scholarly journals Hardy Inequalities, Observability, and Control for the Wave and Schrödinger Equations with Singular Potentials

2009 ◽  
Vol 41 (4) ◽  
pp. 1508-1532 ◽  
Author(s):  
J. Vancostenoble ◽  
E. Zuazua
Keyword(s):  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Singular Potentials ◽  
Hardy Inequalities ◽  
And Control

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT UNBOUNDED SINGULAR POTENTIALS

2011 ◽  
Vol 23 (08) ◽  
pp. 823-838 ◽  
Author(s):  
KENJI YAJIMA
Keyword(s):  
Evolution Equations ◽  
Schrödinger Operators ◽  
Time Variable ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Singular Potentials ◽  
Spatial Infinity ◽  
Abstract Theorem ◽  
Magnetic Potentials

We consider time-dependent perturbations by unbounded potentials of Schrödinger operators with scalar and magnetic potentials which are almost critical for the selfadjointness. We show that the corresponding time-dependent Schrödinger equations generate a unique unitary propagator if perturbations of scalar and magnetic potentials are differentiable with respect to the time variable and they increase at the spatial infinity at most quadratically and at most linearly, respectively, where both have mild local singularities. We use time-dependent gauge transforms and apply Kato's abstract theorem on evolution equations.

On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains

2004 ◽  
Vol 56 (3) ◽  
pp. 655-672 ◽  
Author(s):  
Xiangxing Tao ◽  
Henggeng Wang
Keyword(s):  
Neumann Problem ◽  
Schrodinger Equations ◽  
Lipschitz Domains ◽  
Schrödinger Equations ◽  
Singular Potentials ◽  
Uniform Estimates ◽  
The Neumann Problem ◽  
Reverse Hölder ◽  
The Given ◽  
Nontangential Maximal Function

AbstractWe consider the Neumann problem for the Schrödinger equations –Δu + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class ℬn, in a connected Lipschitz domain Ω Rn. Given boundary data g in Hp or Lp for 1 – ɛ < p ≤ 2, where 0 < ɛ < , it is shown that there is a unique solution, u, that solves the Neumann problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂Ω). Moreover, the uniform estimates are found.

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