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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; Kalantarov, Varga; Faculty Member; Department of Mathematics; 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 LLC
Publication Metadata only A note on the algebra of p-adic multi-zeta values(International Press of Boston, 2015) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871We prove that the algebra of p-adic multi-zeta values, as defined in [4] or [2], are contained in another algebra which is defined explicitly in terms of series. The main idea is to truncate certain series, expand them in terms of series all of which are divergent except one, and then take the limit of the convergent one. The main result is Theorem 3.12.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.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; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; 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 Metadata only Dynamical theory of scattering, exact unidirectional invisibility, and truncated z e(-2ik0x) potential(Iop Publishing Ltd, 2016) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231The dynamical formulation of time-independent scattering theory that is developed in (2014 Ann. Phys., NY 341 70-85) offers simple formulas for the reflection and transmission amplitudes of finite-range potentials in terms of the solution of an initial-value differential equation. We prove a theorem that simplifies the application of this result and use it to give a complete characterization of the invisible configurations of the truncated z e(-2ik0x) potential to a closed interval, [0, L], with k(0) being a positive integer multiple of pi/L. This reveals a large class of exact unidirectionally and bidirectionally invisible configurations of this potential. The former arise for particular values of z that are given by certain zeros of Bessel functions. The latter occur when the wavenumber k is an integer multiple of pi/L but not of k(0). We discuss the optical realizations of these configurations and explore spectral singularities of this potential.Publication Metadata only Existence of the transfer matrix for a class of nonlocal potentials in two dimensions(Institute of Physics (IOP) Publishing, 2022) Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrodinger equation, (-del(2) + v)psi = k(2)psi) for a local potential v, they arise in dimensions greater than one and are generally present regardless of the details of v. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing v with a certain energy-dependent nonlocal potential (V) over bar (k). We present a dynamical formulation of the stationary scattering for (V) over bar (k) in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on v, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for (V) over bar (k) as a densely-defined operator acting in C-2 circle times L-2(-k, k).Publication Metadata only Generalized unitarity and reciprocity relations for PT-symmetric scattering potentials(Iop Publishing Ltd, 2014) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, R-l/r(k) and T(k), of general PT-symmetric scattering potentials. We use these identities to give a general proof of the relations, vertical bar T(-k)vertical bar=vertical bar T(k)vertical bar and vertical bar R-r(- k)vertical bar=vertical bar R-l(k)vertical bar, conjectured in Ahmed (2012 J. Phys. A: Math. Theor. 45 032004), and the generalized unitarity relation: R-l/r(k) R-l/r(-k)+vertical bar T(k)vertical bar 2=1, and show that it is a common property of both real and complex PT-symmetric potentials. The same holds for T(-k)=T(k)* and vertical bar R-r(- k)vertical bar=vertical bar R-l(k)vertical bar.Publication Open Access Generalized unitarity relation for linear scattering systems in one dimension(Springer, 2019) Department of Physics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; 4231We derive a generalized unitarity relation for an arbitrary linear scattering system that may violate unitarity, time-reversal invariance, PT - symmetry, and transmission reciprocity.Publication Metadata only Geometric phases, symmetries of dynamical invariants and exact solution of the Schrodinger equation(Iop Publishing Ltd, 2001) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We introduce the notion of the geometrically equivalent quantum systems (GEQSs) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQSs. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T-periodic, the corresponding class of GEQSs includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQSs that include a system with a cranked Hamiltonian H (t) = e(-iKt)H(0)e(iKt). We show that die cranking operator K also belongs to this class. Hence, in spite of the fact that it is time independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.