Interests

My main mathematical interests are in applications of harmonic analysis to partial differential equations and mathematical physics. I am particularly interested in oscillatory and singular integrals and integral operators that arise in the analysis of dispersive PDEs such as the Schrödinger, wave and Dirac equations.
Starting September 2017 my work will be supported by the Simons Foundation under grant #511825.

Links

Google Scholar Profile
MathSciNet (Subscription Required)
arXiv

Publications

  1. (with M.B. Erdogan) "Dispersive estimates for the Schrödinger equation for C$^{\frac{n-3}{2}}$ potentials in odd dimensions," International Mathematics Research Notices 2010:13, 2532-2565. [preprint], [Journal Version]
  2. "Dispersive estimates for the matrix and scalar Schrödinger equation in dimension five," Illinois J. Math. Volume 56, Number 2 (2012), 307-341. [preprint], [Journal Version].
  3. "Abel's theorem simplifies reduction of order." College Mathematics Journal, Vol. 42, No. 5, November 2011 , pp. 399-402(4). [preprint], [Journal Version].
  4. "Using Abel's theorem to explain repeated roots of the characteristic equation." CODEE Journal, July 2011. URL http://www.codee.org/ref/CJ11-0679.
  5. (with M.B. Erdogan) "Dispersive estimates for two-dimensional Schrödinger operators with obstructions at zero energy." Trans. Amer. Math. Soc. 365 (2013), 6403-6440. [preprint], [Journal Version].
  6. (with H.I. Dwyer) "Integrating factors and repeated roots of the characteristic equation." CODEE Journal, January 2012. URL http://www.codee.org/ref/CJ12-0930.
  7. (with M.B. Erdogan) "A weighted dispersive estimate for Schrödinger operators in dimension two." Comm. Math. Phys. 319 (2013), 791--811. [preprint]. [Journal Version].
  8. (with M.B. Erdogan) "Dispersive estimates for matrix Schrödinger operators in dimension two." DCDS-A. Pages: 4473 - 4495, Volume 33, Issue 10, October 2013. [preprint]. [Journal Version].
  9. (with Dirk Hundertmark) "Exponential decay for dispersion managed solitons for general dispersion profiles," Letters in Mathematical Physics, Volume 106, Issue 2, 221-249. [preprint]. [Journal Version].
  10. "Time decay estimates for the wave equation with potential in dimension two," Journal of Differential Equations 257 (2014) pp. 868-919. [preprint], [Journal Version]
  11. (with M.B. Erdogan and M.J. Goldberg) "Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy." Comm. PDE 39 (2014), no. 10, 1936-1964. [preprint], [Journal Version]
  12. (with M.J. Goldberg) "Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd dimensional case," J. Funct. Anal. 269 (2015) no. 3, 633-682. [preprint], [Journal Version]
  13. (with M.J. Goldberg) "Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues II: The even dimensional case," J. Spectr. Theory 7 (2017), 33-86. [preprint], [Journal Version].
  14. (with M.J. Goldberg) "The $L^p$ boundedness of wave operators for higher dimensional Schrödinger Operators with threshold singularities," Advances in Mathematics, 303 (2016), 360-389. [preprint], [Journal Version]
  15. (with E. Toprak) "Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy." Differential and Integral Equations, Volume 30, Number 5/6 (2017), 329-386. [preprint], [Journal Version].
  16. (with M.B. Erdogan) "The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions." Comm. Math. Phys. 352 (2017), no. 2, 719-757. [preprint], [Journal Version]
  17. (with M.J. Goldberg) "On the $L^p$ boundedness for four-dimensional wave operators for Schrödinger Operators with a threshold eigenvalue," Ann. Henri Poincaré 18 (2017), no.4, 1269-1288. [preprint], [Journal Version] , [Free Journal Version].
  18. (with M.B. Erdogan and E. Toprak) "Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies" [preprint], submitted, 36 pp.
  19. (with M.B. Erdogan and M.J. Goldberg) "On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger Operators with threshold obstructions," [preprint], submitted, 21pp.
  20. (with M.B. Erdogan and M.J. Goldberg) "Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions," [preprint], submitted, 26 pp.
  21. (with M.B. Erdogan and E. Toprak) "Dispersive estimates for massive Dirac operators in dimension two" [preprint], submitted, 40 pp.





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