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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication Open Access Generalized adiabatic product expansion: a nonperturbative method of solving the time-dependent Schrodinger equation(American Institute of Physics (AIP) Publishing, 1999) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field ℰ→(t) through the Stark Hamiltonian H = λ(J→ · ℰ→)2.Publication Open Access On the distributions of sigma(N)/N and N/Phi(N)(Rocky Mountain Mathematics Consortium, 2013) Department of Mathematics; Department of Mathematics; Alkan, Emre; Faculty Member; College of Sciences; 32803We prove that the distribution functions of sigma(n)/n and n/phi(n) both have super-exponential asymptotic decay when n ranges over certain subsets of integers, which, in particular, can be taken as the set of l-free integers not divisible by a thin subset of primes.Publication Open Access Differential realization of pseudo-Hermiticity: a quantum mechanical analog of Einstein's field equation(American Institute of Physics (AIP) Publishing, 2006) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231For a given pseudo-Hermitian Hamiltonian of the standard form: H=p(2)/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator eta satisfying H(dagger)=eta H eta(-1) to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of eta. We apply our general results to calculate eta for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general eta up to second-order terms in the coupling constants.Publication Open Access The intersection problem for PBD(5*,3)s(Elsevier, 2008) Department of Mathematics; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; College of Sciences; 105252For every v equivalent to 5 (mod 6) there exists a pairwise balanced design (PBD) of order v with exactly one block of size 5 and the rest of size 3. We will refer to such a PBD as a PBD(5*, 3). A flower in a PBD(5*, 3) is the set of all blocks containing a given point. If (S, B) is a PBD(5*, 3) and F is a flower, we will write F* to indicate that F contains the block of size 5. The intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v containing the same block b of size 5 such that vertical bar(B-1\b) boolean AND (B-2\b)vertical bar = k. The flower intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v having a common flower F* such that vertical bar(B-1\F*) boolean AND (B-2\F*)vertical bar = k. In this paper we give a complete solution of both problems.Publication Open Access Embedding partial Latin squares in Latin squares with many mutually orthogonal mates(Elsevier, 2020) Donovan, Diane; Grannell, Mike; Department of Mathematics; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; College of Sciences; 27432In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.Publication Open Access Pseudounitary operators and pseudounitary quantum dynamics(American Institute of Physics (AIP) Publishing, 2004) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of i=root-1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2x2 pseudounitary matrices and discuss an example of a quantum system with a 2x2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudounitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.Publication Open Access Symplectic and lagrangian surfaces in 4-Manifolds(Rocky Mountain Mathematics Consortium, 2008) Department of Mathematics; Department of Mathematics; Etgü, Tolga; Faculty Member; College of Sciences; 16206This is a brief summary of recent examples of isotopically different symplectic and Lagrangian surfaces representing a fixed homology class in a simply-connected symplectic 4-manifold.Publication Open Access Embeddings of P-3-designs into bowtie and almost bowtie systems(Elsevier, 2009) Lindner, Curt; Quattrocchi, Gaetano; Department of Mathematics; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; College of Sciences; 105252This paper determines for each admissible w, the set of all n such that every P-3-design of order w can be embedded in an (almost) bowtie system of order n.Publication Open Access Erratum: Geometric phase, bundle classification, and group representation(American Institute of Physics (AIP) Publishing, 1999) Department of Mathematics; Department of Physics; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231Publication Open Access Quantum mechanics of a photon(American Institute of Physics (AIP) Publishing, 2017) Department of Physics; Department of Mathematics; Department of Physics; Department of Mathematics; Babaei, Hassan; Mostafazadeh, Ali; Faculty Member; Graduate School of Sciences and Engineering; N/A; 4231A first-quantized free photon is a complex massless vector field A = (A(mu)) whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.