The bound state at positive energy or the bound state in the continuum (BIC) was first predicted by von Neumann and Wigner in 1929. Unlike conventional bound states, which can only exist at discrete negative energies below continuum spectra of scattering states with positive energies, BICs make use of nonlocality of potential to confine themselves in localized areas by preventing outgoing waves from escaping.

BICs have diverse applications in many fields, including photonics and acoustics. However, despite extensive research, they have not yet been observed in quantum systems.

Addressing this gap, a new study published in Progress of Theoretical and Experimental Physics presents a general theory of constructing potentials that support BICs. The theory shows that BICs can be found only in nonlocal potentials among all possible potentials localized in coordinate space. Interestingly, the study reveals that within this set, mostly consisting of non-local potentials, BICs are as common as negative-energy bound states.

The study introduces a method for constructing all possible Hermitian potentials supporting BICs, which through a process called SB-decomposition, can be expressed in terms of potentials V_s and V_B that operate on scattering and bound state spaces, respectively.

Through numerical examples, the study illustrates that for a given Hermitian potential V, through SB-decomposition followed by analyzing the potential of bound states <k’|V_B|k>, we can determine whether it supports a BIC. Additionally, through coordinate representation, the study shows that any potential that supports a BIC is necessarily non-local in coordinate space.

Furthermore, the study outlines methods for searching BICs in real systems with possible nonlocal potentials.

This groundbreaking research, by serving as a guide for harnessing BICs in real quantum systems, opens the door to new technologies with far-reaching implications.

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