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 equation.
Publications:
 (with M.B. Erdogan)
"Dispersive estimates for the Schrödinger equation for
C$^{\frac{n3}{2}}$ potentials in odd dimensions,"
International Mathematics
Research Notices
2010:13, 25322565.
[preprint], [Journal
Version]

"Dispersive estimates for the matrix and scalar Schrödinger equation in
dimension five," Illinois J. Math. Volume 56, Number 2 (2012), 307341.
[preprint],
[Journal Version].

"Abel's theorem simplifies reduction of order."
College Mathematics Journal,
Vol. 42, No. 5, November 2011 , pp. 399402(4).
[preprint], [Journal
Version].

"Using Abel's theorem to explain repeated roots
of the characteristic equation." CODEE Journal, July 2011.
URL http://www.codee.org/ref/CJ110679.
 (with M.B. Erdogan)
"Dispersive estimates for twodimensional Schrödinger operators with
obstructions at zero energy."
Trans. Amer. Math. Soc. 365 (2013), 64036440.
[preprint], [Journal Version].

(with H.I. Dwyer)
"Integrating factors and repeated roots of the characteristic equation."
CODEE Journal, January 2012. URL http://www.codee.org/ref/CJ120930.

(with M.B. Erdogan)
"A weighted dispersive estimate for Schrödinger
operators in dimension two."
Comm. Math. Phys. 319 (2013), 791811.
[preprint]. [Journal Version].

(with M.B. Erdogan)
"Dispersive estimates for matrix Schrödinger
operators in dimension two."
DCDSA. Pages: 4473  4495, Volume 33, Issue 10, October 2013.
[preprint]. [Journal Version].

(with Dirk
Hundertmark)
"Exponential decay for dispersion managed solitons for general dispersion profiles,"
Letters in Mathematical Physics, Volume 106, Issue 2, 221249.
[preprint]. [Journal Version].

"Time decay estimates for the wave equation with potential in dimension two,"
Journal of Differential Equations 257 (2014)
pp. 868919.
[preprint],
[Journal Version]

(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, 19361964.
[preprint],
[Journal Version]

(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, 633682. [preprint],
[Journal Version]

(with M.J. Goldberg)
"Dispersive Estimates for higher
dimensional Schrödinger Operators with threshold
eigenvalues II: The even dimensional case," to appear in
J. Spectral Theory, 48 pp. [preprint].

(with M.J. Goldberg)
"The $L^p$ boundedness of wave operators for higher
dimensional Schrödinger Operators with threshold
singularities," Advances in Mathematics, 303 (2016), 360389.
[preprint], [Journal Version]

(with E. Toprak)
"Decay estimates for four dimensional Schrödinger, KleinGordon and wave equations with
obstructions at zero energy."
[preprint], submitted, 53 pp.

(with M.B. Erdogan)
"The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions"
[preprint], submitted, 38 pp.

(with M.J. Goldberg)
"On the $L^p$ boundedness for fourdimensional wave operators for Schrödinger Operators with a threshold
eigenvalue," [preprint], submitted, 20 pp.
William Green's Homepage