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Recent publications

• 1D scattering through time dependent media with memory. with Zhen Huang and Maciej Zworski. arXiv2602.19981.
• Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis. with Euan Spence arXiv2511.15287 , to appear in Acta Numer.
• Helmholtz boundary integral methods and the pollution effect. with Manas Rachh and Euan Spence arXiv2507.2297
• Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems. with Martin Averseng and Euan Spence arXiv2506.15630
• Surface Plasmons in Metamaterial Cavities: Scattering by Obstacles with Negative Wave Speed. with Yan-Long Fang. arXiv2505.22253.
• Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems. with Euan Spence arXiv2501.11060, to appear in SIAM J. Numer. Anal.
• Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations. with Theophile Chaumont-Frelet and Euan Spence arXiv2408.04507 to appear in Math. Comp.
• Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems. with Shihua Gong, Ivan Graham, David Lafontaine, and Euan Spence arXiv2404.02156
• Classical-Quantum correspondence in Lindblad evolution with Maciej Zworski and Zhen Huang. 2403.09345. J. Math. Phys., 66(9):Paper No. 091503, 33, 2025

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Numerical analysis of the Helmholtz Equation

• Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis. with Euan Spence arXiv2511.15287 , to appear in Acta Numer.
• Helmholtz boundary integral methods and the pollution effect. with Manas Rachh and Euan Spence arXiv2507.2297
• Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems. with Martin Averseng and Euan Spence arXiv2506.15630
• Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems. with Euan Spence arXiv2501.11060, to appear in SIAM J. Numer. Anal.
• Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations. with Theophile Chaumont-Frelet and Euan Spence arXiv2408.04507 to appear in Math. Comp.
• Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems. with Shihua Gong, Ivan Graham, David Lafontaine, and Euan Spence arXiv2404.02156
• Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies. with Martin Averseng and Euan Spence arXiv2304.14737 .
• Sharp preasymptotic error bounds for the Helmholtz h-FEM. with Euan Spence arXiv2301.03574 . SIAM J. Numer. Anal., 63(1):1–22, 2025.
• Lower bounds for piecewise polynomial approximations of oscillatory functions. arXiv2211.04757 . Journal of Approximation Theory, 305:106100, 2025.
• The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect . with David Lafontaine, Euan Spence, and Jared Wunsch. 2207.05542 . Commun. Math. Sci., 22(7):1761–1816, 2024.
• Does the Helmholtz boundary element method suffer from the pollution effect?. with Euan Spence. arXiv2201.09721. SIAM Rev. 65(2023), no.3, 806–828.
• Perfectly-matched-layer truncation is exponentially accurate at high frequency with David Lafontaine and Euan Spence. arXiv2105.07737 . SIAM J. Math. Anal., 55(4):3344–3394, 2023.
• Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?. with Pierre Marchand, Alastair Spence, Euan Spence. arXiv2102.05367. Adv. Comput. Math. 48 (2022), no. 4, Paper No. 37, 63 pp.. Math.
• Eigenvalues of the truncated Helmholtz solution operator under strong trapping. with Pierre Marchand and Euan Spence. arXiv2101.02116 SIAM J. Math. Anal. 53 (2021), no. 6, 6724-6770.
• Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method with David Lafontaine, Euan Spence, and Jared Wunsch. arXiv2102.13081. SIAM J. Math. Anal. 55 (2023), no. 4, 3903–3958.
• Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves with David Lafontaine and Euan Spence. arXiv2101.02154. IMA J. Numer. Anal., 44(4):1946–2069, 2024.
• Wavenumber-explicit analysis for the Helmholtz h-BEM error estimates and iteration counts for the Dirichlet problem with Eike Muller and Euan Spence. arXiv1608.01035. Numer. Math, 142(2):329-357, 2019.

Semiclassical asymptotics in quantum mechanics

• Classical-Quantum correspondence in Lindblad evolution with Maciej Zworski and Zhen Huang. 2403.09345. J. Math. Phys., 66(9):Paper No. 091503, 33, 2025
• Propagation for Schrodinger operators with potentials singular along a hypersurface with Jared Wunsch. arXiv2302.08154. Arch. Ration. Mech. Anal. 248 (2024), no. 3, Paper No. 37, 28 pp.
• Asymptotics for the spectral function on Zoll manifolds with Yaiza Canzani and Blake Keeler. arXiv2211.09644. J. Spectr. Theory, 15(3):995–1044, 2025.
• Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant with Yaiza Canzani. arXiv2204.11921.
• Weyl remainders: an application of geodesic beams with Yaiza Canzani. arXiv2010.03969. Invent. Math., 232(3):1195–1272, 2023.
• Growth of high L^p norms for eigenfunctions: an application of geodesic beams with Yaiza Canzani. arXiv2003.04597. Anal. PDE, 16(10):2267–2325, 2023.
• Lower bounds for Cauchy data on curves in a negatively curved surface with Steve Zelditch. arXiv2002.09456. Israel J. Math., 244(2):971–1000, 2021.
• Eigenfunction concentration via geodesic beams with Yaiza Canzani. arXiv1903.08461. J. Reine Angew. Math., 775:197–257, 2021.
• Pointwise bounds for joint eigenfunctions of quantum completely integrable systems with John Toth. arXiv1810.04232. Comm. Math. Phys. 375(2):915-947, 2020.
• Improvements for eigenfunction averages: An application of geodesic beams with Yaiza Canzani. arXiv1809.06296. Journal of Differential Geometry, 124(3):443 – 522, 2023.
• A microlocal approach to eigenfunction concentration. arXiv1809.08677. Journées équations aux dérivées partielles (2018): 1-14.
• Control from an interior hypersurface with M. Leautaud. arXiv1711.03939. Trans. Amer. Math. Soc. 273(5):3177-3233, 2020.
• On the growth of eigenfunction averages: microlocalization and geometry with Yaiza Canzani. arXiv1710.07972. Duke Math. J. 168(16):2991–3055, 2019.
• Averages of eigenfunctions over hypersurfaces with Yaiza Canzani and John Toth. arXiv1705.09595. Comm. Math. Phys., 360(2):619-637, 2018.
• Defect measures of eigenfunctions with maximal L^\infty growth . arXiv1704.01452. Annales de L'institut Fourier 69(4):1757--1798, 2019.
• Eigenfunction scarring and improvements in L^\infty growth arXiv1703.10248. with John Toth. Anal. PDE, 11(3):801-812, 2018.
• The L^2 behavior of eigenfunctions near the glancing set. arXiv1604.01699. Comm. Partial Differential Equations, 41(10):1619-1648, 2016.

Asymptotics of Steklov eigenvalues and eigenfunctions

• Lower bounds for Steklov eigenfunctions. with John Toth. arXiv2112.11415 . Pure Appl. Math. Q. 19 (2023), no. 4, 1873–1898.
• Domains without dense Steklov nodal sets with Oscar Bruno. arXiv1908.03307 . J. Fourier Anal. Appl. 26(3):45, 2020.
• Pointwise bounds for Steklov eigenfunctions with John Toth. arXiv1611.05363 . J. Geom. Anal., 29:142-193, 2019.

Mathematics of solid state physics

• An abstract formulation of the flat band condition. with Maciej Zworski. arXiv2307.04896. to appear in Contemp. Math., 2023.

Mathematical scattering theory

• 1D scattering through time dependent media with memory. with Zhen Huang and Maciej Zworski. arXiv2602.19981.
• Surface Plasmons in Metamaterial Cavities: Scattering by Obstacles with Negative Wave Speed. with Yan-Long Fang. arXiv2505.22253.
• The scattering phase: seen at last. with Pierre Marchand, Jian Wang, and Maciej Zworski. arXiv2210.09908 . SIAM J. Appl. Math. 84 (2024), no. 1, 246–261.
• Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case. with Leonid Parnovski and Roman Shterenberg. arXiv2207.08245 . Geom. Funct. Anal. 33 (2023), no. 6, 1454–1538.
• Semiclassical resolvent bounds for compactly supported radial potentials. with Kiril Datchev and Jacob Shapiro. arXiv2112.15133, . J. Funct. Anal. 284 (2023), no. 7, Paper No. 109835, 28 pp.
• Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrodinger operators in dimension one. arXiv2011.09245. J. Spectr. Theory 12 (2022), no. 1, 105–142.
• Semiclassical resolvent bounds for long range Lipschitz potentials with Jacob Shapiro. arXiv2010.01166. Int. Math. Res. Not. IMRN(2022), no. 18, 14134–14150.
• Outgoing solutions via Gevrey-2 properties with Maciej Zworski. arXiv2004.07868 Ann. PDE 7 (2021), no. 1, Paper No. 5, 13 pp.
• Analytic hypoellipticity of Keldysh operators with Maciej Zworski. arXiv2003.08106. Proc. Lond. Math. Soc. (3) 123 (2021), no. 5, 498–516.
• Semiclassical resolvent bounds for weakly decaying potentials with Jacob Shapiro. arXiv2003.02525. Math. Res. Lett. 29 (2022), no. 2, 373–397.
• Viscosity limits for 0th order pseudodifferential operators with Maciej Zworski. arXiv1912.09840. Comm. Pure Appl. Math. 75 (2022), no. 8, 1798–1869.
• An introduction to complex microlocal deformations with Maciej Zworski. arXiv1912.09845 (an expository companion to arXiv1912.09840).
• Optimal constants in non-trapping resolvent estimates and applications in numerical analysis with Euan Spence and Jared Wunsch. arXiv1810.13426 . Pure and Applied Analysis. 2(1): 157-202, 2020.
• On non-diffractive cones with Jared Wunsch. arXiv1807.05043. J. Differential Geom. 120 (2022), no. 3, 505–518.
• Fractal Weyl laws and wave decay for general trapping with Semyon Dyatlov. arXiv1703.06515. Nonlinearity, 30(12):4301-4343. 2017.
• A quantitative Vainberg method for black box scattering. arXiv1511.05894. Comm. Math. Phys.. 349(2):527-549, 2017/
• The quantum Sabine law for resonances in transmission problems. arXiv1511.05091. Pure and Applied Analysis, 1(1):27-100, 2019.
• Resonances for thin barriers on the circle. arXiv1410.0340. J. Phys. A, 49(12):125205, 22, 2016..
• Distribution of resonances in scattering by thin barriers. arXiv1404.3709. Mem. Amer. Math. Soc., 259(1248):ix + 152, 2019.
• Restriction bounds for the free resolvent and resonances in lossy scattering with H. Smith. arXiv1401.6243. Int. Math. Res. Not., (16):7473-7509, 2015.

Analysis of boundary integral operators and boundary layer potentials

• High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem. with Pierre Marchand, Euan Spence. arXiv2109.06017. Integral Equations Operator Theory 94 (2022), no. 4, Paper No. 36, 68 pp.
• Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators. with Euan Spence. arXiv1807.09719. Integral Equations and Operator Theory, 91(1):Art. 6, 35, 2019.
• Appendix in Semiclassical single and double layer potentials: boundedness and sharpness by X. Han and M. Tacy. arXiv1403.6576. J. Funct. Anal. 269:2890-2926, 2015.

Quantum chaos

• Arbitrarily small perturbations of Dirichlet Laplacians are quantum unique ergodic with Sourav Chatterjee. arXiv 1603.00597. J. Spectr. Theory. 8(3):909-947, 2018.
• Quantum ergodicity for a class of mixed systems. arXiv1209.2968. J. Spectr. Theory, 4(1):65-85, 2014.

Pseudospectral effects in non-self-adjoint problems

• Pseudospectra of semiclassical boundary value problems. arXiv1211.3366. J. Inst. Math. Jussieu, 14(2):405-449, 2015.
• Nonlinear instability in a semiclassical problem. arXiv1109.0773. Comm. Math. Phys., 316(3):705-722, 2012.

• Distribution of Resonances in Scattering by Thin Barriers. Ph. D. dissertation (May 2015), written under the supervision of Maciej Zworski.
• Synthesis of size-controlled and shaped copper nanoparticles by D. Mott, J. Galkowski, L. Wan, J. Luo, and C.J. Zhong. Langure, 23(10): 5740-5745, 2007.