Structure of uniform estimates in partial phase deformation

Author(s):  
V. Karpushkin
2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Li Chen ◽  
Jinyeop Lee ◽  
Matthew Liew

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.


Author(s):  
ALEXANDER BRUDNYI

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .


2007 ◽  
Vol 80 (2) ◽  
pp. 171-181 ◽  
Author(s):  
Zhengmin Zhang
Keyword(s):  

2014 ◽  
Vol 31 (5) ◽  
pp. 1517-1530
Author(s):  
Takefumi Kondo ◽  
Tetsu Toyoda

Author(s):  
Yusuke Ochiai ◽  
Takuya Obi ◽  
Yuuki Tsuruoka ◽  
Tokushi Kizuka

2018 ◽  
Vol 52 (2) ◽  
pp. 567-593 ◽  
Author(s):  
Li Chen ◽  
Simone Göttlich ◽  
Stephan Knapp

In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.


2014 ◽  
Vol 35 (2) ◽  
pp. 127-136 ◽  
Author(s):  
R. Mishra ◽  
S. Phapale ◽  
P. Samui ◽  
A. Nagaraj ◽  
S. R. Dharwadkar

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