How would exhaust heat recovery into the combustion chamber affect the efficiency of a jet engine? So, a jet engine basically sucks in air, heats it up and spits it out, right?
The hotter the air gets, the faster the exhaust velocity and thus the more efficient (Higher specific impulse).

Now my question is how this all gets affected if you use a heat recovery system, either a heat pump or a simple heat exchanger. Let's ignore the annoying practicalities of metals melting and all that stuff and just look at the theoretical side.

If you recover heat from the exhaust and put it back into the combustion chamber how does this affect the efficiency of the engine? If we were to apply the heat into the combustion chamber using a heat pump, pumping heat from the exhaust into the chamber could we get even higher efficiencies? While "Yes/No" answers would suffice I would be much more interested in a thermodynamic explanation and possibly even some estimations of the theoretical efficiency which could be reached using the Carnot cycle for pumping the heat.

Onto the practical, has anything like this been experimented with? To my knowledge we don't have heat pumps capable of operating at 2000 degrees Celsius and pumping heat quickly enough to compare to any type of engine, but maybe someone has experimented with simple heat exchangers like how rocket engines use regenerative cooling (for different purposes though, but still similar to this).
Extracting heat from the exhaust will cool and compress it, slowing it down and reducing thrust. Recirculating the heat back into or downstream of combustion will raise the temperature at that point and on downstream to the extraction point. The inevitable thermodynamic losses mean that the returned heat does not quite compensate for the heat drawn out and overall efficiency will fall a little.

What does improve efficiency is to draw through additional air mass in order to increase the exhaust mass flow, even at the expense of lower overall velocity and temperature. This is what the bypass turbofan does.

A more radical modification is to swap the combustion chamber for a heat exchanger and heat the gas via hot fluid from an external source; the hotter the fluid the better. I have seen this proposed for the hydrogen turbopump in some versions at least of the SABRE air-breathing rocket engine. It was also proposed for nuclear-powered bombers in the 1950s, though I cannot recall if those included turbojets as well as ramjets.