We know about a differentiation and integration operations and also other ones can be simulated by an analog way using electric circuits with operational amplifiers (here are articles of the differetiator and integrator for a reference).
Let's suppose we've decided to simulate a kinetic energy operator (formulas 1 and 2) and let it act on a wave function (or in general any function) but not this way but another: mainly using either a single crystal as a converter, or a cascade of thin films on a single-crystal substrate, or a cluster.
A question arises: could there exist a single crystal (or things mentioned above) that would be an analogue of the kinetic energy operator? Passing a certain electric, magnetic, or electromagnetic field, the crystal converts this field such that an input field would be related to an output field as a mathematical function and its second derivative function.
One-dimensional operator: $$ \hat{T} = -\frac{1}{2m}\frac{d^2}{dx^2} \tag{1} $$ Three-dimensional one: $$ \hat{T} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m}\nabla^2 \tag{2} $$
GptChat assumes this problem to be hard so I guess it's probably true. So which mathematical operators would you try to simulate at first if you would set this kind of problem? Perhaps the use of quantum dots as a variant can help reveal the essence of this problem. But a quantum dots are not a single crystal.
