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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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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 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 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.Publication Open Access Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials(American Institute of Physics (AIP) Publishing, 2013) Eden, A.; Zelik, S. V.; Department of Mathematics; Department of Mathematics; Kalantarov, Varga; Faculty Member; college of sciences; 117655We study initial boundary value problems for the unstable convective Cahn-Hilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is well-known that without the convective term, the solutions of the unstable CH equation ?t u + ? 4xu + ?2x(|u|pu) = 0 may blow up in ?nite time for anyp > 0. In contrast to that, we show that the presence of the convective term u?xuin the Cahn-Hilliard equation prevents blow up at least for 0 < p <49. We alsoshow that the blowing up solutions still exist if p is large enough (p ? 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.Publication Open Access Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian(American Institute of Physics (AIP) Publishing, 2002) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.Publication Open Access Is weak pseudo-Hermiticity weaker than pseudo-Hermiticity?(American Institute of Physics (AIP) Publishing, 2006) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231For a weakly pseudo-Hermitian linear operator, we give a spectral condition that ensures its pseudo-Hermiticity. This condition is always satisfied whenever the operator acts in a finite-dimensional Hilbert space. Hence weak pseudo-Hermiticity and pseudo-Hermiticity are equivalent in finite-dimensions. This equivalence extends to a much larger class of operators. Quantum systems whose Hamiltonian is selected from among these operators correspond to pseudo-Hermitian quantum systems possessing certain symmetries.Publication Open Access Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum(American Institute of Physics (AIP) Publishing, 2002) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.Publication Open Access Qualitative properties of solutions for nonlinear Schrodinger equations with nonlinear boundary conditions on the half-line(American Institute of Physics (AIP) Publishing, 2016) Özsarı, Türker; Department of Mathematics; Department of Mathematics; Kalantarov, Varga; Faculty Member; College of Sciences; 117655In this paper, we study the interaction between a nonlinear focusing Robin type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrodinger equations posed on the infinite half-line. We construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in the H-1-sense. We obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow-up solutions. In addition to the blow-up property, we also discuss the stabilization property and the critical exponent for this model. (C) 2016 AIP Publishing LLC.Publication Open Access A note on a strongly damped wave equation with fast growing nonlinearities(American Institute of Physics (AIP) Publishing, 2015) Zelik, Sergey; Department of Mathematics; Department of Mathematics; Kalantarov, Varga; Faculty Member; College of Sciences; 117655A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved, the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function. (C) 2015 AIP Publishing LLCPublication Open Access Pseudo-Hermiticity and generalized PT- and CPT-symmetries(American Institute of Physics (AIP) Publishing, 2003) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (P), time-reversal (T), and charge-conjugation (C) operators and establish the PT- and CPT-invariance of H.