Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work. The fascinating wave phenomenon of ‘bound states in the continuum’ spans different material and wave systems, including electron, electromagnetic and mechanical waves. In this Review, we focus on the common physical mechanisms underlying these bound states, whilst also discussing recent experimental realizations, current applications and future opportunities for research.

Bound states in the continuum

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.

https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-von-oppen/papers/nlin_CD_0605028.pdf

Quantum graphs: Applications to quantum chaos and universal spectral statistics

A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Graph models for waves in thin structures

Sturm–Liouville differential operators on compact graphs without cycles (i.e. on trees) are studied. We establish properties of the spectral characteristics and investigate two inverse problems of recovering the operator from the so-called Weyl functions and from a system of spectra. For these inverse problems, we prove the uniqueness theorems and obtain a procedure for constructing the solution by the method of spectral mappings.

http://iopscience.iop.org/article/10.1088/0266-5611/21/3/017/pdf

Inverse spectral problems for Sturm–Liouville operators on graphs

The spectrum of small vibrations of a graph consisting of three joint smooth strings with the free ends fixed can be reduced to the Sturm--Liouville boundary problem on a graph. This problem occurs...

Inverse Problem for the Sturm--Liouville Equation on a Simple Graph

It is shown that the potential of the Sturm-Liouville equation on interval [0,a] may be restored by the spectra of three boundary problems generated by the equation on the intervals [0,a], [0, 1/2a] and [1/2a,a], respectively. The algorithm of construction is given as well as the sufficient conditions for three sequences of real numbers to be the spectra of the mentioned boundary problems. The problem on [0,a] describes small vibrations of a smooth string with fixed ends. The problems on the half-intervals describe vibrations of the same string clamped at the point of equilibrium.

An inverse sturm-Liouville problem by three spectra

The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

The problem of small vibrations of a graph consisting of n smooth inhomogeneous stretched strings joined at the vertex with the pendant ends fixed is reduced to the Sturm–Liouville boundary problem on a star-shaped graph. The obtained problem occurs also in quantum mechanics. The spectrum of such a problem which consists of normal eigenvalues accumulating at infinity is investigated in comparison with the union of spectra of the Dirichlet–Dirichlet problems on the edges of the graph. It is shown that the eigenvalues of the spectra interlace in certain sense, thus an analogue of Sturm theorem is established. If the (n + 1) spectra (the spectrum of the boundary problem on the graph and the n spectra of the mentioned Dirichlet–Dirichlet problems) do not intersect the inverse problem of recovering the potentials on the edges from the (n + 1) spectra is uniquely solvable. The procedure of recovering of the potentials is presented. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Inverse problem for the Sturm–Liouville equation on a star‐shaped graph

A method of recovering the potential of the Sturm–Liouville equation on a half-interval using a known potential on another half-interval and the spectrum of the Dirichlet–Dirichlet problem on the whole interval is proposed.

https://iopscience.iop.org/article/10.1088/0266-5611/26/3/035011/pdf

Vyacheslav Pivovarchik

Papers

Inverse Problem for the Sturm--Liouville Equation on a Simple Graph

An inverse sturm-Liouville problem by three spectra

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

Inverse problem for the Sturm–Liouville equation on a star‐shaped graph

On the Hochstadt–Lieberman theorem