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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication Open Access Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions(American Physical Society (APS), 2021) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We 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.Publication Open Access Transfer matrices as nonunitary S matrices, multimode unidirectional invisibility, and perturbative inverse scattering(American Physical Society (APS), 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We show that in one dimension the transfer matrix M of any scattering potential v coincides with the S matrix of an associated time-dependent non-Hermitian 2 x 2 matrix Hamiltonian H(tau). If v is real valued, H(tau) is pseudo-Hermitian and its exceptional points correspond to the classical turning points of v. Applying time-dependent perturbation theory to H(tau) we obtain a perturbative series expansion for M and use it to study the phenomenon of unidirectional invisibility. In particular, we establish the possibility of having multimode unidirectional invisibility with wavelength-dependent direction of invisibility and construct various physically realizable optical potentials possessing this property. We also offer a simple demonstration of the fact that the off-diagonal entries of the first-order Born approximation for M determine the form of the potential. This gives rise to a perturbative inverse scattering scheme that is particularly suitable for optical design. As a simple application of this scheme, we construct an infinite-range unidirectionally invisible potential.Publication Open Access Comment on “Identical motion in classical and quantum mechanics”(American Physical Society (APS), 1999) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231Makowski and Konkel [Phys. Rev. A 58, 4975 (1998)] have obtained certain classes of potentials which lead to identical classical and quantum Hamilton-Jacobi equations. We obtain the most general form of these potentials.Publication Open Access Erratum: Geometric phase, bundle classification, and group representation(American Institute of Physics (AIP) Publishing, 1999) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231Publication Open Access On a Z(3)-graded generalization of the Witten index(Elsevier, 2002) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We construct a realization of the algebra of the Z(3)-graded topological symmetry of type (1, 1, 1) in terms of a pair of operators D-1 : H-1 --> H-2, D-2 : H-2 --> H-3 satisfying [D1D1dagger, (D2D2)-D-dagger] = 0. We show that the sequence of the restriction of these operators to the zero-energy subspace forms a complex and establish the equality of the corresponding topological invariants with the analytic indices of these operators. (C) 2002 Published by Elsevier Science B.V.Publication Open Access Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications(Institute of Physics (IOP) Publishing, 2022) Loran, Farhang; Department of Physics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; 4231In 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.Publication Open Access Statistics and characterization of matrices by determinant and trace(Springer, 2017) Department of Mathematics; Department of Physics; Yörük, Ekin Sıla; Alkan, Emre; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; N/A; 32803Answering a question of Erdös, Komlós proved in 1968 that almost all n×n Bernoulli matrices are nonsingular as n→∞. In this paper, we offer a new perspective on the question of Erdös by studying n×n matrices with prime number entries in an almost all sense. Precisely, it is shown that, as x→∞, the probability of randomly choosing a nonsingular n×n matrix among all n×n matrices with prime number entries that are ≤x is 1. If A is a unitary matrix, then it is well known that |detA|=1. However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality.Publication Open Access Transfer matrix in scattering theory: a survey of basic properties and recent developments(TÜBİTAK, 2020) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.Publication Open Access Time-dependent pseudo-Hermitian Hamiltonians and a hidden geometric aspect of quantum mechanics(Multidisciplinary Digital Publishing Institute (MDPI), 2020) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231A non-Hermitian operator H defined in a Hilbert space with inner product may serve as the Hamiltonian for a unitary quantum system if it is ηpseudo-Hermitian for a metric operator (positive-definite automorphism) η. The latter defines the inner product of the physical Hilbert space Hh of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.Publication Open Access Effects of dilute Zn impurities on the uniform magnetic susceptibility of YBa2Cu3O7-δ(American Physical Society (APS), 2000) Department of Mathematics; Bulut, Nejat; Faculty Member; Department of Mathematics; College of SciencesThe effects of dilute Zn impurities on the uniform magnetic susceptibility are calculated in the normal metallic state for a model of the spin fluctuations of the layered cuprates. It is shown that scatterings from extended impurity potentials can lead to a coupling of the q∼ (π, π) and the q∼0 components of the magnetic susceptibility χ(q). Within the presence of antiferromagnetic correlations, this coupling can enhance the uniform susceptibility. The implications of this result for the experimental data on Zn substituted YBa2Cu3O7-δ are discussed.