A Spectral Theory of Scalar Volterra Equations

TL;DR

This work develops a unified spectral theory for scalar Volterra equations across five classes—(gCM), (gPD), discrete-time (dPD), and their regularized delay/fractional variants (rPD, rCM). It constructs three interconversion maps, $\mathcal{B}$ on the circle, $\mathcal{B}_{\mathbb{R}}$ on the real line, and $\mathcal{B}_{\mathrm{reg}}$ in the regularized setting, providing explicit closed-form expressions that translate kernels and resolvents into each other and yield analytic solutions for all five classes. The framework recovers and extends classic results (e.g., Abel-type solvability, Loy–Anderssen interconversions) and yields new formulas for fractional and delay equations, including Mittag–Leffler kernels and regularized Hilbert transforms, with strong continuity/topology guarantees. It unifies viscoelastic, signal-processing, and quantum-system results under a common spectral lens, enabling stable numerical schemes and pen-and-paper solvability for a broad range of Volterra problems. Overall, the theory broadens the set of tractable Volterra equations, connects disparate domains, and provides practical tools for deconvolution and time-domain modeling.

Abstract

This work aims to bridge the gap between pure and applied research on scalar, linear Volterra equations by examining five major classes: integral and integro-differential equations with completely monotone kernels, such as linear viscoelastic models; equations with positive definite kernels, such as partially observed quantum systems; difference equations with discrete, positive definite kernels; a generalized class of delay differential equations; and a generalized class of fractional differential equations. We develop a general, spectral theory that provides a system of correspondences between these disparate domains. As a result, we see how 'interconversion' (operator inversion) arises as a natural, continuous involution within each class, yielding a plethora of novel formulas for analytical solutions of such equations. This spectral theory unifies and extends existing results in viscoelasticity, signal processing, and analysis, and makes progress on an open question of Abel regarding the solution of integral equations of the first kind. Finally, it reduces simple Volterra equations of all classes to pen-and-paper calculation, and offers promising applications to the numerical solution of Volterra equations more broadly.

Figures (19)

• Figure 1: Our system of correspondences between the five classes of Volterra equations under consideration, with a summary of how it unifies and extends existing results. For detail on particular elements of this correspondence, click the relevant hyperlinks in the figure. For detail on the interconversion maps ($\mathcal{B}$, $\mathcal{B}_\mathbb R$, and $\mathcal{B}_\mathrm{reg}$) and embeddings ($\Psi$ and $\Psi_\mathrm{reg}$) that make up these correspondences, see \ref{['sec:main']}. For detail on existing literature, see \ref{['sec:history']}.
• Figure 2: Simple examples of four classes of Volterra equations studied in this paper: \ref{['eq:integrodiff_dPD']}, \ref{['eq:integrodiff_CM']}, \ref{['eq:integrodiff_PD']}, and \ref{['eq:integrodiff_rPD']}. The first column shows the measures $\lambda$ and $\mu$ that correspond to the spectrum of the original equation and its resolvent, respectively; these measures are defined on $S^1$ for \ref{['eq:integrodiff_dPD']} and on $\mathbb R$ for the remaining examples. The second column depicts the Volterra integral kernels $K$ and $J$ associated with each spectrum; for instance, in the gCM context, we have $K=\mathcal{L}[\lambda]$ and $J=\mathcal{L}[\mu]$. In the third column, we confirm that the predictions of our theory in \ref{['ex:dPD', 'ex:gCM', 'ex:gPD', 'ex:gPD_complex', 'ex:rPD']} correctly solve the corresponding Volterra equations. Namely, we show that, given a Volterra equation with kernel $K$, input $x$, and output $y$, the interconverted Volterra equation with kernel $J$ accurately reconstructs the input $\widehat{x}\approx x$ from $y$.
• Figure 3: A five-box reservoir model for the global carbon cycle Keeling1973. In such models, large environmental reservoirs of CO$_2$ are hypothesized to be well-mixed, such that the transport of CO$_2$ between them is determined by the total quantity in each. Such models have been in use since the 1950s https://doi.org/10.1111/j.2153-3490.1957.tb01848.xKeeling1973, with subsequent developments introducing more reservoirs https://doi.org/10.1111/j.2153-3490.1967.tb01509.xBjorkstrom1986, refined diffusion effects https://doi.org/10.1111/j.2153-3490.1975.tb01671.x, and more. They have seen extensive use in understanding anthropogenic effects on the global carbon cycle; for instance, they have been used recently to estimate historical carbon budgets KAMIUTO1994825 and the impacts of radiative forcing CHOI2022424 and of burning biomass CHOI2020106942 on global temperatures.
• Figure 4: The Kelvin--Voigt and Maxwell models of viscoelasticity describe materials as (potentially-infinite) collections of springs and dashpots, connected in series or in parallel, respectively. The spring-dashpot elements in each model can be indexed by a position variable $x$, giving rise to a position-dependent strain (displacement gradient) $\epsilon(x,t)$ and stress (force gradient) $\sigma(x,t)$. The map from average stress ${\overline{\epsilon}}(t)$ to average strain ${\overline{\sigma}}(t)$ in a Kelvin--Voigt material is a CM integral equation of either the first or second kind, while for Maxwell materials, the map from average strain to average stress is either a CM integral equation of the second kind or a CM integro-differential equation.
• Figure 5: Visualization of the Cauchy transform $Q$ given by \ref{['eq:cauchycirc']}. By adding an imaginary component to the Cauchy transform, we recover the one-parameter family of $\sigma$-Cauchy transforms $Q_\sigma$, given by \ref{['eq:cauchygeneral']}; these transforms allow us to capture the Cauchy transforms on the circle and real line (and in fact, any smooth Jordan curve) using the same theory. By taking the imaginary trace of $Q$ and $Q_\sigma$ along the unit circle, we recover the Hilbert and $\sigma$-Hilbert transforms, respectively.

Theorems & Definitions (101)

• Definition 1.1
• Definition 1.2
• Definition 3.1: Sets of Measures
• Lemma 3.2: Bernstein Widder1931NecessaryASGripenberg_Londen_Staffans_1990
• Lemma 3.3: Bochner reed1975ii
• Definition 3.4: Integral Transforms on $S^1$
• Definition 3.5
• Proposition 3.6
• Corollary 3.7: Shifted Cauchy and Hilbert Transforms
• Remark 3.8