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Publication Metadata only A dynamical formulation of one-dimensional scattering theory and its applications in optics(Academic Press Inc Elsevier Science, 2014) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.Publication Metadata only A Hamiltonian formulation of the Pais-Uhlenbeck oscillator that yields a stable and unitary quantum system(Elsevier Science Bv, 2010) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 105131We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. Furthermore it yields the classical Pais-Uhlenbeck oscillator in the classical limit. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real description of this complex system, derive an integral of motion for it, and explore its quantization.Publication Metadata only A new class of adiabatic cyclic states and geometric phases for non-Hermitian Hamiltonians(Elsevier Science Bv, 1999) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(O). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field.Publication Metadata only A new correlation coefficient for bivariate time-series data(Elsevier Science Bv, 2014) Erdem, Orhan; Varlı, Yusuf; Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of SciencesThe correlation in time series has received considerable attention in the literature. Its use has attained an important role in the social sciences and finance. For example, pair trading in finance is concerned with the correlation between stock prices, returns, etc. In general, Pearson's correlation coefficient is employed in these areas although it has many underlying assumptions which restrict its use. Here, we introduce a new correlation coefficient which takes into account the lag difference of data points. We investigate the properties of this new correlation coefficient. We demonstrate that it is more appropriate for showing the direction of the covariation of the two variables overtime. We also compare the performance of the new correlation coefficient with Pearson's correlation coefficient and Detrended Cross-Correlation Analysis (DCCA) via simulated examples. (C) 2014 Elsevier B.V. All rights reserved.Publication Metadata only A physical realization of the generalized PT-, C-, and CPT-symmetries and the position operator for Klein-Gordon fields(World Scientific Publ Co Pte Ltd, 2006) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Generalized parity (P), time-reversal (T), and charge-conjugation (C) operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein-Gordon fields and show that in this realization PT and C operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein-Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton-Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein-Gordon fields with a genuine probabilistic interpretation.Publication Open Access Active invisibility cloaks in one dimension(American Physical Society (APS), 2015) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We outline a general method of constructing finite-range cloaking potentials which render a given finite-range real or complex potential, v(x), unidirectionally reflectionless or invisible at a wave number, k(0), of our choice. We give explicit analytic expressions for three classes of cloaking potentials which achieve this goal while preserving some or all of the other scattering properties of v(x). The cloaking potentials we construct are the sum of up to three constituent unidirectionally invisible potentials. We discuss their utility in making v(x) bidirectionally invisible at k(0) and demonstrate the application of our method to obtain antireflection and invisibility cloaks for a Bragg reflector.Publication Open Access Addendum to 'Unidirectionally invisible potentials as local building blocks of all scattering potentials'(American Physical Society (APS), 2014) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231In [Phys. Rev. A 90, 023833 (2014)], we offer a solution to the problem of constructing a scattering potential v(x) which possesses scattering properties of one's choice at an arbitrarily prescribed wave number. This solution involves expressing v(x) as the sum of n <= 6 finite-range unidirectionally invisible potentials. We improve this result by reducing the upper bound on n from 6 to 4. In particular, we show that we can construct v(x) as the sum of up to n = 3 finite-range unidirectionally invisible potentials, unless if it is required to be bidirectionally reflectionless.Publication Metadata only Alternative descriptions and multipartite compound quantum systems(Soc Italiana Fisica, 2009) Masillo, F.; Solombrino, L.; Department of Mathematics; Scolarici, Giuseppe; Researcher; Department of Mathematics; College of Sciences; N/AWe give a necessary and sufficient kinematical condition for compound quantum systems provided with alternative inner products to be described in terms of k subsystems. Moreover, we propose some methods to characterize the dynamics ruled by quasi-Hermitian Hamiltonians associated with multipartite compound quantum systems.Publication Metadata only Application of pseudo-Hermitian quantum mechanics to a complex scattering potential with point interactions(Iop Publishing Ltd, 2010) Mehri-Dehnavi, Hossein; Department of Mathematics; N/A; Mostafazadeh, Ali; Batal, Ahmet; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 4231; 232890We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p(2)/2m + zeta(-)delta(x + alpha) + zeta(+)delta(x - alpha), where zeta(+/-) and alpha are respectively complex and real parameters and delta(x) is the Dirac delta function. For regions in the space of coupling constants zeta(+/-) where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator eta and the corresponding equivalent Hermitian Hamiltonian h. eta turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, (sic)(zeta(+)) = -(sic)(zeta(-)). This in particular contains the PT-symmetric case: zeta(+) = zeta*. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of the PT - symmetry.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.