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).
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