arXiv subjects

O. Costin

Publications and source records attributed to O. Costin.

47 records · Page 3Linked to original sources

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↗

arXiv2006

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{ωt}$ with $U(x)=2δ(x-a)$ or $U(x)= 2δ(x-a)-2δ(x+a)$. The particle is initially in a bound state produced by the binding potential $-2δ(x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $ω$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $ω$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,ω$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

math-ph↗

arXiv2006

Nonlinear Stokes phenomena in first or second order differential equations

We study singularity formation in nonlinear differential equations of order $m\leqslant 2$, $y^{(m)}=A(x^{-1},y)$. We assume $A$ is analytic at $(0,0)$ and $\partial_y A(0,0)=λ\ne 0$ (say, $λ=(-1)^m$). If $m=1$ we assume $A(0,\cdot)$ is meromorphic and nonlinear. If $m=2$, we assume $A(0,\cdot)$ is analytic except for isolated singularities, and also that $\int_{s_0}^\infty |Φ(s)|^{-1/2}d|s|<\infty$ along some path avoiding the zeros and singularities of $Φ$, where $Φ(s)=\int_{0}^s A(0,τ)dτ$. Let $H_α=\{z:|z|>a>0,\arg(z)\in (-α,α)\}$. If the Stokes constant $S^+$ associated to $\RR^+$ is nonzero, we show that all $y$ such that $\lim_{x\to +\infty}y(x)=0$ are singular at $2πi$-quasiperiodic arrays of points near $i\RR^+$. The array location determines and is determined by $S^+$. Such settings include the Painlevé equations $P_I$ and $P_{II}$. If $S^+=0$, then there is exactly one solution $y_0$ without singularities in $H_{2π-ε}$, and $y_0$ is entire iff $y_0=A(z,0)\equiv 0$. The singularities of $y(x)$ mirror the singularities of the Borel transform of its asymptotic expansion, $\mathcal{B}\tilde{y}$, a nonlinear analog of Stokes phenomena. If $m=1$ and $A$ is a nonlinear polynomial with $A(z,0)\not\equiv 0$ a similar conclusion holds even if $A(0,\cdot)$ is linear. This follows from the property that if $f$ is superexponentially small along $\RR^+$ and analytic in $H_π$, then $f$ is superexponentially unbounded in $H_π$, a consequence of decay estimates of Laplace transforms.

math.CA↗

arXiv2006

Singularity barriers and Borel plane analytic properties of 1+ difference equations

The paper addresses generalized Borel summability of ``$1^+$'' difference equations in ``critical time''. We show that the Borel transform $Y$ of a prototypical such equation is analytic and exponentially bounded for $\Re(p)<1$ but there is no analytic continuation from 0 toward $+\infty$: the vertical line $\ell:=\{p:\Re(p)=1\}$ is a singularity barrier of $Y$. There is a unique natural continuation through the barrier, based on the Borel equation dual to the difference equation, and the functions thus obtained are analytic and decaying on the other side of the barrier. In this sense, the Borel transforms are analytic and well behaved in $\CC\setminus\ell$. The continuation provided allows for generalized Borel summation of the formal solutions. It differs from standard ``pseudocontinuation'' \cite{Shapiro}. This stresses the importance of the notion of cohesivity, a comprehensive extension of analyticity introduced and thoroughly analyzed by Écalle. We also discuss how, in some cases, Écalle acceleration can provide a procedure of natural continuation beyond a singularity barrier.

math.CA↗

arXiv2006

Complex Singularity Analysis for a nonlinear PDE

We introduce a method of rigorous analysis of the location and type of complex singularities for nonlinear higher order PDEs as a function of the initial data. The method is applied to determine rigorously the asymptotic structure of singularities of the modified Harry-Dym equation $$ H_t + H_y = - {1/2} H^3 + H^3 H_{yyy} : H(y, 0) = y^{-1/2} $$ for small time at the boundaries of the sector of analyticity. Previous work \cite{CPAM}, \cite{invent03} shows existence, uniqueness and Borel summability of solutions of general PDEs. It is shown that the solution to the above initial value problem is represented convergently by a series in a fractional power of $t$ down to a small annular neighborhood of a singularity of the leading order equation. We deduce that the exact solution has a singularity nearby having, to leading order, the same type.

math.AP↗

arXiv2006

Nonlinear evolution PDEs in R^+ \times C^d: existence and uniqueness of solutions, asymptotic and Borel summability

We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form $${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left( {\bf x}, t, \{\partial_{\bf x}^{\bf j} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) = {\bf {u}}_I({\bf x})$$ with $\mathbf{u}\in\CC^{r}$, for $ t\in (0,T)$ and large $|{\bf x}|$ in a poly-sector $S$ in $\mathbb{C}^d$ ($\partial_{\bf x}^{\bf j} \equiv \partial_{x_1}^{j_1} \partial_{x_2}^{j_2} ...\partial_{x_d}^{j_d}$ and $j_1+...+j_d\le n$). The principal part of the constant coefficient $n$-th order differential operator $\mathcal{P}$ is subject to a cone condition. The nonlinearity ${\bf g}$ and the functions $\mb u_I$ and $\mb u$ satisfy analyticity and decay assumptions in $S$.The paper shows existence and uniqueness of the solution of this problem and finds its asymptotic behavior for large $|\bf x|$. Under further regularity conditions on $\mb g$ and $\mb u_I$ which ensure the existence of a formal asymptotic series solution for large $|\mb x|$ to the problem, we prove its Borel summability (and automatically its asymptoticity) to an actual solution $\mb u$.In special cases motivated by applications we show how the method can be adapted to obtain short-time existence, uniqueness and asymptotic behavior for small $t$,without size restriction on the space variable.

math.AP↗

arXiv2006

Borel summability and Lindstedt series

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining $C^\infty$ functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in the case of two-dimensional rotation vectors with Diophantine exponent $τ=1$ (e. g. with ratio of the two independent frequencies equal to the golden mean).

math-ph↗

arXiv2004

On the Construction of Particle Distributions with Specified Single and Pair Densities

We discuss necessary conditions for the existence of probability distribution on particle configurations in $d$-dimensions i.e. a point process, compatible with a specified density $ρ$ and radial distribution function $g({\bf r})$. In $d=1$ we give necessary and sufficient criteria on $ρg({\bf r})$ for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case $g(r) = 0, r < D$ and $g(r) = 1, r > D$, if and only if $ρD \leq e^{-1}$: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for $ρg(r)$ to specify a determinantal point process for which all $n$-particle densities, $ρ_n({\bf r}_1, ..., {\bf r}_n)$, are given explicitly as determinants. We give several examples.

cond-mat.stat-mech↗

arXiv2002

Resonance Theory for Schroedinger Operators

Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solutions of the Schrödinger equation. \keywords{Resonances; Time-dependent Schrödinger equation}

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↗

arXiv2000

On the complete ionization of a periodically perturbed quantum system

We analyze the time evolution of a one-dimensional quantum system with zero range potential under time periodic parametric perturbation of arbitrary strength and frequency. We show that the projection of the wave function on the bound state vanishes, i.e. the system gets fully ionized, as time grows indefinitely.

math-ph↗