Journal ArticleDOI
Existence of solutions for Schrödinger evolution equations
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TLDR
In this paper, the authors studied the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrodinger equation, and provided sufficient conditions on V(t,x) such that the equation generates a unique unitary propagatorU( t,s) and such that U(t and s)uAbstract:
We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrodinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u
0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u
0eC
1(ℝ,L
2) ∩C
0(ℝH
2(ℝ
n
)) foru
0eH
2(ℝ
n
). The conditions are general enough to accommodate moving singularities of type ∣x∣−2+ɛ(n≧4) or ∣x∣−n/2+ɛ(n≧3).read more
Citations
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Journal ArticleDOI
Endpoint Strichartz estimates
Markus Keel,Terence Tao +1 more
TL;DR: In this paper, an abstract Strichartz estimate for the wave equation (in dimension n ≥ 4) and for the Schrodinger equation (n ≥ 3) was proved.
MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Journal ArticleDOI
The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s
Thierry Cazenave,F. B. Weissler +1 more
Journal Article
Oscillatory integrals and regularity of dispersive equations
TL;DR: In this article, the authors studied the relationship between local and global smoothing properties of dispersive equations and their application to nonlinear problems and their link with restriction theorems for the Fourier transform and pointwise convergence results.
Book ChapterDOI
Generalized Strichartz Inequalities for the Wave Equation
J. Ginibre,Giorgio Velo +1 more
TL;DR: Strichartz inequalities for the wave equation are estimates of the solution u of the Cauchy problem for that equation, in the form of space-time integral norms, in terms of similar norms of the inhomogeneity f and of suitable norm of the initial data.
References
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Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Journal ArticleDOI
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
TL;DR: In this paper, the authors give a complete solution when S is a quadratic surface given by the duality argument for the special case S {(x, y) yZ xz I} and give the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic invariant norm.
Book ChapterDOI
Wave Operators and Similarity for Some Non-selfadjoint Operators
TL;DR: In this paper, the authors developed a method for establishing the similarity of a perturbed operator T(ϰ), formally given by T + ϰV, to the unperturbed operator, T. It is basically a small perturbation theory, since the parameter ϰ is assumed to be sufficiently small.