  We generalize the theory of nuclear decay and capture of Gamow that is based
on tunneling through the barrier and internal oscillations inside the nucleus.
In our formalism an additional factor is obtained, which describes distribution
of the wave function of the $\alpha$ particle inside the nuclear region. We
discover new most stable states (called quasibound states) of the compound
nucleus (CN) formed during the capture of $\alpha$ particle by the nucleus.
With a simple example, we explain why these states cannot appear in traditional
calculations of the $\alpha$ capture cross sections based on monotonic
penetrabilities of a barrier, but they appear in a complete description of the
evolution of the CN. Our result is obtained by a complete description of the CN
evolution, which has the advantages of (1) a clear picture of the formation of
the CN and its disintegration, (2) a detailed quantum description of the CN,
(3) tests of the calculated amplitudes based on quantum mechanics (not realized
in other approaches), and (4) high accuracy of calculations (not achieved in
other approaches). These peculiarities are shown with the capture reaction of
$\alpha + ^{44}{\rm Ca}$. We predict quasibound energy levels and determine
fusion probabilities for this reaction. The difference between our approach and
theory of quasistationary states with complex energies applied for the $\alpha$
capture is also discussed. We show (1) that theory does not provide
calculations for the cross section of $\alpha$ capture (according to modern
models of the $\alpha$ capture), in contrast with our formalism, and (2) these
two approaches describe different states of the $\alpha$ capture (for the same
$\alpha$-nucleus potential).
