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Rodica D. Costin

Publications and source records attributed to Rodica D. Costin.

15 recordsLinked to original sources
arXiv2024

Parabolic cylinder functions revisited using the Laplace transform

In this paper we gather and extend classical results for parabolic cylinder functions, namely solutions of the Weber differential equations, using a systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of the Weber differential equation \begin{equation*} w''(z)-\left(\frac{z^2}{4}+a\right)w(z)=0 \end{equation*} and provide representations by Laplace integrals extended to include all values of the complex parameter $a$; we find an integral integral representation for the function $V$; none was previously available. For the Weber equation in the form \begin{equation*} u''(x)+\left(\frac{x^2}{4}-a\right)u(x)=0, \end{equation*} we define a new fundamental system $E_\pm$ which is analytic in $a\in\mathbb{C}$, based on asymptotic behavior; they appropriately extend and modify the classical solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for the approach.

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arXiv2019

Solution of the time dependent Schrödinger equation leading to Fowler-Nordheim field emission

We solve the time-dependent Schrödinger equation describing the emission of electrons from a metal surface by an external electric field $E$, turned on at $t=0$. Starting with a wave function $ψ(x,0)$, representing a generalized eigenfunction when $E=0$, we find $ψ(x,t)$ and show that it approaches, as $t\to\infty$, the Fowler-Nordheim tunneling wavefunction $ψ_E$. The deviation of $ψ$ from $ψ_E$ decays asymptotically as a power law $t^{-\frac32}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.

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arXiv2018

Jacobi series for general parameters and applications

Representation of analytic functions as convergent series in Jacobi polynomials $P_n^{(a,b)}$ is reformulated using a unified approach for almost all complex $a, b$. The coefficients of the series are given as usual integrals in the classical case (when $\Re a, \Re b >-1$), or by the Hadamard principal part of these integrals when they diverge. As an application it is shown that inhomogeneous hypergeometric equations do generically have a unique solution which is analytic at both singular points in the complex plane.

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arXiv2018

Truncated Solutions of Painlevé Equation ${\rm P}_{\rm V}$

We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter they represent tri-truncated solutions, analytic in almost the full complex plane, for large independent variable. A brief historical note, and references on truncated solutions of the other Painlevé equations are also included.

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arXiv2017

Ionization by an Oscillating Field: Resonances and Photons

We describe new exact results for a model of ionization of a bound state, induced by an oscillating potential. In particular we have obtained exact expressions, in the form of readily computable rapidly convergent sums, for the energy distribution of the emitted particles as a function of time, frequency and strength of the oscillating potential. Going beyond perturbation theory, these show resonances in the energy distribution which look like single or multi-photon absorption, similar to those observed in laser induced electron emission from solids or atoms. This is particularly so when the strength of the oscillating potential is small compared to the binding energy but is still visible for large fields, and even for time-periods of a few oscillations. We have also obtained the space-time structure of the wave function. Our model exhibits a form of stabilization; the ionization probability is not monotone in the strength of the oscillating potential.

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arXiv2017

Nonperturbative time dependent solution of a simple ionization model

We present a non-perturbative solution of the Schrödinger equation $iψ_t(t,x)=-ψ_{xx}(t,x)-2(1 +α\sinωt) δ(x)ψ(t,x)$, written in units in which $\hbar=2m=1$, describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of $α,ω,t$ for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large $t$ and small $α$ the energy distribution has sharp peaks at energies which are multiples of $ω$, corresponding to photon capture. We obtain small $α$ expansions that converge for all $t$, unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in different emission processes.

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arXiv2015

The Weber equation as a normal form with applications to top of the barrier scattering

In the paper we revisit the basic problem of tunneling near a nondegenerate global maximum of a potential on the line. We reduce the semiclassical Schrödinger equation to a Weber normal form by means of the Liouville-Green transform. We show that the diffeomorphism which effects this stretching of the independent variable lies in the same regularity class as the potential (analytic or infinitely differentiable) with respect to both variables, i.e., space and energy. We then apply the Weber normal form to the scattering problem for energies near the potential maximum. In particular we obtain a representation of the scattering matrix which is accurate up to multiplicative factors of the form 1 + o(1).

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arXiv2009

The first return map for planar vector fields with nilpotent linear part with a center or a focus

The return map for planar vector fields with nilpotent linear part (having a center or a focus and under an assumption generically satisfied) is found as a convergent power series whose terms can be calculated iteratively. The first nontrivial coefficient is the value of an Abelian integral, and the following ones are explicitly given as iterated integrals built with algebraic functions.

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arXiv2009

Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part

Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when the real parts of a,b are greater than -1)

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arXiv2008

Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.

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arXiv2008

Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal "correction" of the nonlinear part so that the "corrected" system is formally linearizable. Normal forms of these systems are found, providing also their classification.

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arXiv2008

Analytic linearization of nonlinear perturbations of Fuchsian systems

Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable.

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arXiv2008

Matrix valued polynomials generated by the scalar-type Rodrigues' formulas

The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators. The conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type is proved for Q(x)=x and Q(x)=x^2-1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials.

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arXiv2008

A class of matrix-valued polynomials generalizing Jacobi Polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a two-step recurrence relation, integral inter-relations, and quasi-orthogonality relations.

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