The rigorous formula of overlap integrals of continuum stationary states with their asymptotic expressions in potentials of finite widths are derived. Those of energies E_1 and E_2 consist of diagonal terms that are proportional to δ(E_1-E_2) and nondiagonal terms. Owing to the composition of nondiagonal terms, superpositions of stationary states have time-dependent norms and finite probability currents. These do not represent isolate states. In various exceptional potentials and in free theory, nondiagonal terms do not exist, and the superpositions of states with different energies represent isolate particles that exactly describe scattering processes.
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Kenzo Ishikawa (2024) studied this question.
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