arXiv subjects

Ethan Sussman

Publications and source records attributed to Ethan Sussman.

At least 19 recordsLinked to original sources

arXiv2026

The asymptotic structure of forward scattering

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose--Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

math.AP↗

arXiv2026

Asymptotics in all regimes for the Schrödinger equation with time-independent coefficients

Using the recent analysis of the output of the low-energy resolvent of Schrödinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for the solutions of the initial-value problem for the Schrödinger equation (assuming Schwartz initial data). Asymptotics are calculated in all joint large-radii large-time regimes, these corresponding to the boundary hypersurfaces of a particular compactification of spacetime.

math.AP↗

arXiv2026

Lecture notes on non-elliptic Fredholm theory

These are lecture notes from the Austral Winter School on Microlocal Analysis and Non-elliptic Fredholm Theory, held at the Australian National University, Canberra, June 30 -- July 11, 2025.

math.AP↗

arXiv2025

On the convergence of the Born series for Coulomb potentials

We provide a short proof of the convergence of the Born series on asymptotically conic manifolds, at sufficiently high energy. The potential is allowed to have multiple Coulomb singularities. This is handled using powerful semiclassical estimates recently proven by Hintz for the case of a single dipole (or better) singularity. The potential is also allowed to be long-range, like the actual Coulomb potential $1/r$; long-range potentials are handled using anisotropic semiclassical (sc-) Sobolev spaces. As a consequence of the above estimates, we show the existence of a resonance-free region for Hamiltonians with multiple screened Coulomb singularities.

math-ph↗

arXiv2025

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics

This is the less technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrödinger equation. As shown in the companion paper, combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. In this paper, we derive asymptotics from those estimates. Our framework differs from those in previous works in that ours is based on spacetime phase-space analysis.

math.AP↗

arXiv2025

The essential self-adjointness of the wave operator on radiative spacetimes

We prove the essential self-adjointness of the d'Alembertian $\square_g$, allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in closely related settings, that all tempered distributions $u$ satisfying $\square_g u = λu +f$ for $λ\in \mathbb{C}\backslash \mathbb{R}$ and $f$ Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the ``de,sc-'' machinery recently developed by the third author.

math.SP↗

arXiv2025

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates

This is the more technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $Ψ_{\natural}$ (a variant of the semiclassical scattering calculus), $Ψ_{\natural\mathrm{res}}$, and $Ψ_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box.

math.AP↗

arXiv2025

Complete asymptotic analysis of low energy scattering for Schrodinger operators with a short-range potential

Recent work by Hintz--Vasy provides a partial asymptotic analysis of the low-energy limit of scattering for Schrödinger operators with a short-range potential. Using a slight refinement of Hintz's algorithm, we complete the asymptotic analysis by providing full asymptotic expansions in every possible asymptotic regime. Moreover, the analysis is done in any dimension $d\geq 3$, for any asymptotically conic manifold, and we keep track of partial multipole expansions. Applications include full asymptotic analyses of the Schrödinger, wave, and Klein--Gordon equations, one of these being described in a companion paper. Using previous work, only partial asymptotic analyses were possible.

math.AP↗

arXiv2025

Full semiclassical asymptotics near transition points

We construct complete asymptotic expansions of solutions of the 1D semiclassical Schrödinger equation near transition points. There are three main novelties: (1) transition points of order $κ\geq 2$ (i.e.\ trapped points -- the simple turning point is $κ=1$, the simple pole is $κ=-1$) are handled, (2) various terms in the operator are allowed to have controlled singularities of a form compatible with the geometric structure of the problem (some applications are given in the text), and (3) the term-by-term differentiability of the expansions with respect to the semiclassical parameter is included. We prove that any solution to the semiclassical ODE with initial data of exponential type is of exponential-polyhomogeneous type on a suitable manifold-with-corners compactifying the $h\to 0^+$ regime. Consequently, such a solution has an atlas of full asymptotic expansions in terms of elementary functions, and these expansions are well-behaved. The Airy and Bessel functions show up in the expected way, as the asymptotic data at one boundary edge. We are able to handle cases that Langer--Olver could not because the framework of polyhomogeneous functions on manifolds-with-corners provides more flexibility (two matched $h\to 0^+$ expansions, possibly with logarithms, in this case) than that employed by Langer--Olver (one uniform $h\to 0^+$ expansion without logarithms). We work entirely in the $C^\infty$ category. No analyticity is ever assumed, nor proven.

math.CA↗

arXiv2025

Hydrogen-like Schrödinger Operators at Low Energies

Consider a Schrödinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian approach," we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\pm i 0) = \lim_{ε\to 0^+} R(E\pm i ε)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to $E=0$. In fact, we will construct a compactification of $(0,1]_E\times X$ on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as $r\to\infty$ times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime.

math.AP↗

arXiv2024

Massive wave propagation near null infinity

We study, fully microlocally, the propagation of massive waves on the octagonal compactification \[\mathbb{O}=[\overline{\mathbb{R}^{1,d}};\mathscr{I};1/2]\] of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti-Shubin-Melrose's sc-calculus) and, more novelly, at null infinity, denoted $\mathscr{I}$. The analysis is closely related to Hintz-Vasy's recent analysis of massless wave propagation at null infinity using the ``e,b-calculus'' on $\mathbb{O}$. We prove several elementary corollaries regarding the Klein-Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, $Ψ_{\mathrm{de,sc}}(\mathbb{O})$, the ``de,sc-calculus'' on $\mathbb{O}$. The `de' refers to the structure (``double edge'') of the calculus at null infinity, and the `sc' refers to the structure (``scattering'') at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein-Gordon equation. Unlike hyperbolic coordinates, the de,sc-boundary fibration structure is Poincaré invariant.

math.AP↗

arXiv2024

The singularities of Selberg- and Dotsenko-Fateev-like integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ's minimal models of 2D CFT as described by Felder and Silvotti and Dotsenko and Fateev (the ``Coulomb gas formalism''). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call ``DF-symmetric,'' we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.

math-ph↗

arXiv2023

Massive waves gravitationally bound to static bodies

We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the Klein--Gordon equation having Schwartz initial data. In fact, there exist infinitely many independent such solutions. The proof is a variational argument based on the long range nature of the effective potential. We give two sets of test functions which serve to verify the hypothesis of the variational argument. One set consists of cutoff versions of the hydrogen bound states and is used to prove the existence of eigenvalues near the hydrogen spectrum.

gr-qc↗

arXiv2023

The regularization of Dotsenko--Fateev integrals

We discuss the regularization of certain hypergeometric integrals appearing in 2D CFT, a step needed in the construction of the BPZ minimal models via the Coulomb gas formalism. The method is a generalization of Pochhammer's regularization of the Euler Beta-function. The constructions of the relevant homology classes are inspired by a recent singular-geometric analysis of the Dotsenko--Fateev integrand.

math-ph↗

arXiv2023

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $Φ$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(Φ)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(Φ)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

math.SP↗

arXiv2023

Long time behavior of the half-wave trace and Weyl remainders

Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace $\operatorname{HWT}_{M,g}(τ) \in \mathscr{S}'(\mathbb{R}_τ)$. From the asymptotics of the half-wave trace as $τ\to 0$, Hörmander deduces the now standard remainder $\smash{O(σ^{d-1}) = O(λ^{d/2-1/2})}$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o(σ^{d-1})$. By examining the Tauberian step in the argument, we show how a quantitative version \[N(σ) = Z(σ) + O(σ^{d-1}\mathcal{R}(σ)^{-1/2})\] of the Duistermaat-Guillemin result follows under slightly stronger hypotheses, these implying that the $(d-1)$-fold regularized half-wave trace \[\langle D_τ\rangle^{1-d} \operatorname{HWT}_{M,g}(τ)\] is in $\smash{L^{1,1}_\mathrm{loc}(\mathbb{R}\backslash \{0\})}$. Here $Z(σ)\in \mathbb{R}[σ]$ is a polynomial and $\mathcal{R}(σ):\mathbb{R}^+\to \mathbb{R}^+$ is an $(M,g)$-dependent nondecreasing function with $\lim_{σ\to\infty} \mathcal{R}(σ)=\infty$, specified in terms of the growth rate of $\langle D_τ\rangle^{1-d} τ^{-1}\operatorname{HWT}_{M,g}(τ)$ as measured in $L^{1,1}$. Per Duistermaat-Guillemin, this hypothesis is implied by geometric conditions that hold ``generically'' for $d\geq 3$. Thus, we clarify the relation between the error term in Weyl's law and the long time behavior of the half-wave trace.

math.SP↗

arXiv2023

A strengthened Orlicz-Pettis theorem via Itô-Nisio

In this note we deduce a strengthening of the Orlicz-Pettis theorem from the Itô-Nisio theorem. The argument shows that given any series in a Banach space which isn't summable (or more generally unconditionally summable), we can construct a (coarse-grained) subseries with the property that -- under some appropriate notion of "almost all" -- almost all further subseries thereof fail to be weakly summable. Moreover, a strengthening of the Itô-Nisio theorem by Hoffmann-Jorgensen allows us to replace `weakly summable' with `$τ$-weakly summable' for appropriate topologies $τ$ weaker than the weak topology. A treatment of the Itô-Nisio theorem for admissible $τ$ is given.

math.FA↗

arXiv2022

Half-waves and spectral Riesz means on the 3-torus

For a full rank lattice $Λ\subset \mathbb{R}^d$ and $\mathbf{A} \in \mathbb{R}^d$, consider $N_{d,0;Λ,\mathbf{A}}(Σ) = \# ([Λ+\mathbf{A}] \cap Σ\mathbb{B}^d) = \# \{\mathbf{k}\in Λ: |\mathbf{k}+\mathbf{A}| \leq Σ\}$. Consider the iterated integrals \[ N_{d,k+1;Λ,\mathbf{A}}(Σ) = \int_0^ΣN_{d,k;Λ,\mathbf{A}}(σ) \,\mathrm{d} σ, \] for $k\in \mathbb{N}$. After an elementary derivation via the Poisson summation formula of the sharp large-$Σ$ asymptotics of $N_{3,k;Λ,\mathbf{A}}(Σ)$ for $k\geq 2$ (these having an $O(Σ)$ error term), we discuss how they are encoded in the structure of the Fourier transform $\mathcal{F}N_{3;Λ,\mathbf{A}}(τ)$. The analysis is related to Hörmander's analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrödinger operator on the flat $3$-torus. That the $N_{3,k;Λ,\mathbf{A}}(Σ)$ obey an asymptotic expansion to $O(Σ^2)$ is a special case of a general result holding for all magnetic Schrödinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at $τ=0$. The improvement to $O(Σ)$ for $k\geq 2$ follows from a bound on the growth rate of the half-wave trace at late times.

math.NT↗