About

Rodica Costin is a professor in the Department of Mathematics at The Ohio State University. She holds a PhD in Applied Mathematics from Rutgers University, obtained in 1997. Her areas of expertise include Differential and Partial Differential Equations, Orthogonal Polynomials, and Asymptotic Analysis. Her research focuses on these mathematical fields, contributing to the understanding and development of methods related to differential equations and asymptotic behaviors. She is actively involved in the academic community, serving as a faculty member within the department and engaging in various educational and research activities.

Research topics

• Mathematics
• Mathematical analysis
• Pure mathematics
• Physics
• Mathematical physics

Selected publications

• Global rational approximations of functions with factorially divergent asymptotic series

  Journal of Approximation Theory · 2025-04-23

  articleSenior author
  PublisherDOI

• Long time evolution of the Hénon-Heiles system for small energy

  arXiv (Cornell University) · 2024-11-25

  preprintOpen access

  The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.

  PublisherOA PDFDOI

• Parabolic cylinder functions revisited using the Laplace transform

  arXiv (Cornell University) · 2024-07-29

  preprintOpen access1st authorCorresponding

  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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• Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field

  Reports on Progress in Physics · 2024-06-20 · 9 citations

  articleOpen access

  Abstract The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential, V , will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound, E N , on the absolute value of the prediction error, e N , of a SH series model, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , truncated at some maximum degree, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mo movablelimits="true">max</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math> . When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> . Costin’s formula is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>B</mml:mi><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> , where R is the radius of the Brillouin sphere. This formula depends on two positive parameters: b , which controls the decay of error amplitude as a function of N when r is fixed, and a scale factor B . We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound, A n on the absolute value of the TS coefficients, a n , for the same radial line. This formula, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>K</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> , depends on degree, n , and two positive parameters, k and K , that are analogous to b and B . We use synthetic planets, for which we can compute the potential, V , and also the radial component of gravitational acceleration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>∂</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>∂</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> , to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscript V refer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscript g to the coefficients and predictions errors associated with g r . For polyhedral planets of uniform density we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>7</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle, α , between that radial line and the singular radial line. We also derive useful identities connecting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>K</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>B</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mi>g</mml:mi></mml:msup></mml:mrow></mml:math> , and B g . These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities are α and R . The phenomenology of ‘series divergence’ and prediction error (when r &lt; R ) can be described as a function of the truncation degree, N , or the depth, d , beneath the Brillouin sphere. For a fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>r</mml:mi><mml:mtext>⩽</mml:mtext><mml:mi>R</mml:mi></mml:mrow></mml:math> , as N increases from very low values, the upper error bound E N shrinks until it reaches its minimum (best) value when N reaches some particular or optimum value, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction error grows as N continues to increase. Eventually, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≫</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction errors increase exponentially with rising N . If we fix the value of N and allow <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth, d , beneath the Brillouin sphere. Because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> everywhere, divergence driven prediction error intensifies more rapidly for g r than for V , both in terms of its dependence on N and d . If we fix both N and d , and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as does B ) as we approach high-amplitude topography.

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• Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

  Communications in Mathematical Physics · 2023-06-23

  article
  PublisherDOI

• Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field

  arXiv (Cornell University) · 2022-09-15

  preprintOpen access

  We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space $x\leqslant 0$, the Schrödinger equation of the system is $i\partial_tψ=-\frac12\partial_x^2ψ+Θ(x) (U-E x \cosωt)ψ$, $t&gt;0$, $x\in\mathbb R$, where $Θ(x)$ is the Heaviside function and $U&gt;0$ is the effective confining potential (we choose units so that $m=e=\hbar=1$). The amplitude $E$ of the external electric field and the frequency $ω$ are arbitrary. We prove existence and uniqueness of classical solutions of this equation for general initial conditions $ψ(x,0)=f(x)$, $x\in\mathbb R$. When the initial condition is in $L^2$ the evolution is unitary and the wave function goes to zero at any fixed $x$ as $t\to\infty$. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-$L^2$ initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all $t&gt;0$. For these we show that the solution approaches in the large $t$ limit a periodic state that satisfies an infinite set of equations formally derived by Faisal, et. al. Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior. It shows a steep increase in the current as the frequency passes a threshold value $ω=ω_c$, with $ω_c$ depending on the strength of the electric field. For small $E$, $ω_c$ represents the threshold in the classical photoelectric effect.

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• On the Domain of Convergence of Spherical Harmonic Expansions

  Communications in Mathematical Physics · 2022-01-01 · 11 citations

  article
  PublisherDOI

• Exact solution of the 1D time-dependent Schrödinger equation for the emission of quasi-free electrons from a flat metal surface by a laser

  Journal of Physics A Mathematical and Theoretical · 2020-07-01 · 1 citations

  article

  Abstract We solve exactly the one-dimensional Schrödinger equation for ψ ( x , t ) describing the emission of electrons from a flat metal surface, located at x = 0, by a periodic electric field E cos( ωt ) at x &gt; 0, turned on at t = 0. We prove that for all physical initial conditions ψ ( x , 0), the solution ψ ( x , t ) exists, and converges for long times, at a rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> , to a periodic solution considered by Faisal et al (2005 Phys. Rev. A 72 023412). Using the exact solution, we compute ψ ( x , t ), for t &gt; 0, via an exponentially convergent algorithm, taking as an initial condition a generalized eigenfunction representing a stationary state for E = 0. We find, among other things, that: (i) the time it takes the current to reach its asymptotic state may be large compared to the period of the laser; (ii) the current averaged over a period increases dramatically as ℏω becomes larger than the work function of the metal plus the ponderomotive energy in the field. For weak fields, the latter is negligible, and the transition is at the same frequency as in the Einstein photoelectric effect; (iii) the current at the interface exhibits a complex oscillatory behavior, with the number of oscillations in one period increasing with the laser intensity and period. These oscillations get damped strongly as x increases.

  PublisherDOI

• Exact solution of the Schrodinger equation for photoemission from a metal

  arXiv (Cornell University) · 2019-11-01

  preprintOpen access

  We solve rigorously the time dependent Schrödinger equation describing electron emission from a metal surface by a laser field perpendicular to the surface. We consider the system to be one-dimensional, with the half-line $x&lt;0$ corresponding to the bulk of the metal and $x&gt;0$ to the vacuum. The laser field is modeled as a classical electric field oscillating with frequency $ω$, acting only at $x&gt;0$. We consider an initial condition which is a stationary state of the system without a field, and, at time $t=0$, the field is switched on. We prove the existence of a solution $ψ(x,t)$ of the Schrödinger equation for $t&gt;0$, and compute the surface current. The current exhibits a complex oscillatory behavior, which is not captured by the "simple" three step scenario. As $t\to\infty$, $ψ(x,t)$ converges with a rate $t^{-\frac32}$ to a time periodic function with period $\frac{2π}ω$ which coincides with that found by Faisal, Kamiński and Saczuk (Phys Rev A 72, 023412, 2015). However, for realistic values of the parameters, we have found that it can take quite a long time (over 50 laser periods) for the system to converge to its asymptote. Of particular physical importance is the current averaged over a laser period $\frac{2π}ω$, which exhibits a dramatic increase when $\hbarω$ becomes larger than the work function of the metal, which is consistent with the original photoelectric effect.

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• Jacobi Series for General Parameters via Hadamard Finite Part and Application to Nonhomogeneous Hypergeometric Equations

  International Journal of Mathematics and Mathematical Sciences · 2019-05-26

  articleOpen access1st authorCorresponding

  The representation of analytic functions as convergent series in Jacobi polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math> is reformulated using the Hadamard principal part of integrals for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">C</mml:mi><mml:mo>∖</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mo>≠</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo>,</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:math>. The coefficients of the series are given as usual integrals in the classical case (when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi mathvariant="fraktur">R</mml:mi><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="fraktur">R</mml:mi><mml:mi>β</mml:mi><mml:mo>&gt;</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math>) or by their Hadamard principal part when they diverge. As an application it is shown that nonhomogeneous differential equations of hypergeometric type do generically have a unique solution which is analytic at both singular points in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow></mml:math>.

  PublisherOA PDFDOI

Frequent coauthors

• Ovidiu Costin

  The Ohio State University

  37 shared

• Joel L. Lebowitz

  12 shared

• Ian Jauslin

  Rutgers, The State University of New Jersey

  6 shared

• Min Huang

  China University of Geosciences

  6 shared

• Wilhelm Schlag

  Yale University

  3 shared

• Hyejin Park

  Korea Institute of Science & Technology Information

  3 shared

• A. Rokhlenko

  Rutgers, The State University of New Jersey

  2 shared

• Marina David

  KU Leuven

  2 shared

Awards & honors

• Graduate Teaching Awards