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268
Universal superposition of coherent states and selfsimilar potentials
 Physical Review A
, 2007
"... A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A = P(d/dx + x) / √ 2, where P is the parity operator. Such A ar ..."
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Cited by 18 (2 self)
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A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A = P(d/dx + x) / √ 2, where P is the parity operator. Such A arises naturally in the q → −1 limit for a symmetry operator of a specific selfsimilar potential obeying the qWeyl algebra, AA † −q 2 A † A = 1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states ǫkα〉, k = 0,1,...,N −1, where ǫ is a primitive Nth root of unity, ǫN = 1. A class of qcoherent states related to the bilateral qhypergeometric series and Ramanujan type integrals is described. It includes a curious set of coherent states of the free nonrelativistic particle which is interpreted as a qalgebraic system without discrete spectrum. A special degenerate form of the symmetry algebras of selfsimilar potentials is found to provide a natural qanalog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.
From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
, 707
"... This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. T ..."
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Cited by 17 (2 self)
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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is
Haag’s theorem and its implications for the foundations of quantum field theory
 Erkenntnis
, 2006
"... Abstract: Although the philosophical literature on the foundations of quantum
eld theory recognizes the importance of Haags theorem, it does not provide a clear discussion of the meaning of this theorem. The goal of this paper is to make up for this de
cit. In particular, it aims to set out the im ..."
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Cited by 15 (1 self)
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Abstract: Although the philosophical literature on the foundations of quantum
eld theory recognizes the importance of Haags theorem, it does not provide a clear discussion of the meaning of this theorem. The goal of this paper is to make up for this de
cit. In particular, it aims to set out the implications of Haags theorem for scattering theory, the interaction picture, the use of nonFock representations in describing interacting
elds, and the choice among the plethora of the unitarily inequivalent representations of the canonical commutation relations for free and interacting elds. 1
Variational Analysis for a Generalized Spiked Harmonic Oscillator
, 1999
"... A variational analysis is presented for the generalized spiked harmonic oscillator Hamiltonian operator − d2 dx2 +Bx2 + A x2 + λ xα, where α is a real positive parameter. The formalism makes use of a basis provided by exact solutions of Schrödinger’s equation for the Gol’dman and Krivchenkov Hamilto ..."
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Cited by 6 (1 self)
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A variational analysis is presented for the generalized spiked harmonic oscillator Hamiltonian operator − d2 dx2 +Bx2 + A x2 + λ xα, where α is a real positive parameter. The formalism makes use of a basis provided by exact solutions of Schrödinger’s equation for the Gol’dman and Krivchenkov Hamiltonian, and the corresponding matrix elements that were previously found. For all the discrete eigenvalues the method provides bounds which improve as the dimension D of the basis set is increased. Extension to the Ndimensional case in arbitrary angularmomentum subspaces is also presented. By minimizing over the free parameter A, we are able to reduce substantially the number of basis functions needed for a given accuracy. PACS 03.65.GeVariational analysis for... page 2 I.
COMMENT Comment on ‘On the Coulomb potential in one dimension’
, 1997
"... Abstract. The mathematical analysis in 1996 J. Phys. A: Math. Gen. 29 1767, is not sufficient to decide whether in one dimension the singularities of the potentials −γ =x and −γ =jxj split the corresponding oneparticle quantum systems at the origin into two completely decoupled subsystems. In fact, ..."
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Abstract. The mathematical analysis in 1996 J. Phys. A: Math. Gen. 29 1767, is not sufficient to decide whether in one dimension the singularities of the potentials −γ =x and −γ =jxj split the corresponding oneparticle quantum systems at the origin into two completely decoupled subsystems. In fact, it is argued that this question cannot be answered by mathematical considerations alone. In [1] Pavel Kurasov constructed quantum Hamiltonians as selfadjoint Schrödinger operators on the Hilbert space L2.R / for a point particle moving along the real line R under the influence of the potential −γ =x or −γ =jxj, where γ; x 2 R. Employing physical units in which twice the mass of the particle equals the square of Planck’s constant, these operators are formally given by the differential expressions − d 2 dx2 − γ x (1) − d
Depopulation After Unification? Population Prospects for East Germany, 19902010
"... East Germany is currently experiencing a fundamental demographic change. Between 1989 and 1993 it lost 1 million people due to migration. Births and marriages declined by 65 percent. Is this region facing further depopulation in the coming decades? This study examines the prospects for future popula ..."
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Cited by 3 (0 self)
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population development in East Germany until the year 2010. The authors assume that the high level of net emigration of the years 198992 was a unique phenomenon. Net emigration from East Germany has already substantially declined. After 2000 a migration balance of zero is assumed. The birth decline should
An algebraic approach to the Quantization of Constrained Systems: Finite Dimensional Examples
 Ch.5 Ph.D Dissertation Syracuse University grqc/9304043
, 1992
"... General relativity has two features in particular, which make it difficult to apply to it existing schemes for the quantization of constrained systems. First, there is no background structure in the theory, which could be used, e.g., to regularize constraint operators, to identify a “time ” or to de ..."
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Cited by 12 (0 self)
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General relativity has two features in particular, which make it difficult to apply to it existing schemes for the quantization of constrained systems. First, there is no background structure in the theory, which could be used, e.g., to regularize constraint operators, to identify a “time ” or to define an inner product on physical states. Second, in the Ashtekar formulation of general relativity, which is a promising avenue to quantum gravity, the natural variables for quantization are not canonical; and, classically, there are algebraic identities between them. Existing schemes are usually not concerned with such identities. Thus, from the point of view of canonical quantum gravity, it has become imperative to find a framework for quantization which provides a general prescription to find the physical inner product, and is flexible enough to accommodate noncanonical variables. In this dissertation I present an algebraic formulation of the Dirac approach to the quantization of constrained systems. The Dirac quantization program is augmented by a general principle to find the inner product on physical states. Essentially, the Hermiticity conditions on physical operators determine this inner product. I also clarify the role in
Geometrodynamics: spacetime or space
, 2004
"... To the memory of my Father, who taught me mathematics and much more. To Claire for her caring support, and to my Mother. To all my friends, with thanks. Yves, Becca, Suzy, Lynnette and Mark helped me survive my Cambridge years. Yves, Becca and Suzy have stayed in touch, while I often enjoyed Lynnett ..."
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Cited by 12 (12 self)
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To the memory of my Father, who taught me mathematics and much more. To Claire for her caring support, and to my Mother. To all my friends, with thanks. Yves, Becca, Suzy, Lynnette and Mark helped me survive my Cambridge years. Yves, Becca and Suzy have stayed in touch, while I often enjoyed Lynnette and Chris’s company though my PhD years, and Mark was a familiar and welcome figure at QMUL. I also thank the good friends I made toward the end of my Cambridge years: Ed, Matt, Bjoern, Alex, Angela and Alison. And Bryony, for being a good friend during the difficult last year. Thanks also to all my other friends, office mates, and kind and entertaining people I have crossed paths with. To Professors Malcolm MacCallum and Reza Tavakol, with thanks for agreeing to supervise me, and for their encouragement and wisdom. I also thank Dr James Lidsey for supervision, and both Dr Lidsey and Professor Tavakol for active collaborations. I also thank Dr. Julian Barbour and Professor Niall Ó Murchadha for teaching me many things and for collaboration, and the Barbour family for much hospitality. I also thank Brendan Foster and Dr. Bryan Kelleher for many discussions and for collaboration, and Dr. Harvey Brown
NonBoolean Descriptions for MindMatter Problems
"... A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmat ..."
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Cited by 10 (0 self)
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A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmatter distinction. Such a description in terms of a locally Boolean but globally nonBoolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely nonBoolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a nonBoolean structure and can be encompassed into a single nonBoolean description. Tensed and tenseless time can be synchronized by holistic correlations. 1.
Results 11  20
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