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On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory
, 2009
"... Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accele ..."
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Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the selfcontradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.
TRANSSERIES FOR BEGINNERS
, 801
"... From the simplest point of view, transseries are a new kind of expansion for realvalued functions. But transseries constitute much more than that—they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, a ..."
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From the simplest point of view, transseries are a new kind of expansion for realvalued functions. But transseries constitute much more than that—they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. This paper is an exposition for the nonspecialist mathematician.
Transseries: Ratios, Grids, and Witnesses
, 2009
"... More remarks and questions on transseries. In particular we deal with the system of ratio sets and grids used in the gridbased formulation of transseries. This involves a “witness ” concept that keeps track of the ratios required for each computation. There are, at this stage, questions and missing ..."
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More remarks and questions on transseries. In particular we deal with the system of ratio sets and grids used in the gridbased formulation of transseries. This involves a “witness ” concept that keeps track of the ratios required for each computation. There are, at this stage, questions and missing proofs in the development.
GAMOW VECTORS AND BOREL SUMMATION
, 902
"... Abstract. We analyze the detailed time dependence of the wave function ψ(x, t) for one dimensional Hamiltonians H = − ∂ 2 x + V (x) where V (for example modeling barriers or wells) and ψ(x, 0) are compactly supported. We show that the dispersive part of ψ(x, t), its asymptotic series in powers of t ..."
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Abstract. We analyze the detailed time dependence of the wave function ψ(x, t) for one dimensional Hamiltonians H = − ∂ 2 x + V (x) where V (for example modeling barriers or wells) and ψ(x, 0) are compactly supported. We show that the dispersive part of ψ(x, t), its asymptotic series in powers of t −1/2, is Borel summable. The remainder, the difference between ψ and the Borel sum, is a convergent expansion of the form P ∞ k=0 gkΓk(x)e −γ kt, where Γk are the Gamow vectors of H, and γk are the associated resonances; generically, all gk are nonzero. For large k, γk ∼ const · k log k + k 2 π 2 i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way. The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions. The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x, t) turns out to be C ∞ in t but nowhere analytic on R +. In fact, ψ is t−analytic in a sector in the lower half plane and has the whole of R + a natural boundary. Extension to other types of potentials, for instance analytic at infinity, is briefly discussed, and in the process we study the singularity structure of the Green’s function in a neighborhood of zero, in energy space.