the explanation, could be other complementary variables), if stops moving in finite time, there would be exist some moment where both are completely determined at the same time (position will be some fixed constant and speed will be zero), differently from classic solutions to linear differential equation which at best could vanish at infinity (think in a power series solution, matching a constant in a non-zero measure interval would violate the Identity theorem).
Question______
So my doubt if what would happen with the Uncertainty Principle with a system of finite duration?
Somehow, my intuition tell me that any inequality would be broken at some time as the quantities becomes lower and lower: thinking in the Gabor's limit, as I integrate the solution for a finite extinction time $T\to 0$, the energy would go to zero in the time variable, as it would be in the frequency due Plancherel Theorem, so I believe it would be a time $T^*$ when any inequaliy $\Delta E_t\cdot\Delta E_f \not> \text{any constant greater than zero}$ (this as a crude example of what I mean with some math what my intuition tells me, not to be taken literally).
Also, is it posible that it is just not possible to define un Uncertainty Principle to begin with for the solutions of finite duration: let say as example, it is not possible to define a finite bandwidth to begin with (still thinking in Gabor's limit, just as example). This because the frequency spectrum of a compact-supported signal is an entire function due the Paley–Wiener theorem (but in the other hand, for a real valued function you expect the frequency spectrum to vanish at infinity due the Riemann-Lebesgue Lemma, and is not obvious for me which one would be more relevant).
I hope you could use this system $u(x,t)$ of the example to show or disprove it is possible to have an Uncertainty Principle for the mentioned solution: my initial plan was to find its Fourier Transform, then take the second moment in time and the frequencies, and evaluate if it fulfill Gabor's limit, by I got stuck in the very beginning.
Motivation________
I am trying to explore if the way we are modelling system through Lipschitz differential equation could be limiting the result/interpretation we are taking from the models, as example what I realized here, and now I want to know if the Uncertainty Principles could be some math created limitation or not if I allow systems to have a finite duration (this require having a Non-Lipschitz term at least in 1st and 2nd order ODEs).
Gabor's paper section I would like to reproduce
