Wave Transport and Localization in Prime Number Landscapes

In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.

On integral representation of the solutions of a model $\vec{2b}$-parabolic boundary value problem

A general boundary value problem for Eidelman type $\overrightarrow{2b}$-parabolic system of equation without minor terms in the equations and boundary conditions, and with constant coefficients in the group of major terms is considered in the region $$\{(t,x_1,\dots,x_n)\in \mathbb{R}^{n+1}|t\in(0,T], x_j\in\mathbb{R}, j\in\{1,\dots,n-1\}, x_n>0\},$$ $T>0$, $n\ge 2$. It is assumed that the boundary conditions are connected with the system of equations by the complementing condition, which is analogous to the Lopatynsky complementing condition. Integral representations of the solutions for such a problem are derived. The kernels of the integrals from this representation form the Green's matrix of the problem. It is revealed that, in general, not all the elements of the Green's matrix are ordinary functions. Some of them contain terms that are linear combinations of Dirac delta functions and their derivatives. This occurs in cases when the boundary conditions include derivatives with respect to the variables $t$ and $x_n$ of orders that are equal or greater than the highest orders of derivatives with respect to these variables in the equations of the system. The obtained results are important, in particular, for the establishing of the correct solvability and integral representation of solutions for more general $\overrightarrow{2b}$-parabolic boundary value problems.

Some Properties of Green’s Matrix of Nonlinear Boundary Value Problem of First Order Differential

This paper discusses Green’s matrix of nonlinear boundary value problem of first-order differential system with rectangular coeffisients, especially about its properties. In this case, the differential equation of the form with boundary conditions of the form and which is a real matrix with whose entries are continuous on and . , are nonsingular matrices such that and are constant vectors. To get the Green’s matrix and the assosiated generalized Green’s matrix, we change the boundary condition problem into an equivalent differential equation by using the properties of the Moore-Penrose generalized inverse, then its solution is found by using method of variation of parameters. The last we prove that the defined matrices satisfy the properties of green’s function. The result is the corresponding the Green’s matrix and the assosiated generalized Green’s matrix have the property of Green’s functions with the jump-discontinuity.

GREEN’S MATRICES FOR FIRST ORDER DIFFERENTIAL SYSTEMS WITH NONLOCAL CONDITIONS∗

In this paper, we investigate the linear system of first order ordinary differential equations with nonlocal conditions. Green’s matrices, their explicit representations and properties are considered as well. We present the relation between the Green’s matrix for the system and the Green’s function for the differential equation. Several examples are also given.