No, it is not true that only electrons in continuum/scattering/asymptotic states can undergo quantum tunneling. Quantum tunneling is a phenomenon that can occur for particles, such as electrons, regardless of their initial state. In quantum mechanics, tunneling refers to the ability of particles to pass through potential energy barriers that would be classically impassable.
In a quantum tunneling scenario, particles can penetrate a barrier even if their energy is less than the height of the barrier. This is because the wave nature of particles allows them to exhibit a probability of existing on the other side of the barrier, despite classically lacking sufficient energy to overcome it.
While quantum tunneling is often discussed in the context of scattering or asymptotic states, it is not limited to these scenarios. Tunneling can occur for particles in various states, including bound states and even localized states within a potential well. The probability of tunneling depends on factors such as the barrier height, barrier width, and the energy of the particle.
It's worth noting that electrons in bound states, such as those within an atom or a molecule, can also exhibit tunneling behavior. For example, electrons in a quantum well or in a molecule can tunnel through energy barriers or potential wells, allowing for phenomena such as electron tunneling microscopy or tunneling transport in electronic devices.
Therefore, the ability to undergo quantum tunneling is not restricted to electrons in continuum/scattering/asymptotic states but applies to electrons and other particles in a broader range of states.