Daniel-Crespin, It is well known among physicists that the Schrodinger equation (STDE) is incomplete. For example, it is not compatible with relativity. This is why quantum field theory introduces elements that allow for the creation and destruction of quantum states, while the Schrodinger equation does not.
You said above: "The theory claims that electrons preserve their energies while moving. Spectroscopy tells that when electrons move they typically change their energy; otherwise no photons could be radiated nor absorbed, let alone photons for the spectral lines."
What you offered here is a good example of where the Schrodinger equation is incomplete. However, quantum physicists know this. What the Schrodinger equation represents is what happens to quantum states and superposition states when they are not involved in exchanges of energy. Only linear changes are possible to quantum states. Superpositions never collapse to one actuality under these conditions. Whenever measurement is involved, however, there are non-linear effects, such as your example above.
Quantum formalism says that the use of the Schrodinger equation is only valid for states that are not being observed or measured. We can also say that these are only valid for states where there is no exchange of energy or momentum. The STDE is only for showing how "coherent" quantum states evolve over time.
Quantum formalism uses a different step in the process to evaluate the results of measurements, where there is an exchange of energy or momentum.
If you want to challenge the foundations of quantum mechanics, I wouldn't start with challenging the Schrodinger equation. It would be more compelling if you could explain why we see a wave-like interference pattern when electrons or photons are sent, one at a time, through two-slits. And why does the interference pattern disappear when we measure which slit the electrons or photons go through?
Or why is it that we can know for sure what the result of measuring the spin of an electron will be if we measured the electron just before we measure it again? Whatever the result of the first measurement is, it will always be the same with the second measurement as long as the second measurement happens soon enough after the first, and the spin axis being measured is the same for the first and second measurements. However, when the second measurement is made using a different axis, then there is no way to know for sure what the result of an individual electron will be. The results can only be predicted statistically. Why is that?
If you can explain these two issues using only classical physics, then you can start to make the case that perhaps a fully deterministic classical physics could replace quantum mechanics.
Hopefully, this is the kind of feedback you are looking for.
In the spirit of open and respectful dialogue,
Doug Marman