89 research outputs found
A Spectral Multiplier Theorem associated with a Schr\"odinger Operator
We establish a spectral multiplier theorem associated with a Schr\"odinger
operator H=-\Delta+V(x) in \mathbb{R}^3. We present a new approach employing
the Born series expansion for the resolvent. This approach provides an explicit
integral representation for the difference between a spectral multiplier and a
Fourier multiplier, and it allows us to treat a large class of Schr\"odinger
operators without Gaussian heat kernel estimates. As an application to
nonlinear PDEs, we show the local-in-time well-posedness of a 3d quintic
nonlinear Schr\"odinger equation with a potential
A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations
We consider the one-dimensional cubic fractional nonlinear Schr\"odinger
equation where and the operator is the fractional Laplacian of
symbol . Despite of lack of any Galilean-type invariance, we
construct a new class of traveling soliton solutions of the form
by a rather involved variational argument
Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity
In this paper, we establish the unconditional uniqueness of solutions to the
cubic Gross-Pitaevskii hierarchy on in a low regularity Sobolev
type space. More precisely, we reduce the regularity down to the currently
known regularity requirement for unconditional uniqueness of solutions to the
cubic nonlinear Schr\"odinger equation ( if and
if ). In such a way, we extend the recent work of
Chen-Hainzl-Pavlovi\'c-Seiringer.Comment: 26 pages, 1 figur
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