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{{Английская Википедия/Панель перехода}}
[[File:Bikwiki0.png|thumb|250px|Schematic picture of energy levels and examples of different states. Discrete spectrum states<ref group=nb>Technically, bound states belong to the pure point spectrum, not the discrete spectrum. See [[Decomposition of spectrum (functional analysis)#Quantum physics|Decomposition of spectrum (functional analysis)]] and "{{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 | page=3| s2cid=16913591 }}"</ref> (green), resonant states (blue dotted line) <ref name = "resonant">{{cite journal | doi=10.1119/1.3276053 | title=Resonant states and transmission coefficient oscillations for potential wells and barriers | date=2010 | last1=Maheswari | first1=A. Uma | last2=Prema | first2=P. | last3=Shastry | first3=C. S. | journal=American Journal of Physics | volume=78 | issue=4 | pages=412–417 | bibcode=2010AmJPh..78..412M }}</ref> and bound states in the continuum (red). Partially reproduced from <ref name = "hsu" /> and <ref>Koshelev, K .; Bogdanov, A .; Kivshar, Y. Engineering with bound states in the continuum. Opt. Photonics News 2020, 31, 38−45</ref>]]

A '''bound state in the continuum''' (BIC) is an [[Quantum state|eigenstate]] of some particular [[Quantum mechanics|quantum]] system with the following properties:

# [[Energy]] lies in the [[continuous spectrum]] of propagating modes of the surrounding space;
# The state does not interact with any of the states of the continuum (it cannot emit and cannot be excited by any wave that came from the infinity); 
# Energy is real and [[Q factor]] is infinite, if there is no absorption in the system.

BICs are observed in electronic, [[photonics|photonic]], [[acoustics|acoustic]] systems. [[Bound state]]s in the forbidden zone, where there are no finite solutions at infinity, are widely known ([[atoms]], [[quantum dot]]s, defects in [[semiconductors]]). For solutions in a continuum that are associated with this continuum, resonant states<ref name = "resonant" />  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.<ref name = "hsu">{{cite journal | doi=10.1038/natrevmats.2016.48 | title=Bound states in the continuum | date=2016 | last1=Hsu | first1=Chia Wei | last2=Zhen | first2=Bo | last3=Stone | first3=A. Douglas | last4=Joannopoulos | first4=John D. | last5=Soljačić | first5=Marin | journal=Nature Reviews Materials | volume=1 | issue=9 | page=16048 | bibcode=2016NatRM...116048H | hdl=1721.1/108400 | s2cid=123778221 | hdl-access=free }}</ref>

==Classification of bound states in the continuum by mechanism of occurrence ==

Source:<ref name="hsu"/>

=== BICs arising when solving the inverse problem ===

==== Wigner-von Neumann's BIC (Potential engineering)====
The wave function of one of the continuum states is modified to be normalizable and the corresponding potential is selected for it.

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In the work of von Neumann and Wigner <ref name="wigner">{{Cite journal |author = J. von Neumann, E.P. Wigner |editor=  |format= |url=  |title= Über merkwürdige diskrete Eigenwerte |type=  |orig-year= | agency =  |journal = Phys. Z.  |year= 1929 |volume= 30 |issue=  |number= |pages = 465–467 |series=  |issn =  |doi =  |bibcode =  |arxiv =  |pmid =  |language=  |quote=  }}</ref> two types of spherically symmetric potentials are considered, with the potential of
<math> V(r)=\frac{2}{r^2}-9r^4 </math> 
localization occurs due to the "reflection" from infinity (since the material point goes to infinity for a finite time, rolling down from this potential, and the probability of being there tends to zero). A similar example in 1D considers the exponential dependence of the potential.<ref>Zafar Ahmed et al 2019 Phys. Scr. 94 105214</ref> The second type of potential<ref>Simon, B. On positive eigenvalues of one-body Schrödinger operators. Commun. Pure Appl. Math. 22, 531-538 (1969)</ref><ref name="Stillinger, F. H. 1975">Stillinger, F. H. & Herrick, D. R. Bound states in the continuum. Phys. Rev. A 11, 446-454 (1975)</ref> has a complex shape, but is periodically modulated with period <math>\pi</math>. A wave function with period <math> 2\pi </math> is reflected from such a potential to the center in analogy to the reflection from a [[Distributed Bragg reflector|Bragg mirror]]. In the one-dimensional case, such potentials can be realized using superlattices.<ref>D. R. Herrick, "Construction of bound states in the continuum for epitaxial heterostructure superlattices," Physica B 85, 44-50 (1977).</ref><ref>Molina, M. I., Miroshnichenko, A. E. & Kivshar, Y. S. Surface bound states in the continuum. Phys. Rev. Lett. 108, 070401 (2012)</ref>
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==== Hopping rate engineering ====
In the [[tight binding]] approximation, the jump rates are modified so that the state becomes localized

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BICs of this type have been realized in an array of coupled optical [[Waveguide (optics)|waveguides]],<ref>Corrielli, G., Della Valle, G., Crespi, A., Osellame, R. & Longhi, S. Observation of surface states with algebraic localization. Phys. Rev. Lett. 111, 220403 (2013)</ref> with the hopping rate modulated by the spacing between the waveguides. Experimentally, a state with following rates

: <math> \kappa_{n}=\left\{\begin{array}{ll}
\kappa & n \neq 2l \\
\left(\frac{l+1}{l}\right) \kappa & n=2l,(l=1,2,3, \ldots)
\end{array}\right.</math>

was realized in the system which consist of 40 waveguides. There are also examples of BICs in [[Non-Hermitian quantum mechanics|PT-symmetric]] systems.<ref>Stefano Longhi. Non-Hermitian tight-binding network engineering. Phys. Rev. A 93, 022102</ref><ref>Stefano Longhi, "Bound states in the continuum in PT-symmetric optical lattices, " Opt. Lett. 39, 1697—1700 (2014)</ref>
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==== Boundary shape engineering ====
Sources for BICs of different types, e.g. Fabry - Perot type are replaced by scatterers so as to create BIC of the same type.

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BICs of this type are observed for water waves.<ref>McIver, M. An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257—266 (1996)</ref><ref>Kuznetsov, N. & McIver, P. On uniqueness and trapped modes in the water-wave problem for a surface-piercing axisymmetric body. Q. J. Mech. Appl. Math. 50, 565—580 (1997)</ref><ref>Porter, R. & Evans, D. V. Water-wave trapping by floating circular cylinders. J. Fluid Mech. 633, 311—325 (2009).</ref><ref>{{cite journal | doi=10.1017/S0022112010004222 | title=Experimental study on water-wave trapped modes | date=2011 | last1=Cobelli | first1=P. J. | last2=Pagneux | first2=V. | last3=Maurel | first3=A. | last4=Petitjeans | first4=P. | journal=Journal of Fluid Mechanics | volume=666 | pages=445–476 | bibcode=2011JFM...666..445C | s2cid=55836054 | url=https://hal.science/hal-03994541/file/art_59.pdf }}</ref>
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=== BICs arising due to parameter tuning===
==== Fabry-Perot BICs====
For resonant structures, the reflection coefficient near resonance can reach unity. Two such structures can be arranged in such a way that they radiate in antiphase and compensate each other.
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In addition to the waves on water mentioned above, states of this type are found in many systems, such as those of pairs of quantum dots coupled to a waveguide,<ref>Cattapan, G. & Lotti, P. Bound states in the continuum in two-dimensional serial structures. Eur. Phys. J. B 66, 517–523 (2008)</ref><ref>Sadreev, A. F., Bulgakov, E. N. & Rotter, I. Trapping of an electron in the transmission through two quantum dots coupled by a wire. JETP Lett. 82, 498–503 (2005)</ref> double chains of copper atoms,<ref>Díaz-Tendero, S., Borisov, A. G. & Gauyacq, J.-P. Extraordinary electron propagation length in a metallic double chain supported on a metal surface. Phys. Rev. Lett. 102, 166807 (2009)</ref> waveguide pairs of Z- and П- shapes <ref>Sadreev, A. F., Maksimov, D. N. & Pilipchuk, A. S. Gate controlled resonant widths in double-bend waveguides: bound states in the continuum. J. Phys. Condens. Matter 27, 295303 (2015).</ref>
In photonics, states are realized, for example, in pairs of two-dimensional [[Photonic crystal|photonic crystal slabs]],<ref>Suh, W., Yanik, M. F., Solgaard, O. & Fan, S. Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs. Appl. Phys. Lett. 82, 1999—2001 (2003)</ref> double arrays of dielectric cylinders<ref>Ndangali, R. F. & Shabanov, S. V. Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders. J. Math. Phys. 51, 102901 (2010)</ref> etc.<ref>{{Cite journal |title= Bound States in the Continuum in Magnetophotonic Metasurfaces |journal= JETP Letters |year= 2020 |volume= 111 |issue= 1|pages = 46–49 |doi= 10.1134/S0021364020010105|bibcode= 2020JETPL.111...46C|s2cid= 255230442 |last1= Chernyak |first1= A. M. |last2= Barsukova |first2= M. G. |last3= Shorokhov |first3= A. S. |last4= Musorin |first4= A. I. |last5= Fedyanin |first5= A. A. }}</ref>
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==== Friedrich-Wintgen BICs====

Two modes of the same symmetry of one and the same structure approach each other when the parameters of the structure are changed, and at some point an anti-crossing occurs. In this case, BIC is formed on one of the branches, since the modes as if compensate each other, being in antiphase and radiating into the same radiation channel.<ref>Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum. Phys. Rev. A 32, 3231-3242 (1985)</ref><ref>{{cite journal | doi=10.1016/0375-9601(90)90361-q | title=Trapping in competitive decay of degenerate states | date=1990 | last1=Remacle | first1=F. | last2=Munster | first2=M. | last3=Pavlov-Verevkin | first3=V.B. | last4=Desouter-Lecomte | first4=M. | journal=Physics Letters A | volume=145 | issue=5 | pages=265–268 | bibcode=1990PhLA..145..265R }}</ref>

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Friedrich-Wintgen BICs (FW BICs) were considered for a hydrogen atom in a magnetic field<ref>Friedrich, H. & Wintgen, D. Physical realization of bound states in the continuum. Phys. Rev. A 31, 3964-3966 (1985)</ref> and experimentally manifested in inhibition of autoionization of barium atom,<ref>{{cite journal | doi=10.1103/physrevlett.55.1979 | title=Autoionization Inhibited by Internal Interferences | date=1985 | last1=Neukammer | first1=J. | last2=Rinneberg | first2=H. | last3=Jönsson | first3=G. | last4=Cooke | first4=W. E. | last5=Hieronymus | first5=H. | last6=König | first6=A. | last7=Vietzke | first7=K. | last8=Spinger-Bolk | first8=H. | journal=Physical Review Letters | volume=55 | issue=19 | pages=1979–1982 | pmid=10031978 | bibcode=1985PhRvL..55.1979N | url=https://lup.lub.lu.se/record/2259032 }}</ref> topological insulators with a defect <ref>Sablikov, V. A. & Sukhanov, A. A. Helical bound states in the continuum of the edge states in two dimensional topological insulators. Phys. Lett. A 379, 1775—1779 (2015)</ref> and many other quantum systmes.<ref>Sadreev, A. F., Bulgakov, E. N. & Rotter, I. Bound states in the continuum in open quantum billiards with a variable shape. Phys. Rev. B 73, 235342 (2006)</ref><ref>Texier, C. Scattering theory on graphs: II. The Friedel sum rule. J. Phys. A 35, 3389 (2002).</ref>
Also, this type of BIC is found in two-dimensional and three-dimensional cylindrical open acoustic resonators.<ref>Hein, S., Koch, W. & Nannen, L. Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems. J. Fluid Mech. 692, 257—287 (2012)</ref><ref>Lyapina, A. A., Maksimov, D. N., Pilipchuk, A. S. & Sadreev, A. F. Bound states in the continuum in open acoustic resonators. J. Fluid Mech. 780, 370—387 (2015)</ref> In photonics, accidental BICs in periodic structures can sometimes appear as FW BICs,<ref>{{cite journal | doi=10.1103/physreva.98.053840 | title=Avoided crossings and bound states in the continuum in low-contrast dielectric gratings | date=2018 | last1=Bulgakov | first1=Evgeny N. | last2=Maksimov | first2=Dmitrii N. | journal=Physical Review A | volume=98 | issue=5 | page=053840 | arxiv=1808.03180 | bibcode=2018PhRvA..98e3840B }}</ref><ref>{{cite journal | doi=10.1515/nanoph-2020-0346 | title=Bound states in the continuum (BIC) accompanied by avoided crossings in leaky-mode photonic lattices | date=2020 | last1=Lee | first1=Sun-Goo | last2=Kim | first2=Seong-Han | last3=Kee | first3=Chul-Sik | journal=Nanophotonics | volume=9 | issue=14 | pages=4373–4380 | arxiv=2007.00371 | bibcode=2020Nanop...9..346L }}</ref> also modes of this type appeared in Bragg waveguide at interaction of TE and TM waves through anisotropic medium.<ref>{{cite journal | doi=10.1038/s42005-020-0353-z | title=One-dimensional photonic bound states in the continuum | date=2020 | last1=Pankin | first1=P. S. | last2=Wu | first2=B.-R. | last3=Yang | first3=J.-H. | last4=Chen | first4=K.-P. | last5=Timofeev | first5=I. V. | last6=Sadreev | first6=A. F. | journal=Communications Physics | volume=3 | issue=1 | page=91 | bibcode=2020CmPhy...3...91P | doi-access=free }}</ref> A notable example is the theoretical realization of BIC on a layered spherical nanoparticle of specifically selected materials,<ref>Embedded Photonic Eigenvalues in 3D Nanostructures. Francesco Monticone and Andrea Alù. Phys. Rev. Lett. 112, 213903 (2014)</ref><ref>M. G. Silveirinha, Phys. Rev. A 89, 023813 (2014).</ref> in this case BICs appear due to the interaction of different dipole modes of the same particle. By the same mechanism high-Q states can be realized in nonspherical dielectric resonators as well,<ref>{{cite journal | doi=10.1103/physrevlett.119.243901 | title=High- <math>Q</math> Supercavity Modes in Subwavelength Dielectric Resonators | date=2017 | last1=Rybin | first1=Mikhail V. | last2=Koshelev | first2=Kirill L. | last3=Sadrieva | first3=Zarina F. | last4=Samusev | first4=Kirill B. | last5=Bogdanov | first5=Andrey A. | last6=Limonov | first6=Mikhail F. | last7=Kivshar | first7=Yuri S. | journal=Physical Review Letters | volume=119 | issue=24 | page=243901 | pmid=29286713 | hdl=1885/238511 | s2cid=41020099 | hdl-access=free }}</ref><ref>K. Koshelev et al., Science 367, 288—292 (2020).</ref> but in this case each mode consist of an infinite number of multipoles,<ref>S. Gladyshev, K. Frizyuk, A. Bogdanov Phys. Rev. B 102, 075103 — Published 3 August 2020</ref> and the Q-factor doesn't go to infinity.
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==== Single-resonance parametric BICs====
Occur when a single mode can be represented as a sum of contributions,<ref>{{cite journal | doi=10.1038/srep31908 | title=Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs | date=2016 | last1=Gao | first1=Xingwei | last2=Hsu | first2=Chia Wei | last3=Zhen | first3=Bo | last4=Lin | first4=Xiao | last5=Joannopoulos | first5=John D. | last6=Soljačić | first6=Marin | last7=Chen | first7=Hongsheng | journal=Scientific Reports | volume=6 | page=31908 | pmid=27557882 | pmc=4997268 | arxiv=1603.02815 | bibcode=2016NatSR...631908G }}</ref> each of which varies with the structure parameters. At some point, destructive interference of all contributions occurs.

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BICs of this type are usually considered in photonic crystals<ref>Hsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499, 188—191 (2013)</ref> and are also accidental BICs. They can appear in planar structures in <math>xy</math>-plane, which possess <math>{C_2}^z</math> and <math>\sigma_ z</math> symmetries,<ref>B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇci ́c, "Topological Nature of Optical Bound States in the Continuum, " Phys. Rev. Lett. 113, 257401 (2014)</ref> but their existence is not guaranteed by symmetry, they appear for particular values of structure parameters. In addition to photonic periodic structures,<ref>Porter, R. & Evans, D. V. Embedded Rayleigh-Bloch surface waves along periodic rectangular arrays. Wave Motion 43, 29-50 (2005).</ref><ref>Bulgakov, E. N. & Sadreev, A. F. Light trapping above the light cone in a one-dimensional array of dielectric spheres. Phys. Rev. A 92, 023816 (2015)</ref> BICs of this type can appear for water waves,<ref>McIver, M., Linton, C. M., McIver, P., Zhang, J. & Porter, R. Embedded trapped modes for obstacles in two-dimensional waveguides. Q. J. Mech. Appl. Math. 54, 273—293 (2001).</ref> quantum waveguides,<ref>Linton, C. M. & Ratcliffe, K. Bound states in coupled guides. I. Two dimensions. J. Math. Phys. 45, 1359—1379 (2004).</ref> mechanical resonators,<ref>Chen, Y. et al. Mechanical bound state in the continuum for optomechanical microresonators. New J. Phys. 18, 063031 (2016)</ref> as a surface waves<ref>Yamanouchi, K. & Shibayama, K. Propagation and amplification of rayleigh waves and piezoelectric leaky surface waves in LiNbO3 . J. Appl. Phys. 43, 856—862 (1972).</ref> etc.
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=== Symmetry-protected BICs===
Arise when the symmetry of the eigenstate differs from any of the possible symmetries of propagating modes in the continuum.
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In the simple case, states of this type are observed in the Г-point of planar photonic crystals possessing symmetry <math>{C_n}^z</math>.<ref>B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇci ́c, "Topological Nature of Optical Bound States in the Continuum, " Phys. Rev. Lett. 113, 257401 (2014).</ref><ref>Z. Sadrieva, K. Frizyuk, M. Petrov, Yu. Kivshar, and A.Bogdanov «Multipolar origin of bound states in the continuum» Phys. Rev. B 100, 115303</ref><ref>Lee, J. et al. Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs. Phys. Rev. Lett. 109, 067401 (2012)</ref> Often, other types of BICs exist in the same structures. Also symmetry-protected BICs have been observed in coupled waveguide systems.<ref>Dreisow, F. et al. Adiabatic transfer of light via a continuum in optical waveguides. Opt. Lett. 34, 2405—2407 (2009)</ref><ref>Plotnik, Y. et al. Experimental observation of optical bound states in the continuum. Phys. Rev. Lett. 107, 183901 (2011).</ref>
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=== Separable BICs===
Arise when the eigenvalue problem is solved by the [[Separation of Variables|Separation of Variables Method]], and the [[wave function]] is represented, for example, as <math>\psi=\psi_1(x)\psi_2(y)</math>, where both multipliers correspond to localized states, with the total energy lying in the continuum.
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The Hamiltonian for obtaining this type of BIC was first proposed by Robnik<ref>Robnik, M. A simple separable Hamiltonian having bound states in the continuum. J. Phys. A 19, 3845 (1986).</ref> and then studied in various quantum systems<ref>Duclos, P., Exner, P. & Meller, B. Open quantum dots: resonances from perturbed symmetry and bound states in strong magnetic fields. Rep. Math. Phys. 47, 253—267 (2001).</ref><ref>Prodanovic´, N., Milanovic´, V., Ikonic´, Z., Indjin, D. & Harrison, P. Bound states in continuum: quantum dots in a quantum well. Phys. Lett. A 377, 2177—2181 (2013).</ref> and 2D photonic structures.<ref>Čtyroký, J. Photonic bandgap structures in planar waveguides. J. Opt. Soc. Am. A 18, 435—441 (2001).</ref><ref>Watts, M. R., Johnson, S. G., Haus, H. A. & Joannopoulos, J. D. Electromagnetic cavity with arbitrary Q and small modal volume without a complete photonic bandgap. Opt. Lett. 27, 1785—1787 (2002).</ref>
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== Wigner-Von Neumann BICs ==
Bound states in the continuum were first predicted in 1929 by [[Eugene Wigner]] and [[John von Neumann]].<ref name="wigner"/> 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:

: <math>
-\frac{h^{2}}{8 \pi^{2} m} \Delta \psi+(V(r)-E) \psi=0,
</math>

the angle derivatives disappear, since we limit ourselves to considering only spherically symmetric wave functions:

: <math>
\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}=\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r} \frac{\partial}{\partial r},
</math>

For <math>E=0</math> to be the eigenvalue for the spherically symmetric wave function <math>\psi=\psi(r)</math>, the potential must be

: <math>V=\frac{h^{2}}{8 \pi^{2} m}\left(\frac{\psi^{\prime \prime}}{\psi}+\frac{2 \psi^{\prime}}{r \psi}\right)</math>.

We obtain the specific values <math>\psi</math> and <math>V</math> for which the BIC will be observed.

=== First case ===
[[File:Wikibic1.png|thumb|Potential and wave function corresponding to zero energy, for the first case of the Wigner-Von Neumann BIC]]
Let us consider the function <math>\psi=r^\alpha{\sin r^\beta}</math>. While the integral <math>\int_0^\infty 4\pi r^2|\psi(r)|^2\text{d}r=\int_0^\infty 4\pi r^{2\alpha+2}{\sin^2 r^\beta}\text{d}r</math> must be finite, then considering the behavior when <math>r=0</math>, we get that <math>2\alpha+\beta+2>-1</math>, then considering the behavior when  <math>r=\infty</math>, we get <math>2\alpha+2<-1</math>. The regularity <math>V(r)</math> in <math>r\neq0</math> requires <math>2\alpha+\beta+1=0</math>. Finally, we get <math>\alpha=-2, \  \beta=3</math>.

Assuming <math>\psi(r)=\frac{\sin r^3}{r^2}</math>, then the potential will be equal to (discarding the irrelevant multiplier <math>{h^{2}}/{8 \pi^{2} m}</math>):

: <math>
V(r)=\frac{2}{r^2}-9r^4
</math>

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 <math>E=0</math>.

In the work <ref name="wigner" />  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 <math>V=\frac{2}{r^2}-9r^4</math> is described by the following equation:

: <math>
{\frac{m}{2}\left(\frac{\text{d} r}{\text{d} t}\right)^{2}+V(r)=\mathrm{Const}}
</math>

: <math>
{\frac{\text{d} r}{\text{d} t}=\sqrt{\frac{2}{m} \mathrm{Const}-\frac{4}{m} \frac{{1}}{r^{2}}+\frac{18}{m} r^{4}}}
</math>

It's easy to see that when <math>r\rightarrow \infty</math>, <math>\frac{\text{d}r}{\text{d}t}\rightarrow \infty</math>, so the asymptotic is
: <math>
\frac{d r}{d t}=\frac{3 \sqrt{2}}{\sqrt{m}} r^{2};  \ \ \ \ \ \ \  \frac{1}{r}=\frac{3 \sqrt{2}}{\sqrt{m}}\left(t_{0}-t\right) , \ \ 
</math> <math>
r=\frac{\sqrt{m}}{3 \sqrt{2}\left(t_{0}-t\right)}
</math>

that is, for a finite time <math>t=t_0</math> the point goes to infinity. The stationary solution <math>\psi(r)</math> means that the point returns from infinity again, that it is as if it is reflected from there and starts oscillating. The fact that <math>\psi(r)</math> at <math>r=\infty</math> 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 <math>r_{min}</math> to infinity and back) is periodic, it is logical that this quantum mechanical problem has a stationary solution.

=== Second case ===
[[File:Wikibic2.png|thumb|(a) Potential and wave function (on an arbitrary scale along the vertical axis) corresponding to zero energy, for the second case of the Wigner von Neumann SSC, <math>A=1.5</math> (b) <math>A=15</math>.]]

Let's move on to the second example, which can no longer be interpreted from such considerations.

First of all, we take a function <math>\psi=\frac{\sin r}{r}</math>, then <math>V=-1</math>. These are divergent spherical waves, since the energy <math>E=0</math> is greater than the potential <math>V=-1</math>, the classical kinetic energy remains positive. The wave function belongs to a continuous spectrum, the integral <math>\int_0^\infty 4\pi r^2|\psi(r)|^2\text{d}r=4\pi \int_0^\infty \sin^2 r\text{d}r</math> 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:

: <math>\psi=\frac{\sin r}{r}f(r)</math>

If the function <math>f(r)</math> is continuous, and at <math>r \rightarrow \infty</math> the asymptotic is <math>r^\alpha, \ \ \alpha<-1/2</math> then the integral is finite. The potential would then be equal (with the corrected arithmetical error in the original article):<ref name="Stillinger, F. H. 1975"/>

: <math>
V=-1+2 \operatorname{ctg} r \frac{f^{\prime}(r)}{f(r)}+\frac{f^{\prime \prime}(r)}{f(r)}.
</math>

In order for the potential to remain near -1, and at <math>r\rightarrow \infty</math> tend to -1, we must make the functions <math>\operatorname{ctg} r \frac{f^{\prime}(r)}{f(r)}, \ \frac{f^{\prime \prime}(r)}{f(r)}</math> small and at <math>r\rightarrow \infty</math> tend to zero.

In the first case, also <math>\frac{f^{\prime}(r)}{f(r)}</math> should vanish for <math>\operatorname{ctg} r= \infty</math>, namely for <math>r=0,\pi, \ 2\pi, \ 2\pi, \dots</math>, that is for <math>\sin r=0</math>. This is the case when <math>f(r)=\int_0^r \sin^2r \text{d}r=\frac{r}2-\frac14\sin2r</math> or any other function of this expression.

Let assume <math>f(r)=[A^2+(\frac{r}2-\frac14\sin2r)^2]^{-1}</math>, where <math>A</math> is arbitrary (here <math>f(r)</math>  tends to <math>r^\alpha, \ \ \alpha<-1/2</math> when <math>r\rightarrow \infty</math>). Then

: <math>
\psi=\frac{\sin r}{r(A^2+(2r-\sin2r)^2)}.
</math>

The expression for the potential is cumbersome, but the graphs show that for <math>r\rightarrow \infty</math> the potential tends to -1.
Furthermore, it turns out that for any <math>\varepsilon>0</math> one can choose such an ''A'' that the potential is between <math>-1-\varepsilon</math> and <math>-1+\varepsilon</math>.
We can see that the potential oscillates with period <math>\pi</math> and the wave function oscillates with period <math>2\pi</math>. 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 [[Distributed Bragg reflector|Bragg mirror]].

==Notes==
{{Reflist|group=nb}}
{{Reflist}}

== Literature ==

* {{cite journal |doi=10.1038/natrevmats.2016.48 |title=Bound states in the continuum |date=2016 |last1=Hsu |first1=Chia Wei |last2=Zhen |first2=Bo |last3=Stone |first3=A. Douglas |last4=Joannopoulos |first4=John D. |last5=Soljačić |first5=Marin |journal=Nature Reviews Materials |volume=1 |issue=9 |bibcode=2016NatRM...116048H |hdl=1721.1/108400 |s2cid=123778221 |hdl-access=free }}
* {{cite journal |doi=10.1364/OPN.31.1.000038 |title=Engineering with Bound States in the Continuum |date=2020 |last1=Koshelev |first1=Kirill |last2=Bogdanov |first2=Andrey |last3=Kivshar |first3=Yuri |journal=Optics and Photonics News |volume=31 |issue=1 |page=38 |bibcode=2020OptPN..31a..38K |s2cid=213414202 }}
* {{cite journal |doi=10.1002/adom.202001469 |title=Photonic Bound States in the Continuum: From Basics to Applications |date=2021 |last1=Azzam |first1=Shaimaa I. |last2=Kildishev |first2=Alexander V. |journal=Advanced Optical Materials |volume=9 |s2cid=228843072 }}
* {{cite journal |arxiv=2011.01221 |doi=10.1088/1361-6633/abefb9 |title=Interference traps waves in an open system: Bound states in the continuum |date=2021 |last1=Sadreev |first1=Almas F. |journal=Reports on Progress in Physics |volume=84 |issue=5 |pmid=33730696 |bibcode=2021RPPh...84e5901S }}

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[[Category:Quantum optics]]
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