The concept of parity-time symmetric systems is rooted in non-Hermitian quantum mechanics where complex potentials obeying this symmetry could exhibit real spectra. The concept has applications in many fields of physics, e.g., in optics, metamaterials, acoustics, Bose-Einstein condensation, electronic circuitry, etc. The inclusion of nonlinearity has led to a number of new phenomena for which no counterparts exist in traditional dissipative systems. Several examples of nonlinear parity-time symmetric systems in different physical disciplines are presented and their implications discussed.

Nonlinear waves in PT -symmetric systems

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

Linear Differential Operators By Prof. Cornelius Lanczos. Pp. xvi + 564. (London: D. Van Nostrand Co., Ltd.; New York: D. Van Nostrand Company, Inc., 1961.) 80s.

Linear Differential Operators

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

/pdf/an-overview-of-periodic-elliptic-operators-uqeh1n79ah.pdf

An overview of periodic elliptic operators

An explicit derivation of dispersion relations and spectra for periodic Schrodinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.

/pdf/on-the-spectra-of-carbon-nano-structures-19871ricg0.pdf

On the Spectra of Carbon Nano-Structures

The spectra of Schrodinger and Dirac operators with periodic potentials on the real line have a band structure, that is, the intervals of continuous spectrum alternate with spectral gaps, or instability zones. The sizes of these zones decay, and the rate of decay depends on the smoothness of the potential. In the opposite direction, one can make conclusions about the smoothness of a potential based on the rate of decay of the instability zones. In the 1960s and 1970s this phenomenon was understood at the level of infinitely differentiable or analytic functions in the case of Schrodinger operators. However, only recently has the relationship between the smoothness of the potential and the rate of decay of the instability zones become completely understood and analyzed - for a broad range of classes of differentiable functions, - for Dirac operators and not just for Hill-Schrodinger operators, - in both the self-adjoint and non-self-adjoint cases. This paper is devoted to a survey of these results, mostly with complete proofs based on an approach developed by the authors.

Instability zones of periodic 1-dimensional Schrödinger and Dirac operators

Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-potential $v={pmatrix} 0 & P Q & 0 {pmatrix},$ subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in $L^p$-spaces and other rearrangement invariant function spaces.

One-dimensional Dirac operators 

L-bc(v)y = i(1 0 0 -1) dy/dx + v(x)y, y = (y(1)y(2)), x is an element of [0, pi], 

considered with L-2-potentials v (x) = (0 Q(x) P(x)0) and subject to regular boundary conditions (bc), have discrete spectrum. 

For strictly regular bc, it is shown that every eigenvalue of the free operator L-bc(0) is simple and has the form lambda(0)(k,alpha) = k + tau(alpha), where alpha is an element of {1, 2}, k is an element of 2Z and tau(alpha) = tau(alpha)(bc); if vertical bar k vertical bar > N(v, bc), each of the discs D-k(alpha) = {z : vertical bar z - lambda(0)(k,alpha) vertical bar < rho = rho(bc)}, alpha is an element of {1, 2}, contains exactly one simple eigenvalue lambda(k,alpha)< of L-bc (v) and (lambda(k,alpha) - lambda(0)(k,alpha))(k is an element of 2Z) is an l(2)-sequence. Moreover, it is proven that the root projections P-n,P-alpha = 1/(2 pi i) integral(partial derivative Dn)alpha (z - L-bc (v))(-1) dz satisfy the Bari-Markus condition 

Sigma(vertical bar n vertical bar>N) parallel to P-n,P-alpha - P-n,alpha(0)parallel to(2) <infinity, n is an element of 2Z, 

where P-n,alpha(0) are the root projections of the free operator L-bc(0). Hence, for strictly regular bc, there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular bc-then in general there is no Riesz basis consisting of root functions, but we prove that the corresponding system of two-dimensional root projections is a Riesz basis of projections.

Plamen Djakov

Papers

Instability zones of periodic 1-dimensional Schrödinger and Dirac operators

Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions