In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.

Thesis (MSc)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: We are studying a possible implementation of an appropriate framework for a proper non- Hermitian quantum theory. We present the case where for a non-Hermitian Hamiltonian with real eigenvalues, we define a new inner product on the Hilbert space with respect to which the non-Hermitian Hamiltonian is Quasi-Hermitian. The Quasi-hermiticity of the Hamiltonian introduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates, in which case the metric becomes a basis transformation. We use the non-Hermitian quadratic Hamiltonian to show that such a metric is not unique but can be uniquely defined by requiring to hermitize all elements of one of the irreducible sets defined on the set of all observables. We compare the constructed metric with specific known examples in the literature in which cases a unique choice is made. / AFRIKAANSE OPSOMMING: Ons ondersoek die implementering van n gepaste raamwerk virn nie-Hermitiese kwantumteorie. Ons beskoun nie-Hermitiese Hamilton-operator met reele eiewaardes en definieer in gepaste binneproduk ten opsigtewaarvan die operator kwasi-Hermitiese is. Die kwasi- Hermities aard van die Hamilton operator lei dan tot n stel bi-ortogonale toestande. Ons konstrueer n basistransformasie wat die linker en regter eietoestande van hierdie stel koppel. Hierdie transformasie word dan gebruik omn nuwe binneproduk op die Hilbert-ruimte te definieer. Die oorspronklike nie-HermitieseHamilton-operator is danHermitiesmet betrekking tot hierdie nuwe binneproduk. Ons gebruik die nie-Hermitiese kwadratieseHamilton-operator omte toon dat hierdie metriek nie uniek is nie, maar wel uniek bepaal kan word deur verder te vereis dat dit al die elemente van n onherleibare versameling operatoreHermitiseer. Ons vergelyk hierdie konstruksiemet die bekende voorbeelde in die literatuur en toon dat diemetriek in beide gevalle uniek bepaal kan word.

PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 69-71).

No description available.

The spatial numerical range of an operator on a normed linear space and the algebra numerical range of an element of a unital Banach algebra, as developed by G. Lumer and F. F. Bonsall, are considered and the theory of such numerical ranges applied to Banach algebra. The first part of the thesis is largely expository as in it we introduce the basic results on numerical ranges. For an element of a unital Banach algebra, the question of approximating its spectrum by numerical ranges has been considered by F. F. Bonsall and J. Duncan. We give an alternative proof that the convex hull of the spectrum of an element may be approximated by its numerical range defined with respect to equivalent renormings of the algebra. In the particular case of operators on a Hilbert space, this leads to a sharper version of a result by J. P. Williams. An element is hermitian if it has real numerical range. Such an element is characterized in terms of the linear subspace spanned by the unit, the element and its square. This is used to characterize Banach*–algebras in which every self–adjoint element is hermitian. From this an elementary proof that such algebras are B*-algebras in an equivalent norm is given. As indicated by T. W. Palmer, a formula of L. Harris is then used to show that the equivalent renorming is unnecessary, thus giving a simple proof of Palmer's characterization of B*-algebras among Banach algebras. The closure properties of the spatial numerical range are studied. A construction of B. Berberian is extended to normed linear spaces, however because the numerical range need not be convex, the result obtained is weaker than that of Berberian for Hilbert spaces. A Hilbert space or an Lp-space, for p between one and infinity, is seen to be finite dimensional if and only if all the compact operators have closed spatial numerical range. The spatial numerical range of a compact operator, on a Hilbert space or an Lp-space, for p between one and infinity, is shown to contain all the non–zero extreme points of its closure. So, for a compact operator on a Hilbert space the spatial numerical range is closed if and only if it contains the origin, thus answering a question of P. R. Halmos. Operators that attain their numerical radius are also considered. A result of D. Hilbert is extended to a class of Banach spaces. In a Hilbert space the hermitian operators, which attain their numerical radius, are shown to be dense among all the hermitian operators. This leads to a stronger form of a result by J. Lindenstrauss in the spatial case of operators on a Hilbert space. / PhD Doctorate

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Orientadores: Fernando Eduardo Torres Orihuela, Ercílio Carvalho da Silva / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T19:08:58Z (GMT). No. of bitstreams: 1 TeheranHerrera_ArnoldoRafael_D.pdf: 1331567 bytes, checksum: 7885ebc0ee3a5a3c7ddbc40bca6def1e (MD5) Previous issue date: 2014 / Resumo: Apresentamos algumas aplicações, especialmente usaremos as curvas construídas para calcular alguns AG códigos num ponto racional; estes serão construídos usando certo semigrupo telescópico no ponto racional da curva correspondente. Finalmente compararemos os parâmetros obtidos de nossos exemplos, com os parâmetros dos códigos existentes na literatura / Abstract: In this thesis we work out exemples of maximal curve wich are not covered by the corresponding Hermitian curve. These exemples arise as covered curves of the called GK curve. We also construct exemples of maximal array which cannot be Galois covered by the corresponding Hermitian curve. Finally we stay some applications to coding theory / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Indiana University-Purdue University Indianapolis (IUPUI) / Over the last two decades a new theory has been developed and intensively investigated in quantum physics. The theory stipulates that a non-Hermitian Hamiltonian can also represents a physical system as long as its energy spectra can be purely real in certain regime depending on the parameters of the Hamiltonian. It was demonstrated that the reality of the eigenenergy was conditioned by a certain kind of symmetry embedded in the actual non-Hermitian system. Indeed, such systems have a combined reflection (parity) symmetry (P) and time-reversal symmetry (T), PT-symmetry. The theory opens the door to new features particularly in open systems in which there could be gain and/or loss of particle or energy from and/or to the environment. A key property of the theory is the PT-symmetry breaking transition which occurs at the exceptional point (EP). The exceptional points are special degeneracies characterized by a coalescence of not only the eigenvalues but also of the corresponding eigenvectors of the system; and the coalescence happens when the gain-loss strength, a measure of the openness of the system, exceeds the intrinsic energy-scale of the system. In recent years, quantum walks with PT-symmetric non-unitary time evolution have been realized in systems with balanced gain and loss. These systems fall in two categories namely continuous time quantum walks (CTQW) that are characterized by a unitary or non-unitary time evolution Hamiltonian, and discrete-time quantum walks (DTQW) whose dynamic is described by a unitary or non-unitary time evolution operator consisting of a product of shift, coin, and gain-loss operations. In this thesis, we investigate the PT-symmetric phase of CTQW and DTQW in a variety of non-Hermitian lattice systems with both position-dependent and position independent, parity-symmetric tunneling functions in the presence of PT-symmetric impurities located at arbitrary parity-symmetric site on the lattice. Moreover, we explore the topological phase diagram and its novel features in non-Hermitian, homogeneous and non-homogeneous, PT-symmetric DTQW with closed and open boundary conditions. We conduct our study using analytical and numerical approaches that are directly and easily implementable in physical experiments. Among others, we found that, despite their non-unitary evolution, open systems governed by parity-time symmetric Hamiltonian support conserved quantities and that the PT-symmetry breaking threshold depends on the physical structure of the Hamiltonian and its underlying symmetries.

yes / In a quantum system with d-dimensional Hilbert space, the Q-function of a Hermitian positive semide nite operator , is de ned in terms of the d2 coherent states in this system. The Choquet integral CQ( ) of the Q-function of , is introduced using a ranking of the values of the Q-function, and M obius transforms which remove the overlaps between coherent states. It is a gure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q-function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.