Research Project: Zamandan Bağımsız Saçılma Kuramının Dinamik Formülasyonu, Matematiksel Temelleri ve Uygulamaları
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Contributors
Funders
ID
TB.00497
Authors
Mostafazadeh, Ali
Faculty Member
Publications
Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications
(Institute of Physics (IOP) Publishing, 2022) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Department of Physics; Yes; College of Sciences
In two dimensions, the standard treatment of the scattering problem for a delta-function potential, v(r) = 3 delta(r), leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant 3. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors' screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.
Low-frequency scattering defined by the Helmholtz equation in one dimension
(Institute of Physics (IOP) Publishing, 2021) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Department of Physics; Yes; College of Sciences
The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrodinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.
Exceptional points and pseudo-Hermiticity in real potential scattering
(The SciPost Foundation, 2022) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Department of Physics; Yes; College of Sciences
We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator H which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of Hand its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.
Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti- PT -symmetry
(American Physical Society, 2023) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Yes; College of Sciences
We develop a fundamental transfer-matrix formulation of the scattering of electromagnetic (EM) waves that incorporates the contribution of the evanescent waves and applies to general stationary linear media which need not be isotropic, homogenous, or passive. Unlike the traditional transfer matrices whose definition involves slicing the medium, the fundamental transfer matrix is a linear operator acting in an infinite-dimensional function space. It is given in terms of the evolution operator for a nonunitary quantum system and has the benefit of allowing for analytic calculations. In this respect it is the only available alternative to the standard Green's-function approaches to EM scattering. We use it to offer an exact solution of the outstanding EM scattering problem for an arbitrary finite collection of possibly anisotropic nonmagnetic point scatterers lying on a plane. In particular, we provide a comprehensive treatment of doublets consisting of pairs of isotropic point scatterers and study their spectral singularities. We show that identical and PT-symmetric doublets do not admit spectral singularities and cannot function as a laser unless the real part of their permittivity equals that of the vacuum. This restriction does not apply to doublets displaying anti-PT-symmetry. We determine the lasing threshold for a generic anti-PT-symmetric doublet and show that it possesses a continuous lasing spectrum.
Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
(American Physical Society (APS), 2021) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Department of Physics; Yes; College of Sciences
We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 x 2 matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.
