Bound state in the continuum
Bound state in the continuum (BIC)   - is an eigenstate of some particular quantum system with the following properties: BICs are observed in electronic, photonic, acoustic systems. Bound states in the forbidden zone, where there are no finite solutions at infinity, are widely known (atoms, quantum dots, defects in semiconductors). For solutions in a continuum that are associated with this continuum, resonant states are known, which decay (lose energy) over time. They can be excited, for example, by an incident wave with the same energy. The bound states in the continuum have real energy eigenvalues and  therefore do not interact with the states of the continuous spectrum and cannot decay. Wigner-Von Neumann BICs. Bound states in the continuum were first predicted in 1929 by Eugene Wigner and John von Neumann. Two potentials were described, in which BICs appear for two different reasons. In this work, a spherically symmetric wave function is first chosen so as to be quadratically integrable over the entire space. Then a potential is chosen such that this wave function corresponds to zero energy. The potential is spherically symmetric, then the wave equation will be written as follows: the angle derivatives  disappear, since we limit ourselves to considering only spherically symmetric wave functions: For formula_3 to be the eigenvalue for the spherically symmetric wave function formula_4, the potential must be We obtain the specific values formula_6 and formula_7 for which the BIC will be observed. First case. Let us consider the function formula_8. While the integral formula_9 must be finite, then considering the behavior when formula_10, we get that formula_11, then considering the behavior when formula_12, we get formula_13. The regularity formula_14 in formula_15 requires formula_16. Finally, we get formula_17. Assuming formula_18, then the potential will be equal to (discarding the  irrelevant multiplier formula_19): The eigenfunction and the potential curve are shown in the figure. It seems that the electron will simply roll off the potential and the energy will belong to the solid spectrum, but there is a stationary orbit with formula_3. In the work is given the following interpretation: this behavior can be understood from an analogy with classical mechanics (considerations belong to Leo Szilard). The motion of a material point in the potential formula_22 is described by the following equation: It's easy to see that when formula_25, formula_26, so the asymptotic is that is, for a finite  time formula_29 the point goes to infinity. The stationary solution formula_30 means that the point returns from infinity again, that it is as if it is reflected from there and starts oscillating. The fact that formula_30 at formula_12 tends to zero follows from the fact that it rolls down a large potential slide and has an enormous speed and therefore a short lifetime. And since the whole oscillatory process (from formula_33 to infinity and back) is periodic, it is logical that this quantum mechanical problem has a stationary solution. Second case. Let's move on to the second example, which  can no longer be interpreted from such considerations. First of all, we take a function formula_34, then formula_35. These are divergent spherical waves, since the energy formula_3 is greater than the potential formula_35, the classical kinetic energy remains positive. The wave function belongs to a continuous spectrum, the integral formula_38 diverges. Let's try to change the wave function so that the quadratic integral converges and the potential varies near -1. Consider the following ansatz: If the function formula_40 is continuous, and at formula_41 the asymptotic is formula_42 then the integral is finite. The potential would then be equal (with  the corrected arithmetical error in the original article): In order for the potential to remain near -1, and at formula_25 tend to -1, we must make the functions formula_45 small and at formula_25 tend to zero. In the first case, also formula_47 should vanish for formula_48, namely for formula_49, that is for formula_50. This is the case when formula_51 or any other function of this expression. Let assume formula_52, where formula_53 is arbitrary (here formula_40 tends to formula_42 when formula_25). Then The expression for the potential is cumbersome, but the graphs show that for formula_25 the potential tends to -1 . Furthermore, it turns out that for any formula_59 one can choose such an "A" that the potential is between formula_60 and formula_61. We can see that the potential oscillates with period formula_62 and the wave function oscillates with period formula_63. It turns out that all reflected waves from the "humps" of such a potential are in phase, and the function is localized in the center, being reflected from the potential by a mechanism similar to the reflection from a Bragg mirror.