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.
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arXiv subjects
Andrew Hassell
Publications and source records attributed to Andrew Hassell.
The scattering map for the Schrodinger operator on curved spaces
Let $P$ be a Schr\"odinger operator $D_t+\Delta_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$. Here $D_t = -i \partial_t$ and $\Delta_g$ is the positive Laplacian. We consider the scattering map $S$ defined previously by the first author with Gell-Redman and Gomes arXiv:2201.03140, which relates the asymptotic data, as $t \to \pm \infty$, of global solutions $u$ to $Pu = 0$. We show that $S$ is a `1-cusp' Fourier integral operator, where `1-cusp' refers to a pseudodifferential calculus introduced by Vasy and Zachos arXiv:2204.11706 in the completely different setting of inverse problems on asymptotically conic manifolds. Our viewpoint is that 1-cusp geometry is the natural setting for studying the asymptotic data of solutions to Schr\"odinger's equation.
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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 Schr\"odinger 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.
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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, $\Psi_{\natural}$ (a variant of the semiclassical scattering calculus), $\Psi_{\natural\mathrm{res}}$, and $\Psi_{\natural2\mathrm{res}}$, 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.
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The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3
In this article we consider the defocusing nonlinear Schr\"odinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schr\"odinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.
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Geodesic flow and decay of traces on hyperbolic surfaces
We study pseudodifferential operators on a hyperbolic surface using `Zelditch quantization'. We motivate and study the trace of $A_2^* A_1(t)$, where $A_2$ is a fixed operator and the Zelditch symbol of $A_1(t)$ evolves by geodesic flow. We find conditions under which the trace decays exponentially as $t \to \pm \infty$.
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Scattering regularity for small data solutions of the nonlinear Schr\"{o}dinger equation
Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad u(z,t) \sim (4\pi it)^{-n/2} e^{i|z|^2/4t} f\big( \frac{z}{2t} \big), \quad t \to -\infty, $$ where $u : \mathbb{R}^{n+1} \to \mathbb{C}$ is the unknown and $f : \mathbb{R}^n \to \mathbb{C}$ is the asymptotic data. Here $D_t = -i \frac{\partial}{\partial t}$ and $\Delta = \sum_{j=1}^n D_{z_j} D_{z_j}$ is the positive Laplacian, or more generally a compactly supported, nontrapping perturbation of this, $V$ is a smooth compactly supported potential function, and the nonlinear term $N$ is a (suitable) polynomial in $u$, $\partial_{z_j}u$ and their complex conjugates satisfying phase invariance. Our assumption on the asymptotic data $f$ is that it is small in a certain function space $\mathcal{W}^k$ constructed in arXiv:2201.03140, for sufficiently large $k \in \mathbb{N}$, where the index $k$ measures both regularity and decay at infinity (it is similar to, but not quite a standard weighted Sobolev space $H^{k, k}(\mathbb{R}^n)$). We find that for $N[u] = \pm |u|^{p-1} u$, $p$ odd, and $(n,p) \neq (1, 3)$ then if the asymptotic data as $t \to -\infty$ is small in $\mathcal{W}^k$, then the asymptotic data as $t \to +\infty$ is also in $\mathcal{W}^k$; that is, the nonlinear scattering map preserves these spaces of asymptotic data. For a more general nonlinearity involving derivatives of $u$, we show that if the asymptotic data as $t \to -\infty$ is small in $\langle \zeta \rangle^{-1} \mathcal{W}^k_\zeta$, then the asymptotic data as $t \to +\infty$ is also in this space (where $\zeta$ is the argument of $f$).
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Function spaces for decoupling
We introduce new function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that yield a natural reformulation of the $\ell^{q}L^{p}$ decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless $p=q$, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.
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Propagation of singularities and Fredholm analysis for the time-dependent Schr\"odinger equation
We study the time-dependent Schr\"odinger operator $P = D_t + \Delta_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $\Delta_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, by finding pairs of Hilbert spaces between which the operator acts invertibly. Using this invertibility it is straightforward to solve the `final state problem' for the time-dependent Schr\"odinger equation, that is, find a global solution $u(z, t)$ of $Pu = 0$ having prescribed asymptotics as $t \to \infty$. These asymptotics are of the form $$ u(z, t) \sim t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \quad t \to +\infty $$ where $f_+$, the `final state' or outgoing data, is an arbitrary element of a suitable function space $\mathcal{W}^k(\mathbb{R}^n)$; here $k$ is a regularity parameter simultaneously measuring smoothness and decay at infinity. We can of course equally well prescribe asymptotics as $t \to -\infty$; this leads to incoming data $f_-$. We consider the `Poisson operators' $\mathcal{P}_\pm : f_\pm \to u$ and precisely characterize the range of these operators on $\mathcal{W}^k(\mathbb{R}^n)$ spaces. Finally we show that the scattering matrix, mapping $f_-$ to $f_+$, preserves these spaces.
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Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions
We study the nonlinear Helmholtz equation $(\Delta - \lambda^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $\lambda > 0$, $p \in \mathbb{N}$ odd, and more generally $(\Delta_g + V - \lambda^2)u = N[u]$, where $\Delta_g$ is the (positive) Laplace-Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1) > 4$ and $k > (n-1)/2$, for every $f \in H^k(S^{n-1}_\omega)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, \omega) = r^{-(n-1)/2} \Big( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} b(\omega) + O(r^{-\epsilon}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b \in H^k(S_\omega^{n-1})$ and $\epsilon > 0$. That is, the scattering matrix $f \mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work with Zhang, that proved a similar result with a loss of four derivatives.
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$L^{p}$ and $\mathcal{H}^{p}_{FIO}$ regularity for wave equations with rough coefficients
We consider wave equations with time-independent coefficients that have $C^{1,1}$ regularity in space. We show that, for nontrivial ranges of $p$ and $s$, the standard inhomogeneous initial value problem for the wave equation is well posed in Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators introduced recently by the authors and Portal, following work of Smith. In spatial dimensions $n = 2$ and $n=3$, this includes the full range $1 < p < \infty$. As a corollary, we obtain the optimal fixed-time $L^{p}$ regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.
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Off-singularity bounds and Hardy spaces for Fourier integral operators
We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.
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Riesz transforms on a class of non-doubling manifolds II
We consider a class of manifolds $\mathcal{M}$ obtained by taking the connected sum of a finite number of $N$-dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, δ) \times (\mathcal{M}_i, g)$, where $\mathcal{M}_i$ is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions $n_i$ are not all equal. This means that the ends have different `asymptotic dimension', and implies that the Riemannian manifold $\mathcal{M}$ is not a doubling space. In the first paper in this series, by the first and third authors, we considered the case where each $n_i$ is least $3$. In the present paper, we assume that one of the $n_i$ is equal to $2$, which is a special and particularly interesting case. Our approach is to construct the low energy resolvent and determine the asymptotics of the resolvent kernel as the energy tends to zero. We show that the resolvent kernel $(Δ+ k^2)^{-1}$ on $\mathcal{M}$ has an expansion in powers of $1/\log (1/k)$ as $k \to 0$, which is significantly different from the case where all $n_i$ are at least 3, in which case the expansion is in powers of $k$. We express the Riesz transform in terms of the resolvent to show that it is bounded on $L^p(\mathcal{M})$ for $1 < p \leq 2$, and unbounded for all $p > 2$.
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Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics
We study eigenvalues of non-self-adjoint Schrödinger operators on non-trapping asymptotically conic manifolds of dimension $n\ge 3$. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the $L^p$-norm of the potential, while the second one is a Lieb-Thirring type bound controlling sums of powers of eigenvalues in terms of the $L^p$-norm of the potential. We extend the results of Frank (2011), Frank-Sabin (2017), and Frank-Simon (2017) on the Keller and Lieb-Thirring type bounds from the case of Euclidean spaces to that of non-trapping asymptotically conic manifolds. In particular, our results are valid for the operator $Δ_g+V$ on $\mathbb{R}^n$ with $g$ being a non-trapping compactly supported (or suitably short range) perturbation of the Euclidean metric and $V\in L^p$ complex valued.
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Existence and asymptotics of nonlinear Helmholtz eigenfunctions
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} where $Δ= -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $λ$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic. We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} g(ω) + O(r^{-ε}) \Big), \text{ as } r \to \infty, \quad ε> 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.
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Riesz transforms on a class of non-doubling manifolds
We consider a class of manifolds $\mathcal{M}$ obtained by taking the connected sum of a finite number of $N$-dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, δ) \times (\mathcal{M}_i, g)$, where $\mathcal{M}_i$ is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions $n_i$ are not all equal. This means that the ends have different `asymptotic dimension', and implies that the Riemannian manifold $\mathcal{M}$ is not a doubling space. We completely describe the range of exponents $p$ for which the Riesz transform on $\mathcal{M}$ is a bounded operator on $L^p(\mathcal{M})$. Namely, under the assumption that each $n_i$ is at least $3$, we show that Riesz transform is of weak type $(1,1)$, is continuous on $L^p$ for all $p \in (1, \min_i n_i)$, and is unbounded on $L^p$ otherwise. This generalizes results of the first-named author with Carron and Coulhon devoted to the doubling case of the connected sum of several copies of Euclidean space $\mathbb{R}^{N}$, and of Carron concerning the Riesz transform on connected sums.
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Semiclassical scarring on tori in KAM Hamiltonian systems
We show that for almost all perturbations in a one-parameter family of KAM Hamiltonians on a smooth compact surface, for almost all KAM Lagrangian tori $Λ_ω$, we can find a semiclassical measure with positive mass on $Λ_ω$.
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The Heat Kernel on Asymptotically Hyperbolic Manifolds
Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points. Inspired by the work of Davies-Mandouvalos on $\mathbb{H}^{n + 1}$, we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel on such manifolds is comparable to the heat kernel on hyperbolic space of the same dimension (expressed as a function of time $t$ and geodesic distance $r$), uniformly for all $t \in (0, \infty)$ and all $r \in [0, \infty)$. Our approach is microlocal and based on the resolvent on asymptotically hyperbolic manifolds, constructed in the celebrated work of Mazzeo-Melrose, as well as its high energy asymptotic, due to Melrose-Sà Barreto-Vasy.
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