Atkinson-Type Asymptotic Expansion

Updated 18 January 2026

Atkinson-type asymptotic expansion is a family of explicit, uniform expansions that generalize classical spectral and analytic formulas with computable main and subordinate terms.
It provides precise error estimates and explicit coefficient computations for differential, difference, and integral operators, thereby enhancing the accuracy of spectral and scattering analyses.
The methodology extends to measure-valued, discrete, and piecewise systems, underpinning applications in inverse problems, phase retrieval, and analytic number theory.

An Atkinson-type asymptotic expansion is an explicit, uniform asymptotic expansion—in most contexts, of spectral, analytic, or scattering quantities—where the structure and coefficients generalize the classical formulas obtained by F.V. Atkinson. Atkinson-type formulas are characterized by explicit computation of main and subordinate terms, uniform error estimates, and typically broad generality in the class of differential, difference, or integral operators considered. Such expansions appear prominently in spectral theory, analytic number theory, and mathematical physics.

1. Classical Foundations and the Atkinson Paradigm

Atkinson’s original results addressed the spectral function and Weyl–Titchmarsh $m$ -function for one-dimensional Schrödinger operators and specific quadratic mean integrals for the Riemann zeta function. His approach combined explicit computation with analytic control of all error terms, e.g., for the Weyl $m$ -function of

$- u''(x) + q(x) u(x) = z u(x), \qquad x\in (a,b),$

with Dirichlet (or more general) boundary conditions, he obtained the expansion (Luger et al., 2014)

$m(z) = - \sqrt{-z} - \int_a^{x_0} e^{-2\sqrt{-z}y} q(y) \, dy + O(|z|^{-1/2}),$

as $|z|\to\infty$ in nonreal sectors. The Atkinson formula in analytic number theory, for the mean square of $\zeta(s)$ on the critical line, reads (Dona et al., 2021)

$\int_0^T |\zeta(\tfrac12+it)|^2 \,dt = T \log T + (2\gamma -1 -\log 2\pi)T + O(\sqrt{T} \log^2 T).$

These formulas generalize earlier results by explicitly handling singularities, boundary contributions, and error terms.

2. Extensions to Measure-Valued and Discontinuous Potentials

Modern developments have vastly generalized the original Atkinson formula. For Schrödinger operators with measure-valued potentials—that is, permitting both regular functions and singular objects such as Dirac masses—the Weyl function $m(z)$ admits an Atkinson-type expansion (Luger et al., 2014):

$m(z) = - k - \int_{[0,x_0)} e^{-2k y} \mu(dy) + O(|z|^{-1/2}), \quad k=\sqrt{-z}, \ \Re k\geq0,$

where $\mu$ is a locally finite Borel measure. This formulation allows for:

Potentials supported on sets of zero Lebesgue measure (e.g., Dirac deltas),
Uniform error control by the total variation $|\mu|([0,x_0))$ ,
Explicit higher-order coefficients as iterated moments of $\mu$ ,

$\int_{[0,x_0)} e^{-2k y} \mu(dy) = \mu(\{0\}) - 2k \int_{(0,x_0)} y \mu(dy) + \frac{(2k)^2}{2!} \int_{[0,x_0)} y^2\mu(dy) + \cdots$

and a distributional interpretation of the expansion.

3. Asymptotic Expansions in Discrete and Piecewise Operators

Atkinson-type expansions also govern the high-energy or high-index behavior of spectrum in both continuous and discrete Sturm-Liouville problems, including those of “piecewise-degenerate” or Atkinson-type structure (Ren et al., 2019). The archetypal differential operator is blockwise degenerate:

In each “velocity block” $(b_{j-1},a_j)$ , $P^{-1}(x)\equiv0$ , $W(x)>0$ ,
In each “potential block” $(a_j, b_j)$ , $Q(x)\equiv0$ , $P^{-1}(x)>0$ .

The large- $n$ expansion for the $n$ th eigenvalue is given by the Weyl law

$\lambda_n = \left( \frac{n\pi}{L} \right)^2 + \frac{1}{L}\int_a^b \frac{Q(x)}{W(x)} dx - \frac{\pi}{2L^2} [\Phi(b)-\Phi(a)] \frac{1}{n} + O(n^{-2}),$

where $L = \int_a^b \sqrt{W(x)/P(x)} dx$ and $\Phi(x)$ is the Liouville–Green phase shift. The precise spectral count and singularities may, for discrete analogues, be governed by a Hermitian matrix parameterizing boundary conditions (Ren et al., 2019). However, in the large- $n$ regime, the Atkinson-type expansion is independent of these lower-order details.

4. Atkinson-Type Expansions in Analytic Number Theory

The Atkinson formula provides an archetypal explicit asymptotic expansion for long-time averages in analytic number theory. For the mean square integral of the Riemann zeta function on vertical lines, a refined Atkinson-type formula is (Dona et al., 2021)

$\int_0^T |\zeta(\tau+it)|^2 dt = \begin{cases} \frac{\zeta(2-2\tau)}{(2-2\tau)(2\pi)^{1-2\tau}} T^{2-2\tau} + \zeta(2\tau)T + O^*(2.215\,T^{3/2-2\tau} \log^2 T / (1/2-\tau)^2), & \tau\in [1/4,1/2), \[2ex] T\log T + (2\gamma -1 -\log 2\pi)T + O^*(18.169\,\sqrt{T} \log^2 T), & \tau=1/2, \[1ex] \zeta(2\tau)T + \frac{(2\pi)^{2\tau-1} \zeta(2-2\tau)}{2-2\tau} T^{2-2\tau} + O^*(16.839\,\sqrt{T} \log^2 T/( \tau-1/2)^2), & \tau\in (1/2,3/4]. \end{cases}$

with all terms and error bounds explicit. The derivation utilizes integral representations, contour decompositions, explicit convexity bounds, and precise estimates for the remainder.

In expansions for generalized Euler–Jacobi series and allied functions, Atkinson-type formulas yield an explicit breakdown into algebraic (dominant) and exponentially small (subdominant) components (Paris, 2015). For example,

$S_p(a;w) = \sum_{n=1}^\infty n^{-w} e^{-a n^p}$

has, as $a\to 0$ ,

$S_p(a;w) \sim \frac1p \Gamma\left( \frac{1-w}{p} \right) a^{(w-1)/p} + \sum_{k=0}^{K} (-1)^k \frac{\zeta(w - pk)}{k!} a^k + \sum_{r=0}^{N-1} E_r(a;w,p)$

with all error terms and coefficients explicitly computable.

5. Atkinson-Type Expansions in Asymptotic Scattering and Phase Retrieval

In time-harmonic scattering, Atkinson-type asymptotic expansions yield the leading far-field structure for solutions to higher-order elliptic equations, such as the biharmonic equation (Cheng et al., 11 Jan 2026). For the scattered field at large distance, one obtains

$u_H(x) = \frac{e^{ik|x|}}{|x|^{(m-1)/2}} \left\{ \sum_{j=1}^N \frac{f_j(\hat{x})}{|x|^{j-1}} + O(|x|^{-N}) \right\},$

$u_M(x) = \frac{e^{-k|x|}}{|x|^{(m-1)/2}} \left\{ \sum_{j=1}^N \frac{g_j(\hat{x})}{|x|^{j-1}} + O(|x|^{-N}) \right\},$

where $f_j(\hat{x})$ and $g_j(\hat{x})$ are smooth amplitude functions determined by the boundary data. These expansions support explicit inversion schemes for phase retrieval from phaseless measurement, as the coefficients can be reconstructed from modulus data at a set of large radii.

6. General Analytic Theory and Geometric Criteria

Beyond explicit computation, Atkinson-type expansions connect to a general analytic theory of finite asymptotic expansions for differentiable functions (Granata, 2014). The existence of a two-term expansion

$f(x) = a_1 \phi_1(x) + a_2 \phi_2(x) + o(\phi_2(x)), \ x\to x_0,$

can be characterized by convergence of certain improper integrals involving the differential operator $L$ with $\ker L = \operatorname{span}\{\phi_1,\phi_2\}$ :

The coefficients are determined using Wronskian expressions,
Explicit remainder representations and limit theorems link analytic (canonical factorization) and geometric (Wronskian limit) criteria.

This framework unifies classical and operator-theoretic Atkinson-type expansions within a rigorous analytic structure.

7. Methods of Proof and Remainder Estimates

The derivation of Atkinson-type expansions relies on:

Volterra-type integral equations for fundamental solutions (yielding power series in $k^{-1}$ for Schrödinger $m$ -functions) (Luger et al., 2014),
Bootstrapping expansions, matching boundary traces, and explicit use of factorization identities,
Steepest descents and saddle point analysis for integral representations in analytic number theory (Paris, 2021),
Explicit trace formulae and Liouville–Green/WKB methods in spectral theory (Ren et al., 2019),
Uniform distributional or $L^2$ remainder control, often in explicit quantitative form.

Typical error estimates are $O(|\text{parameter}|^{-\alpha})$ , with the exponent $\alpha$ explicit and all dependence on data (potential, measure, or boundary) localized in the coefficients and error constant.

Atkinson-type asymptotic expansions thus constitute a robust toolkit for explicit spectral, analytic, and scattering asymptotics across a wide range of mathematical physics and analysis, characterized by their uniform structure, computability of coefficients, and precise remainder bounds. They underpin methodologies in inverse problems, analytic number theory, and spectral theory, with extension principles to generalized, singular, and higher-order settings (Luger et al., 2014, Dona et al., 2021, Ren et al., 2019, Cheng et al., 11 Jan 2026, Paris, 2015, Granata, 2014).