arXiv subjects

Nicholas Lohr

Publications and source records attributed to Nicholas Lohr.

5 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

Semiclassical measures through Coulomb collisions

We prove that $\mu$ is a semiclassical measure associated to a sequence of eigenfunctions of energy $E<0$ of the attractive Coulomb operator if and only if $\mu$ is a probability measure on the energy (hyper)surface $\Sigma_E$ invariant under the regularized Kepler flow due to Moser. The converse was shown in recent work by the author, and the present article proves the other direction (as well as an independent proof of the converse). We prove the main theorem for a general symbol class allowing certain non-decay at infinity, which implies that semiclassical measure mass entirely reflects off of the origin. In the special case of semiclassical measures of sequences of eigenfunctions of the exact Coulomb operator, this article solves an open problem posed by Keraani. The main tools include the celebrated Moser-Fock map along with a technical operator extension lemma in $\Psi_{\hbar}^0(\mathbb{S}^d)$, which utilizes standard eigenfunction concentration bounds.

math.AP↗

arXiv2024

On the intertwining map between Coulomb and hyperbolic scattering

We construct a unitary operator between Hilbert spaces of generalized eigenfunctions of Coulomb operators and the Laplace-Beltrami operator of hyperbolic space that intertwines their respective Poisson operators on $L^2(\mathbb{S}^{d-1})$. The constructed operator generalizes Fock's unitary transformation, originally defined between the discrete spectra of the attractive Coulomb operator and the Laplace-Beltrami operator on the sphere, to the setting of continuous spectra. Among other connections, this map explains why the scattering matrices are the same in these two different settings, and it also provides an explicit formula for the Poisson operator of the Coulomb Hamiltonian.

math.AP↗

arXiv2023

Semiclassical measures of eigenfunctions of the attractive Coulomb operator

We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator $\widehat{H}_{\hbar}:=-\frac{\hbar^2}{2}\Delta_{\mathbb{R}^3}-\frac{1}{|x|}$. In particular, any Radon probability measure on the fixed negative energy hypersurface $\Sigma_E$ of the Kepler Hamiltonian $H$ in classical phase space that is invariant under the regularized Kepler flow is the semiclassical measure of a sequence of eigenfunctions of $\widehat{H}_{\hbar}$ with eigenvalue $E$ as $\hbar \to 0$. The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of $\widehat{H}_{\hbar}$ and $-\Delta_{\mathbb{S}^3}$. We first prove that for any Kepler orbit $\gamma$ on $\Sigma_E$, there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $\gamma$ as $\hbar \to 0$, and we finish using a density argument in the weak-* topology.

math.AP↗

arXiv2022

Scaling Asymptotics of Wigner Distributions of Harmonic Oscillator Orbital Coherent States

The main result of this article gives scaling asymptotics of the Wigner distributions $W_{\varphi_N^{\gamma},\varphi_N^{\gamma}}$ of isotropic harmonic oscillator orbital coherent states $\varphi_N^{\gamma}$ concentrating along Hamiltonian orbits $\gamma$ in shrinking tubes around $\gamma$ in phase space. In particular, these Wigner distributions exhibit a $\textit{hybrid}$ semi-classical scaling. That is, simultaneously, we have an $\textit{Airy scaling}$ when the tube has radius $N^{-2/3}$ normal to the energy surface $\Sigma_E$, and a $\textit{Gaussian scaling}$ when the tube has radius $N^{-1/2}$ tangent to $\Sigma_E$.

math-ph↗