Research
Publications
Semiclassical asymptotics in quantum mechanics
- Classical-Quantum correspondence in Lindblad evolution
with Maciej Zworski. .
- Propagation for Schrodinger operators with potentials singular along a hypersurface
with Jared Wunsch. 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.
- Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant
with Yaiza Canzani.
- Weyl remainders: an application of geodesic beams with Yaiza Canzani. to appear in Invent. Math.
- Growth of high L^p norms for eigenfunctions: an application of geodesic beams with Yaiza Canzani. . to appear in Anal. PDE.
- Lower bounds for Cauchy data on curves in a negatively curved surface with Steve Zelditch. . to appear in Israel J. Math.
- Eigenfunction concentration via geodesic beams with Yaiza Canzani. . to appear in J. Reine Angew. Math.
- Pointwise bounds for joint eigenfunctions of quantum completely integrable systems with John Toth. Comm. Math. Phys. 375(2):915-947, 2020.
- Improvements for eigenfunction averages: An application of geodesic beams with Yaiza Canzani. to appear in J. Differ. Geom.
- A microlocal approach to eigenfunction concentration. Journées équations aux dérivées partielles (2018): 1-14. .
- Control from an interior hypersurface with M. Leautaud. Trans. Amer. Math. Soc. 273(5):3177-3233, 2020.
- On the growth of eigenfunction averages: microlocalization and geometry with Yaiza Canzani. Duke Math. J. 168(16):2991β3055, 2019.
- Averages of eigenfunctions over hypersurfaces with Yaiza Canzani and John Toth. Comm. Math. Phys., 360(2):619-637, 2018.
- Defect measures of eigenfunctions with maximal L^\infty growth . Annales de L'institut Fourier 69(4):1757--1798, 2019.
- Eigenfunction scarring and improvements in L^\infty growth with John Toth. Anal. PDE, 11(3):801-812, 2018.
- The L^2 behavior of eigenfunctions near the glancing set. Comm. Partial Differential Equations, 41(10):1619-1648, 2016.
Asymptotics of Steklov eigenvalues and eigenfunctions
Numerical analysis of the Helmholtz Equation
- Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations. with Theophile Chaumont-Frelet and Euan Spence
- 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
- Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies. with Martin Averseng and Euan Spence
- Sharp preasymptotic error bounds for the Helmholtz h-FEM. with Euan Spence
- Lower bounds for piecewise polynomial approximations of oscillatory functions.
- 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.
- Does the Helmholtz boundary element method suffer from the pollution effect?. with Euan Spence. , SIAM Rev. 65(2023), no.3, 806β828.
- Perfectly-matched-layer truncation is exponentially accurate at high frequency with David Lafontaine and Euan Spence.
SIAM J. Math. Anal.
- 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. 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. 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. SIAM J. Math. Anal. 55 (2023), no. 4, 3903β3958.
- Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves with David Lafontaine and Euan Spence. to appear in IMA J. Numer. Anal.
- Wavenumber-explicit analysis for the Helmholtz h-BEM error estimates and iteration counts for the Dirichlet problem with Eike Muller and Euan Spence. Numer. Math, 142(2):329-357, 2019.
Mathematics of solid state physics
Mathematical scattering theory
- The scattering phase: seen at last. with Pierre Marchand, Jian Wang, and Maciej Zworski. SIAM J. Appl. Math. 84 (2024), no. 1, 246β261.
- Classical Wave methods and modern gauge transforms: Spectral Asymptotics
in the one dimensional case. with Leonid Parnovski and Roman Shterenberg. Geom. Funct. Anal. 33 (2023), no. 6, 1454β1538.
- Semiclassical resolvent bounds for compactly supported radial potentials. with Kiril Datchev and Jacob Shapiro.
- Complete asymptotic expansions of the spectral function for symbolic
perturbations of almost periodic Schrodinger operators in dimension one. J. Spectr. Theory 12 (2022), no. 1, 105β142.
- Semiclassical resolvent bounds for long range Lipschitz potentials with Jacob Shapiro. tInt. Math. Res. Not. IMRN(2022), no. 18, 14134β14150.
- Outgoing solutions via Gevrey-2 properties with Maciej Zworski. Ann. PDE 7 (2021), no. 1, Paper No. 5, 13 pp.
- Analytic hypoellipticity of Keldysh operators with Maciej Zworski. Proc. Lond. Math. Soc. (3) 123 (2021), no. 5, 498β516.
- Semiclassical resolvent bounds for weakly decaying potentials with Jacob Shapiro. . Math. Res. Lett. 29 (2022), no. 2, 373β397.
- Viscosity limits for 0th order pseudodifferential operators with Maciej Zworski. , Comm. Pure Appl. Math. 75 (2022), no. 8, 1798β1869.
- An introduction to complex microlocal deformations with Maciej Zworski. (an expository companion to arXiv1912.09840).
- Optimal constants in non-trapping resolvent estimates and applications in numerical analysis with Euan Spence and Jared Wunsch. Pure and Applied Analysis. 2(1): 157-202, 2020.
- On non-diffractive cones with Jared Wunsch. J. Differential Geom. 120 (2022), no. 3, 505β518.
- Fractal Weyl laws and wave decay for general trapping with Semyon Dyatlov. Nonlinearity, 30(12):4301-4343. 2017.
- A quantitative Vaniberg method for black box scattering. Comm. Math. Phys.. 349(2):527-549, 2017/
- The quantum Sabine law for resonances in transmission problems. Pure and Applied Analysis, 1(1):27-100, 2019.
- Resonances for thin barriers on the circle. J. Phys. A, 49(12):125205, 22, 2016..
- Distribution of resonances in scattering by thin barriers. Mem. Amer. Math. Soc., 259(1248):ix + 152, 2019.
- Restriction bounds for the free resolvent and resonances in lossy scattering with H. Smith. Int. Math. Res. Not., (16):7473-7509, 2015.
Analysis of boundary integral operators and boundary layer potentials
Quantum chaos
Pseudospectral effects in non-self-adjoint problems
Other Research