  A possible route towards achieving high power microwave (HPM) devices is
through the use of novel slow-wave structures, represented by multiple coupled
transmission lines (MTLs), and whose behavior when coupled to electron beams
has not been sufficiently explored. We present the extension of the
one-dimensional linearized Pierce theory to MTLs coupled to a single electron
beam. We develop multiple formalisms to calculate the k-{\omega} dispersion
relation of the system and find that the existence of a growing wave solution
is always guaranteed if the electron propagation constant is larger than or
equal to the largest propagation constant of the MTL system. We verify our
findings with illustrative examples which bring to light unique properties of
the system in which growing waves were found to exist within finite bands of
the electron propagation constant and also present possible means to improve
the gain. By treating the beam-MTL interaction as distributed dependent current
generators in the MTL, we derive relations characterizing the power flux and
energy exchange between the beam and the MTLs. For the growing wave, we show
that the beam always behaves as an energy source causing power flux along the
transmission lines. The theory developed in this paper is the basis for the
possible use of degenerate band-edges in periodic MTL systems for HPM
amplifiers.
