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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3
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Publication Metadata only Can Nth order Born approximation be exact?(IOP Publishing Ltd, 2024) Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Department of Mathematics; College of SciencesFor the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions, we identify a condition on the scattering interaction under which the Nth order Born approximation gives the exact solution of the scattering problem for some N >= 1.Publication Metadata only Consistent treatment of quantum systems with a time-dependent Hilbert space(MDPI, 2024) Department of Mathematics; Mostafazadeh, Ali; Department of Mathematics; College of SciencesWe consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product. © 2024 by the author.Publication Metadata only The triangle intersection problem for K4 - E designs(Utilitas Mathematica Publishing Inc., 2007) Billington, Elizabeth J.; Lindner, C. C.; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432An edge-disjoint decomposition of the complete graph Kn into copies of K4 - e, the simple graph with four vertices and five edges, is known to exist if and only if n ≡ 0 or 1 (mod 5) and n ≥ 6 (Bermond and Schönheim, Discrete Math. 19 (1997)). The intersection problem for K4 - e designs has also been solved (Billington, M. Gionfriddo and Lindner, J. Statist. Planning Inference 58 (1997)); this problem finds the number of common K4 - e blocks which two K4 - e designs on the same set may have. Here we answer the question: how many common triangles may two K4 - e designs on the same set have? Since it is possible for two K4 - e designs on the same set to have no common K4 - e blocks and yet some positive number of common triangles, this problem is largely independent of the earlier K4 - e intersection result.Publication Metadata only Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator(Iop Publishing Ltd, 1998) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrodinger equation. A simple generalization of this result leads to the reduction of the Schrodinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive red solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.Publication Metadata only Propagation of electromagnetic waves in linear media and pseudo-hermiticity(EPL Association, European Physical Society, 2008) Loran, F.; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We express the electromagnetic field propagating in an arbitrary time-independent non-dispersive medium in terms of an operator that turns out to be pseudo-Hermitian for Hermitian dielectric and magnetic permeability tensors (epsilon) over left right arrow and (mu) over left right arrow. We exploit this property to determine the propagating field. In particular, we obtain an explicit expression for a planar field in an isotropic medium with (epsilon) over left right arrow = epsilon(1) over left right arrow and mu = mu(1) over left right arrow varying along the direction of the propagation. We also study the scattering of plane waves due to a localized inhomogeneity.Publication Metadata only Z(n)-graded topological generalizations of supersymmetry and orthofermion algebra(Iop Publishing Ltd, 2003) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We review various generalizations of supersymmetry and discuss their relationship. in particular, we show how supersymmetry, parasupersymmetry, fractional supersymmetry, orthosupersymmetry, and the Z(n)-graded topological symmetries are related.Publication Metadata only Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti- PT -symmetry(American Physical Society, 2023) Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We 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.Publication Metadata only Hilbert space structures on the solution space of Klein-Gordon-type evolution equations(Iop Publishing Ltd, 2003) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We use the theory of pseudo-Hermitian operators to address the problem of the construction and classification of positive-definite invariant inner-products on the space of solutions of a Klein-Gordon-type evolution equation. This involves dealing with the peculiarities of formulating a unitary quantum dynamics in a Hilbert space with a time-dependent inner product. We apply our general results to obtain possible Hilbert space structures on the solution space of the equation of motion for a classical simple harmonic oscillator, a free Klein-Gordon equation and the Wheeler-DeWitt equation for the FRW-massive-real-scalar-field models.Publication Metadata only Physics of spectral singularities(Trends in Mathematics, 2015) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Spectral singularities are certain points of the continuous spectrum of generic complex scattering potentials. We review the recent developments leading to the discovery of their physical meaning, consequences, and generalizations. In particular, we give a simple definition of spectral singularities, provide a general introduction to spectral consequences of ��-symmetry (clarifying some of the controversies surrounding this subject), outline the main ideas and constructions used in the pseudo-Hermitian representation of quantum mechanics, and discuss how spectral singularities entered in the physics literature as obstructions to these constructions. We then review the transfer matrix formulation of scattering theory and the application of complex scattering potentials in optics. These allow us to elucidate the physical content of spectral singularities and describe their optical realizations. Finally, we survey some of the most important results obtained in the subject, drawing special attention to the remarkable fact that the condition of the existence of linear and nonlinear optical spectral singularities yield simple mathematical derivations of some of the basic results of laser physics, namely the laser threshold condition and the linear dependence of the laser output intensity on the gain coefficient.Publication Metadata only On the dynamical invariants and the geometric phases for a general spin system in a changing magnetic field(Elsevier Science Bv, 2001) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We consider a class of general spin Hamiltonians of the form H-S(t) = H-0(t) + H ' (t), where H-0(t) and H ' (t) describe the dipole interaction of the spins with an arbitrary time-dependent magnetic field and the internal interaction of the spins, respectively. We show that if H ' (t) is rotationally invariant, then H-S(t) admits the same dynamical invariant as H-0(t). A direct application of this observation is a straightforward rederivation of the results of Yan et al. (Phys. Lett. A 251 (1999) 289, 259 (1999) 207) on the Heisenberg spin system in a changing magnetic field.