JEFFREY GALKOWSKI

Research

Publications

Semiclassical asymptotics in quantum mechanics

- Classical-Quantum correspondence in Lindblad evolution with Maciej Zworski. 2403.09345.
- Propagation for Schrodinger operators with potentials singular along a hypersurface with Jared Wunsch. arXiv2302.08154 to appear in Arch. Ration. Mech. Anal.
- Asymptotics for the spectral function on Zoll manifolds with Yaiza Canzani and Blake Keeler. arXiv2211.09644
- 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 to appear in Invent. Math.
- Growth of high L^p norms for eigenfunctions: an application of geodesic beams with Yaiza Canzani. arXiv2003.04597. to appear in Anal. PDE.
- Lower bounds for Cauchy data on curves in a negatively curved surface with Steve Zelditch. arXiv2002.09456. to appear in Israel J. Math.
- Eigenfunction concentration via geodesic beams with Yaiza Canzani. arXiv1903.08461. 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. arXiv1809.06296 to appear in J. Differ. Geom.
- A microlocal approach to eigenfunction concentration. Journées équations aux dérivées partielles (2018): 1-14. arXiv1809.08677.
- 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):2991–3055, 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

- Lower bounds for Steklov eigenfunctions. with John Toth. arXiv2112.11415 to apper in Pure Appl. Math. Q.
- Domains without dense Steklov nodal sets with Oscar Bruno. J. Fourier Anal. Appl. 26(3):45, 2020.
- Pointwise bounds for Steklov eigenfunctions with John Toth. J. Geom. Anal., 29:142-193, 2019.

Numerical analysis of the Helmholtz Equation

- 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
- Lower bounds for piecewise polynomial approximations of oscillatory functions. arXiv2211.04757
- 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
- Does the Helmholtz boundary element method suffer from the pollution effect?. with Euan Spence. arXiv2201.09721, to appear in SIAM Review
- Perfectly-matched-layer truncation is exponentially accurate at high frequency with David Lafontaine and Euan Spence. arXiv2105.07737 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. arXiv2102.05367 to appear in Adv. Comput. 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 to appear in SIAM J. Math. Anal.
- Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves with David Lafontaine and Euan Spence. arXiv2101.02154 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

- An abstract formulation of the flat band condition. with Maciej Zworski. arXiv2307.04896.

Mathematical scattering theory

- The scattering phase: seen at last. with Pierre Marchand, Jian Wang, and Maciej Zworski. arXiv2210.09908 to appear in SIAM J. Appl. Math.
- Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case. with Leonid Parnovski and Roman Shterenberg. to appear in Geom. Funct. Anal. arXiv2207.08245
- Semiclassical resolvent bounds for compactly supported radial potentials. with Kiril Datchev and Jacob Shapiro. arXiv2112.15133, to appear in J. Funct. Anal.
- Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrodinger operators in dimension one. arXiv2011.09245 to appear in J. Spectr. Theory
- Semiclassical resolvent bounds for long range Lipschitz potentials with Jacob Shapiro. arXiv2010.01166 to appear in Int. Math. Res. Not. IMRN
- Outgoing solutions via Gevrey-2 properties with Maciej Zworski. arXiv2004.07868 to appear in Ann. PDE
- Analytic hypoellipticity of Keldysh operators with Maciej Zworski. arXiv2003.08106 to appear in Proc. Lond. Math. Soc..
- Semiclassical resolvent bounds for weakly decaying potentials with Jacob Shapiro. arXiv2003.02525. To appear in Math. Res. Lett.
- Viscosity limits for 0th order pseudodifferential operators with Maciej Zworski. arXiv1912.09840, to appear in Comm. Pure Appl. Math..
- 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. Pure and Applied Analysis. 2(1): 157-202, 2020.
- On non-diffractive cones with Jared Wunsch. arXiv1807.05043 to appear in J. Differ. Geom.
- 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

- High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem. with Pierre Marchand, Euan Spence. arXiv2109.06017, to appear in Integral Equations and Operator Theory.
- Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators. with Euan Spence. 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. J. Funct. Anal. 269:2890-2926, 2015.

Quantum chaos

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

Pseudospectral effects in non-self-adjoint problems

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

Other Research

- 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.