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R. D. Costin

Publications and source records attributed to R. D. Costin.

10 recordsLinked to original sources
arXiv2025

Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are numerically efficient representations. For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays. We prove that relatively general functions, Écalle resurgent ones, possess convergent dyadic expansions. These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).

math.CA↗
arXiv2020

On the domain of convergence of spherical harmonic expansions

Spherical harmonic expansions (SHEs) play an important role in most of the physical sciences, especially in physical geodesy. Despite many decades of investigation, the large order behavior of the SHE coefficients, and the precise domain of convergence for these expansions, have remained open questions. These questions are settled in the present paper for generic planets, whose shape (topography) may include many local peaks, but just one globally highest peak. We show that regardless of the smoothness of the density and topography, short of outright analyticity, the spherical harmonic expansion of the gravitational potential converges exactly in the closure of the exterior of the Brillouin sphere (The smallest sphere around the center of mass of the planet containing the planet in its interior), and convergence below the Brillouin sphere occurs with probability zero. More precisely, such over-convergence occurs on zero measure sets in the space of parameters. A related result is that, in a natural Banach space, SHE convergence of the potential below the Brillouin sphere occurs for potential functions in a subspace of infinite codimension (while any positive codimension already implies occurrence of probability zero). Provided a certain limit in Fourier space exists, we find the leading order asymptotic behavior of the coefficients of SHEs. We go further by finding a necessary and sufficient condition for convergence below the Brillouin sphere, which requires a form of analyticity at the highest peak, which would not hold for a realistic celestial body. Namely, a longitudinal average of the harmonic measure on the Brillouin sphere would have to be real-analytic at the point of contact with the boundary of the planet. It turns out that only a small neighborhood of the peak is involved in this condition.

math.CA↗
arXiv2016

A new type of factorial series expansions and applications

We construct a new type of convergent asymptotic representations, dyadic factorial expansions. Their convergence is geometric and the region of convergence can include Stokes rays, and often extends down to 0^+. For special functions such as Bessel, Airy, Ei, Erfc, Gamma and others, this region is C without an arbitrarily chosen ray effectively providing uniform convergent asymptotic expansions for special functions. We prove that relatively general functions, Ecalle resurgent ones possess convergent dyadic factorial expansions. We show that dyadic expansions are numerically efficient representations. The expansions translate into representations of the resolvent of self-adjoint operators in series in terms of the associated unitary evolution operator evaluated at some prescribed points (alternatively, in terms of the generated semigroup for positive operators).

math.CA↗
arXiv2015

Tronquée solutions of the Painlevé equation \P1

We analyze the one parameter family of tronquée solutions of the Painlevé equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in \cite{inprep} leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.

math.CA↗
arXiv2006

Nonperturbative analysis of a model quantum system under time periodic forcing

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation. We show that for generic forcing which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized for large t irrespective of the magnitude or frequency of the forcing. There are however exceptional, very non-generic forcings that do not lead to full ionization. These include rather simple explicit periodic forcings for which the system evolves to a nontrivial localized stationary state related to eigenfunctions of the Floquet operator.

math-ph↗
arXiv2006

Rigorous WKB for finite order linear recurrence relations with smooth coefficients

We study the $ε\to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(kε,ε)y_{k+j}=0,$ $k\in \zdd$ ($l$ fixed). The $a_j$ are $C^{\infty}$ functions in each variable on $I\times [0,\e_0]$ for a bounded interval $I$ and $\e_0>0$. Under certain regularity assumptions we find the asymptotic behavior of the solutions of such recurrences. In typical cases there exists a fundamental set of solutions in the form $\{\exp(\epi F_m(kε,ε))\}_{m=1... l}$ where the functions $F_m$ are $C^{\infty}$ in each variable on the same domain as the $a_j$, showing in particular that the formal perturbation-series solutions are asymptotic to true solutions of these recurrences. Some applications are also briefly discussed.

math.CA↗
arXiv2006

On the formation of singularities of solutions of nonlinear differential systems in antistokes directions

We determine the position and the type of spontaneous singularities of solutions of generic analytic nonlinear differential systems in the complex plane, arising along antistokes directions towards irregular singular points of the system. Placing the singularity of the system at infinity we look at equations of the form $\mathbf{y}'=\mathbf{f}(x^{-1},\mathbf{y})$ with $\mathbf{f}$ analytic in a neighborhood of $(0,\mathbf{0})$, with genericity assumptions; $x=\infty$ is then a rank one singular point. We analyze the singularities of those solutions $\mathbf{y}(x)$ which tend to zero for $x\to \infty$ in some sectorial region, on the edges of the maximal region (also described) with this property. After standard normalization of the differential system, it is shown that singularities occuring in antistokes directions are grouped in nearly periodical arrays of similar singularities as $x\to\infty$, the location of the array depending on the solution while the (near-) period and type of singularity are determined by the form of the differential system.

math.CA↗
arXiv2006

Time asymptotics of the Schroedinger wave function in time-periodic potentials

We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.

math-ph↗
arXiv2000

Evolution of a model quantum system under time periodic forcing: conditions for complete ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $η(t)$. We show that for generic $η(t)$, which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as $t\to\infty$. This is irrespective of the magnitude or frequency (resonant or not) of $η(t)$. There are however exceptional, very non-generic $η(t)$, that do not lead to full ionization, which include rather simple explicit periodic functions. For these $η(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.

math-ph↗