Wilhelm Schlag

Research Papers

Nonlinear PDEs Linear PDEs Operator and spectral theory Harmonic and classical analysis Survey articles

Nonlinear PDEs

  1. Global dynamics of the nonradial energy-critical wave equation above the ground state energy
    with Joachim Krieger, Kenji Nakanishi, preprint 2011, to appear in Disc. Cont. Dyn. Sys.

  2. Scattering for wave maps exterior to a ball
    with Andrew Lawrie, preprint 2011, to appear in Advances Math.

  3. Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation with Kenji Nakanishi,
    to appear in SIAM Journal of Analysis

  4. Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation with Roland Donninger,
    Nonlinearity 24 (2011) 2547-2562.

  5. Global dynamics above the ground state energy for the one-dimensional NLKG equation (with Joachim Krieger and Kenji Nakanishi),
    has appeared in Math. Z.

  6. Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption (with Kenji Nakanishi),
    to appear in Arch. Rat. Mechanics and Analysis

  7. Global dynamics away from the ground state for the energy-critical nonlinear wave equation (with Joachim Krieger and Kenji Nakanishi),
    to appear in Amer. Journal of Math.

  8. Global dynamics above the ground state energy for the cubic NLS equation in 3D (with Kenji Nakanishi)
    To appear in Calc. of Variations and PDE

  9. Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation (with Kenji Nakanishi)
    Journal Diff. Eq. 250 (2011), 2299-2233.

  10. Concentration compactness for critical wave maps. (with Joachim Krieger)
    preprint 2009, to appear in the series ``Monographs'' by the Publishing House of the EMS.

  11. Renormalization and blow-up for the critical Yang-Mills problem. (with Joachim Krieger and Daniel Tataru)
    Adv. Math. 221 (2009), no. 5, 1445-1521.

  12. Slow blow up solutions for the H^1(R^3) critical focusing semi-linear wave equation (with Joachim Krieger and Daniel Tataru)
    Duke Math. J. 147 (2009), no. 1, 1-53.

  13. Renormalization and blow up for charge one equivariant critical wave maps (with Joachim Krieger and Daniel Tataru)
    Invent. Math. 171 (2008), no. 3, 543-615.

  14. Non-generic blow-up solutions for the critical focusing nonlinear Schrödinger equations in 1-d (with J. Krieger)
    J. Eur. Math. Soc. (JEMS) 11 (2009), no. 1, 1-125.

  15. On the focusing critical semi-linear wave equation (with J. Krieger)
    Amer. J. Math. 129 (2007), no. 3, 843-913.

  16. Asymptotic stability of N-soliton states of nonlinear Schrödinger equations (with Igor Rodnianski and Avy Soffer)
    preprint 2003.

  17. Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension (with Joachim Krieger)
    J. Amer. Math. Soc. 19 (2006), no. 4, 815-920.

  18. Stable manifolds for an orbitally unstable NLS
    Ann. of Math. (2) 169 (2009), no. 1, 139-227.

Linear PDEs

  1. On the spectral properties of L_{+-} in three dimensions
    with Ovidiu Costin, Min Huang, has appeared in Nonlinearity.

  2. Semiclassical low energy scattering for one--dimensional Schroedinger operators with exponentially decaying potentials
    with Ovidiu Costin, Roland Donninger, and Saleh Tanveer, has appeared in Annales Henri Poincare

  3. On pointwise decay for linear waves on a Schwarzschild black hole background (with Roland Donninger and Avy Soffer)
    To appear in Comm. Math. Physics

  4. Decay estimates for the one-dimensional wave equation with an inverse power potential (with Roland Donninger)
    Int. Math. Res. Not. IMRN 2010, no. 22, 4276-4300.

  5. A proof of Price's law on Schwarzschild black hole manifolds for all angular momenta (with Roland Donninger and Avy Soffer)
    Adv. Math. 226 (2011), no. 1, 484-540.

  6. Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials (with Ovidiu Costin, Wolfgang Staubach, and Saleh Tanveer)
    J. Funct. Anal. 255 (2008), no. 9, 2321-2362.

  7. Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II (with Avy Soffer and Wolfgang Staubach)
    Trans. Amer. Math. Soc. 362 (2010), no. 1, 289-318.

  8. Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I (with Avy Soffer and Wolfgang Staubach)
    Trans. Amer. Math. Soc. 362 (2010), no. 1, 19-52.

  9. Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions
    (with Burak Erdogan and Michael Goldberg)
    Forum Math. 21 (2009), no. 4, 687-722.

  10. Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R^3 (with Burak Erdogan and Michael Goldberg)
    J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 507-531

  11. Energy growth in Schrödinger's equation with Markovian forcing (with Burak Erdogan and Rowan Killip)
    Comm. Math. Phys. 240 (2003), no. 1-2, 1-29.

  12. A remark on Littlewood-Paley theory for the distorted Fourier transform
    Proc. Amer. Math. Soc. 135 (2007), no. 2, 437-451.

  13. Dispersive analysis of charge transfer models (with Igor Rodnianski and Avy Soffer)
    Comm. Pure Appl. Math. 58 (2005), no. 2, 149-216.

  14. A Limiting Absorption Principle for the three-dimensional Schrödinger equation with L^p potentials (with Michael Goldberg)
    Int. Math. Res. Not. 2004, no. 75, 4049-4071.

  15. Dispersive estimates for Schrödinger operators in dimensions one and three (with Michael Goldberg)
    Comm. Math. Phys. vol. 251, no. 1 (2004), 157-178.

  16. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials (with Igor Rodnianski)
    Invent. Math. 155 (2004), no. 3, 451-513.

  17. Dispersive estimates for Schrodinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II
    (with M Burak Erdogan)
    J. Anal. Math. 99 (2006), 199-248.

  18. Dispersive estimates for Schrodinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I
    (with Burak Erdogan)
    Dyn. Partial Differ. Equ. 1 (2004), no. 4, 359-379.

  19. Agmon-Kato-Kuroda theorems for a large class of perturbations (with Alexander Ionescu)
    Duke Math. J. 131 (2006), no. 3, 397-440.

  20. Dispersive estimates for Schrödinger operators in dimension two
    Comm. Math. Phys. 257 (2005), no. 1, 87-117.

  21. Energy growth in Schroedinger's equation with Markovian forcing.
    (with Burak Erdogan and Rowan Killip)
    Comm. Math. Phys. 240 (2003), no. 1-2, 1-29.

  22. Schauder and Lp estimates for parabolic systems via Campanato spaces
    Comm. PDE 21, no. 7 and 8, 1141-1175 (1994).


Operator and spectral theory

  1. Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation (with L. Demanet)
    Nonlinearity 19 (2006), no. 4, 829-852.

  2. On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations (with Michael Goldstein)
    Ann. of Math. (2) 173 (2011), no. 1, 337-475.

  3. On Schrödinger operators with dynamically defined potentials (with Michael Goldstein)
    Mosc. Math. J. 5 (2005), no. 3, 577-612, 742-743.

  4. On non-perturbative Anderson localization for C^alpha potentials generated by shifts and skew-shifts (with J. Chan and M. Goldstein)
    preprint 2006.

  5. Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues (with Michael Goldstein)
    Geom. Funct. Anal. 18 (2008), no. 3, 755-869

  6. On the integrated density of states for Schrödinger operators on Z^2 with quasi periodic potential
    Comm. Math. Phys. 223 (2001), no. 1, 47-65.

  7. Anderson localization for Schrödinger operators on Z with potentials given by the skew-shift (with J. Bourgain and M. Goldstein)
    Comm. Math. Phys. 220 (2001), no. 3, 583-621.

  8. Holder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions
    (with M. Goldstein)
    Ann. of Math. (2) 154 (2001), no. 1, 155-203.

  9. Anderson localization for Schrödinger operators on Z with strongly mixing potentials (with J. Bourgain)
    Comm. Math. Phys. 215 (2000), 143-175.

  10. Classical and quantum scattering for a class of long range random potentials (with Igor Rodnianski)
    Int. Math. Res. Notices (2003), no. 5, 243-300.

  11. Frequency concentration and localization lengths for the Anderson model at small disorders (with C. Shubin and T. Wolff)
    Journal d'analyse math. 88 (2002), 173-220.

  12. Anderson localization for Schrödinger operators on Z^2 with quasi-periodic potential (with J. Bourgain and M. Goldstein)
    Acta Mathematica 188 (2002), no. 1, 41-86.

Harmonic and classical analysis

  1. Two Erdoes problems on lacunary sequences: chromatic number and Diophantine approximation (with Yuval Peres)
    Bull. Lond. Math. Soc. 42 (2010), no. 2, 295-300.

  2. On the Hardy-Littlewood majorant problem for random sets (with Gerd Mockenhaupt)
    J. Func. Analysis 256 (2009), no. 4, 1189-1237.

  3. On continuum incidence problems related to harmonic analysis
    J. Func. Analysis 201 (2003), no. 2, 480-521.

  4. Bernoulli convolutions and an intermediate value theorem for entropies of K-partitions (with E. Lindenstrauss and Y. Peres)
    Journal d'analyse math. 87 (2002), 337-367.

  5. On minima of the absolute value of certain random exponential sums
    Amer. J. Math. 122, 483-514 (2000)

  6. On uniformly distribution dilates of finite integer sequences (with S.V. Konyagin and I. Ruzsa)
    J. Number Theory 82, no. 2, 165-187 (2000)

  7. Smoothness of projections, Bernoulli convolutions, and dimensions of exceptions (with Yuval Peres)
    Duke Math. J. 102, no. 2, 193-251 (2000)

  8. A geometric inequality with applications to the Kakeya problem in three dimensions
    Geom. Funct. Anal. 8, no. 3, 606-625 (1998).

  9. Lower bounds of the absolute value of random polynomials on a neighborhood of the unit circle (with S. Konyagin)
    Trans. Amer. Math. Soc. 351, no. 12, 4963-4980 (1999)

  10. A geometric proof of the circular maximal theorem
    Duke Math. J. 93, no. 3, 505-533 (1998)

  11. Local smoothing estimates related to the circular maximal theorem (with C. Sogge)
    Math. Res. Letters 4 (1997), 1-15, (1996)

  12. A generalization of Bourgain's circular maximal theorem
    J. Amer. Math. Soc. 10, no. 1, 103-122 (1997).

  13. Lp to Lq estimates for the circular maximal function
    Ph.D. Thesis, California Institute of Technology, (1996).
    Has also appeared as part of a book: Topics in Analysis and Applications, Selected Theses, World Scientific Press, 2000.

Survey articles

  1. Dispersive estimates for Schrödinger operators: a survey
    Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007

  2. Spectral theory and nonlinear PDE: a survey
    Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 703-723.

  3. On the formation of gaps in the spectrum of Schrödinger operators with quasi-periodic potentials (with Michael Goldstein)
    Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon's 60th birthday, 591-611, Proc.
    Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.

  4. On discrete Schrödinger operators with stochastic potentials
    Proc. of the XIVth Int. Cong. on Math. Phys., 206-215, World Sci. Publ., Hackensack, NJ, 2005.

  5. Sixty years of Bernoulli convolutions (with Y. Peres and B. Solomyak)
    in Fractals and Stochastics, II, Proceedings of the Greifswald 1998 Conference, (ed. Bandt, Graf, and Zähle), 1999.