Takayanagi Kazuo

Abstract We present a general theory of potentials that support bound states at positive energies (bound states in the continuum). On the theoretical side, we prove that, for systems described by nonlocal potentials of the form $V(r,r^{\prime })$, bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form $V(r)$ rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.

Abstract We present the generalized optical theorem and its applications with special emphasis on the roles of bound states. First, we prove the theorem which gives a necessary and sufficient condition for a function $\langle {\boldsymbol {k } }^{\prime } | T | {\boldsymbol {k } } \rangle$ of two variables ${\boldsymbol {k } }^{\prime }$ and ${\boldsymbol {k } }$ to be physically acceptable as a half-on-shell T-matrix, i.e., to have an underlying Hermitian potential V. Secondly, using the theorem, we construct a scattering theory starting from a physically acceptable half-on-shell T-matrix $\langle {\boldsymbol {k } }^{\prime } | T | {\boldsymbol {k } } \rangle$, which in turn introduces a very useful classification scheme of Hermitian potentials. In the end, as an application of our theory, we present the most general solution of the inverse scattering problem with numerical examples.