Обширный труд на 53 страницы, начинающийся с цитирования братьев Вариан. Из недостатков: автор слегка, ИМХО, злоупотребляет математическим языком, все эти леммы и пр. :)))

Analytic theory of multicavity klystron: https://arxiv.org/abs/2105.04410
Alexander Figotin
Multicavity Klystron (MCK) is a high power microwave (HPM) vacuum electronic device used to amplify radio-frequency (RF) signals with numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in klystrons take place in resonant cavities at discrete locations along the beam. Importantly, there is no electromagnetic coupling between cavities, they are coupled only by the bunched electron beam, which drifts from one cavity to the next. We advance here an analytic theory of MCKs operating in voltage amplification mode associated with the maximal gain. This theory features in particular exact formulas for the MCK instability frequencies, its dispersion relations and optimal values of the MCK parameters providing for maximal gain.

Апдейт: частично перекрывающаяся статья того же автора:

Analytic theory of coupled-cavity traveling wave tubes: https://arxiv.org/abs/2109.09945
Alexander Figotin
Coupled-cavity traveling wave tube (CCTWT) is a high power microwave (HPM) vacuum electronic device used to amplify radio-frequency (RF) signals. CCTWTS have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT particularly the amplification mechanism are similar to those of a multicavity klystron (MCK). We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects including the so-called debunching (electron-to-electron repulsion). The corresponding Euler-Lagrange equations are ODEs with coefficients varying periodically in the space. Utilizing the system periodicity we develop the instrumental features of the Floquet theory including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include in particular the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs and our theory applies to them also.