Basic Theory of Ordinary Differential Equations

Hsieh, Po-Fang; Sibuya, Yasutaka

doi:10.1007/978-1-4612-1506-6

Cited by 134 publications

(167 citation statements)

References 21 publications

Supporting

Mentioning

166

Contrasting

Order By: Relevance

“…The next lemma is a reshaped version of Proposition 5.3 stated in [8]. This result is a q-analog of the well known statement that a holomorphic function is exponentially flat of order k on a sector S if and only if its has0 0 as asymptotic expansion of Gevrey order 1=k on S, see [7], Theorem XI-3-2. …”

Section: Proof We First Check I) Let Wðt; Mþmentioning

confidence: 75%

“…The classical Ramis-Sibuya theorem is a cohomological criterion that ensures the k-summability of a given formal series (see [2], Section 4.4 Proposition 2 or [7], Lemma XI-2-6). In this subsection, we present a version of this theorem in the framework of q-Gevrey asymptotics of order 1=k.…”

Section: A Q-analog Of the Ramis-sibuya Theoremmentioning

confidence: 99%

“…Proof. We will be devoted to the same arguments as in Lemma XI-2-6 from [7] with suitable modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform. For all 0 a l a v À 1, we choose a segment…”

Section: A Q-analog Of the Ramis-sibuya Theoremmentioning

confidence: 99%

See 2 more Smart Citations

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

Malek

2017

View full text Add to dashboard Cite

Abstract. We study an inhomogeneous linear q-di¤erence di¤erential Cauchy problem, with a complex perturbation parameter e, whose coe‰cients depend holomorphically on e and on time in the vicinity of the origin in C 2 and are bounded analytic on some horizontal strip in C w.r.t the space variable. This problem is seen as a q-analog of an initial value problem recently investigated by the author and A. Lastra in [9]. Here a comparable result with the one in [9] is achieved, namely we construct a finite set of holomorphic solutions on a common bounded open sector in time at the origin, on the given strip above in space, when e belongs to a well selected set of open bounded sectors whose union covers a neighborhood of 0 in C Ã . These solutions are constructed through a continuous version of a q-Laplace transform of some order k b 1 introduced newly in [6] and Fourier inverse map of some function with q-exponential growth of order k on adequate unbounded sectors in C and with exponential decay in the Fourier variable. Moreover, by means of a q-analog of the classical Ramis-Sibuya theorem, we prove that they share a common formal power series (that generally diverge) in e as q-Gevrey asymptotic expansion of order 1=k.

show abstract

Section: Proof We First Check I) Let Wðt; Mþmentioning

confidence: 75%

Section: A Q-analog Of the Ramis-sibuya Theoremmentioning

confidence: 99%

See 1 more Smart Citation

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

Malek

2017

View full text Add to dashboard Cite

show abstract

“…The coefficients of the 2 × 2 matrix A(t) defining the linear system (39) are analytic in the variable t, and the Liapunov's type numbers [24] of the system are the same as those of the linear system obtained by replacing A(t) with A(∞), where:…”

Section: A the Asymptotic Behaviormentioning

confidence: 99%

“…which are very long times 24 , compared with the characteristic times of a quantum experiment. [26]).…”

Section: Effect Of the Reducing Terms On The Microscopic Dynamicsmentioning

confidence: 99%

Collapse models: analysis of the free particle dynamics

Bassi¹

2005

J. Phys. A: Math. Gen.

125

View full text Add to dashboard Cite

We study a model of spontaneous wavefunction collapse for a free quantum particle. We analyze in detail the time evolution of the single-Gaussian solution and the double-Gaussian solution, showing how the reduction mechanism induces the localization of the wavefunction in space; we also study the asymptotic behavior of the general solution. With an appropriate choice for the parameter λ which sets the strength of the collapse mechanism, we prove that: i) the effects of the reducing terms on the dynamics of microscopic systems are negligible, the physical predictions of the model being very close to those of standard quantum mechanics; ii) at the macroscopic scale, the model reproduces classical mechanics: the wavefunction of the center of mass of a macro-object behaves, with high accuracy, like a point moving in space according to Newton's laws.

show abstract

The Gevrey Asymptotics in the Case of Singular Perturbations

Sibuya

2000

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

Basic Theory of Ordinary Differential Equations

Cited by 134 publications

References 21 publications

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

Collapse models: analysis of the free particle dynamics

The Gevrey Asymptotics in the Case of Singular Perturbations

Contact Info

Product

Resources

About