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arXiv:2410.09647 · Progress of Theoretical and Experimental Physics

General Theory of Constructing Potential with Bound States in the Continuum

Mao Kurino, Kazuo Takayanagi

Published Jun 18, 2024• 1 citation
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Abstract

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.

DOI
10.1093/ptep/ptae092

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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′)V(r,r^{\prime })V(r,r′), 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)V(r)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.
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In this work, we construct a time-dependent step-like potential supporting a normalizable state with energy embedded in the continuum. The potential is allowed to evolve until a stopping time ti, where it becomes static. The normalizable state also evolves but remains localized at every fixed time up to ti. After this time, the probability density of this state freezes becoming a Bound state In the Continuum. Closed expressions for the potential, the freezable bound state in the continuum, and scattering states are given.
An example of a three-dimensional Schrödinger equation is solved exactly and it is shown that the potential can sustain an arbitrary number of bound states. Nevertheless, in the second-order perturbation theory all the contributions for all bound states come from the continuum states if the perturbation is spherically symmetric. This is also true for any central potential that does not show accidental degeneracy and has both bound and unbound states, when the perturbation is another central potential.
A bound state in the continuum (BIC) is a spatially localized energy eigenstate embedded in a continuous spectrum of extended eigenstates. While diverse types of single-particle BICs have been reported in the literature, understanding how such localization persists in interacting many-body systems remains a fundamental problem. Here, we present numerical evidence and a perturbative analysis supporting the existence of many-body BICs in a one-dimensional Bose-Hubbard chain with an attractive impurity potential, a system previously known to host a BIC in the two-particle sector. Furthermore, we demonstrate that the Bethe-type construction of the known two-particle BIC breaks down already for three particles even for arbitrary finite superpositions of Bethe waves. The resulting BICs violate the eigenstate thermalization hypothesis for several local one- and two-body observables, leading to nonthermal dynamics from experimentally accessible initial states.
Abstract Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann’s Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier V ( x ) = − V 0 [ exp ( 2 ∣ x ∣ / a ) − 1 ] which for E < V 0 has a continuum of non-square-integrable, definite-parity, degenerate states. In this continuum, we show a surprising presence of discrete energy, square-integrable, definite-parity, non-degenerate states. For E > V 0 , there is again a continuum of complex scattering solutions ψ ( x ) whose real and imaginary parts though solutions of Schrödinger equation yet their parities cannot be ascertained as Cψ ( x ) is also a solution where C is an arbitrary complex non-real number.