About

Hans Christianson is a Professor and Associate Chair in the Department of Mathematics at the University of North Carolina at Chapel Hill. He earned his B.S. from the University of Minnesota, Twin Cities in 2002 and completed his Ph.D. at the University of California, Berkeley in 2007. Following his doctoral studies, he served as a CLE Moore Instructor at MIT from 2007 to 2010 and was a postdoctoral researcher at MSRI in 2008. He joined UNC Chapel Hill as an Assistant Professor in 2010 and was promoted to Associate Professor in 2016, a position he currently holds. His research focuses on the intersection of classical and quantum systems, particularly through the lens of differential geometry, microlocal analysis, and spectral geometry. His work explores how classical systems, such as geodesic flows in Newtonian mechanics, relate to quantum phenomena like drum vibrations and quantum particle distributions. Christianson is especially interested in chaotic and singular classical systems, which influence the behavior of quantum systems, including wave equidistribution and unstable scattering. His contributions advance the understanding of partial differential equations and their applications in quantum chaos and spectral geometry.

Research topics

• Mathematical analysis
• Mathematics
• Physics
• Geometry
• Applied mathematics
• Computer Science
• Combinatorics
• Quantum mechanics
• Pure mathematics

Selected publications

• Energy distribution for Dirichlet eigenfunctions on right triangles

  Journal of Differential Equations · 2024-06-21

  article1st authorCorresponding
  PublisherDOI

• Quantum Flux and Quantum Ergodicity for Cross Sections

  arXiv (Cornell University) · 2024-04-02

  preprintOpen access1st authorCorresponding

  For sequences of quantum ergodic eigenfunctions, we define the quantum flux norm associated to a codimension $1$ submanifold $Σ$ of a non-degenerate energy surface. We prove restrictions of eigenfunctions to $Σ$, realized using the quantum flux norm, are quantum ergodic. We compare this result to known results from \cite{CTZ} in the case of Euclidean domains and hyperfurfaces. As a further application, we consider complexified analytic eigenfunctions and prove a second microlocal analogue of \cite{CTZ} in that context.

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• Non-concentration estimates for Laplace eigenfunctions on compact $C^{\infty}$ manifolds with boundary

  arXiv (Cornell University) · 2024-12-23

  preprintOpen access1st authorCorresponding

  Let $Ω$ be an $n$-dimensional compact Riemannian manifold $(n \geq 3)$ with $C^\infty$ boundary, and consider $L^2$-normalized eigenfunctions $ - Δϕ_λ = λ^2 ϕ_λ$ with Dirichlet or Neumann boundary conditions . In this note, we extend well-known interior nonconcentration bounds up to the boundary. Specifically, in Theorem \ref{thm1}, using purely stationary local methods, we prove that for such $Ω$ it follows that for {\em any} $x_0 \in \overlineΩ$ (including boundary points) and for all $μ\geq C_Ω λ^{-1}$ with sufficiently large constant $C_Ω >0,$ \begin{equation} \label{nonconbdy} \| ϕ_λ\|_{B(x_0,μ)\cap Ω}^2 = O(μ). \end{equation} In Theorem \ref{thm2} we extend a result of Sogge \cite{So} to manifolds with smooth boundary and show that \begin{equation} \label{SUPBD} \| ϕ_λ\|_{L^\infty(Ω)} \leq C λ^{\frac{n}{2}} \cdot \Big( \sup_{x \in Ω} \| ϕ_λ \|_{L^2( B(x,λ^{-1}) \cap Ω)} \Big). \end{equation} The sharp sup bounds $\| ϕ_λ \|_{L^\infty(Ω)} = O(λ^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.

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• Control estimates for 0th order pseudodifferential operators

  arXiv (Cornell University) · 2023-03-11

  preprintOpen access1st authorCorresponding

  We introduce the control conditions for 0th order pseudodifferential operators $\mathbf{P}$ whose real parts satisfy the Morse--Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i\partial_t-\mathbf{P})u=f$ when $\mathbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty}$. This is motivated by the study of a microlocal model for the damped internal waves.

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• Energy Distribution for Dirichlet Eigenfunctions on Right Triangles

  arXiv (Cornell University) · 2023-01-09

  preprintOpen access1st authorCorresponding

  In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \cite{Chr-tri} and \cite{Chr-simp}. The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being equidistributed on each side. The proof of this result is remarkably simple, using only the radial vector field and a Rellich type integrations by parts. The equidistribution law, including on higher dimensional simplices, agrees with what Quantum Ergodic Restriction would predict. However, distribution of the Neumann data on subsets of a side is not well understood, and elementary methods do not appear to give enough information to draw conclusions. In the present note, we first show that an "obvious" conjecture fails even for the simplest right isosceles triangle using only Fourier series. We then use a result of Marklof-Rudnick \cite{Marklof-Rudnick} in which the authors show an interior {\it spatial} equidistribution law for a density-one subsequence of eigenfunctions to give an estimate on energy distribution of eigenfunctions on the interior. Finally we present some numerical computations suggesting the behaviour of eigenfunctions on almost isosceles triangles is quite complicated.

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• Control Estimates for 0th-Order Pseudodifferential Operators

  International Mathematics Research Notices · 2023 · 1 citations

  1st authorCorresponding
  • Computer Science
  • Mathematics
  • Pure mathematics

  Abstract We introduce the control conditions for 0th-order pseudodifferential operators $\textbf{P}$ whose real parts satisfy the Morse–Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i\partial _{t}-\textbf{P})u=f$ when $\textbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty }$. This is motivated by the study of a microlocal model for the damped internal waves.

  PublisherDOI

• Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains

  arXiv (Cornell University) · 2023-09-19

  preprintOpen access1st authorCorresponding

  Let $Ω$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary and corner points) and any $δ\in [0,1),$ $$ \| ϕ_λ\|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument.

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• Local smoothing for the Schrödinger equation on a multi-warped product manifold with inflection-transmission trapping

  Proceedings of the American Mathematical Society · 2021-04-21

  preprintOpen access1st authorCorresponding

  Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of Christianson and Metcalfe [Indiana Univ. Math. J. 63 (2014), pp. 969–992] on inflection-transmission type trapping on warped product manifolds to the case of <italic>multi</italic> -warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping.

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• Exterior Mass Estimates and L2-Restriction Bounds for Neumann Data Along Hypersurfaces

  UNC Libraries · 2021-08-14 · 2 citations

  articleOpen access

  We study the problem of estimating the L2 norm of Laplace eigenfunctions on a compact Riemannian manifold M when restricted to a hypersurface H. We prove mass estimates for the restrictions of eigenfunctions ϑh, (h2Δ − 1)ϑh = 0, to H in the region exterior to the coball bundle of H, on hϐ-scales (0 ≤ ϐ < 2/3). We use this estimate to obtain an O(1) L2-restriction bound for the Neumann data along H. The estimate also applies to eigenfunctions of semiclassical Schroedinger operators.

  PublisherDOI

• Construction of nonlinear quasimodes near elliptic periodic orbits

  UNC Libraries · 2021-08-28

  articleOpen access

  We consider the nonlinear Schroedinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schroedinger evolution of such a quasimode remains localized near the geodesic, at least for short times.

  PublisherDOI

Recent grants

• Qualitative Properties of Eigenfunctions for some Selfadjoint and Non-selfadjoint partial differential equations

  NSF · $180k · 2015–2019

• Microlocal analysis in nonlinear PDE and PDE on manifolds

  NSF · $157k · 2009–2010

• Microlocal analysis in nonlinear PDE and PDE on manifolds

  NSF · $84k · 2010–2013

Frequent coauthors

• Jeremy L. Marzuola

  22 shared

• Laurent Thomann

  Institut Élie Cartan de Lorraine

  18 shared

• Pierre Albin

  University of Illinois Urbana-Champaign

  18 shared

• John A. Toth

  10 shared

• Jason Metcalfe

  University of North Carolina at Chapel Hill

  10 shared

• Jared Wunsch

  Northwestern University

  9 shared

• Gigliola Staffilani

  5 shared

• Andrew Hassell

  5 shared