Ayman Kachmar

教授

校长学者
教育背景

博士(巴黎第十一大学)

研究领域
数学分析、谱论和偏微分方程,重点关注与量子力学和凝聚态物理中的相变问题相关的领域
学术领域
数学与应用数学
个人网站
电子邮件
akachmar@cuhk.edu.cn
个人简介

Ayman Kachmar教授自2023年9月1日起担任香港中文大学(深圳)理工学院教授。他在法国巴黎第十一大学获得了数学博士学位;之后在法国巴黎和丹麦奥胡斯担任博士后。在加入香港中文大学(深圳)之前,Ayman Kachmar担任黎巴嫩大学教授,并曾在黎巴嫩、法国和瑞典担任客座教授。他研究磁Laplace算子和Robin-Laplace算子及其特征值、量子隧道效应、Ginzburg-Landau 理论及Landau-de Gennes理论中的相变问题等,在同行评审期刊上发表了50多篇论文。

学术著作

Magnetic Laplacian:

(Pure magnetic iso-perimetric inequality):

 

A. Kachmar, V. Lotoreichik. On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter. J. Geometric Anal. Vol. 32, No. 6, Paper No. 182, 20 pp. (2022).

(Spectral asymptotics, semi-classical regime):

W. Assaad, A. Kachmar, B. Helffer. Semi-classical eigenvalue estimates under magnetic steps. To appear in Analysis & PDE

B. Helffer, S. Fournais, A. Kachmar, N. Raymond. Effective operators on an attractive magnetic edge. J. Éc. Polytech., Math. Vol. 10, 917--944, 2023. 

(Sum of eigenvalues, semi-classical regime):

 

S. Fournais. A. Kachmar. On the energy of bound states for magnetic Schrödinger operators. J. London Math. Soc. Vol. 80 (1) 233-255 (2009).

(Accumulation of eigenvalues below the essential spectrum):

M. Goffeng, A. Kachmar, M. P. Sundqvist. Clusters of eigenvalues of the magnetic Laplacian with Robin condition. J. Math. Phys. Vol. 57 (6) article number 063510 (2016).

Robin Laplacian (strong coupling regime):

B. Helffer, A. Kachmar. Eigenvalues for the Robin Laplacian in domains with variable curvature. Transactions of AMS, Vol. 369 (5) 3253-3287 (2017).

B. Helffer, A. Kachmar, N. Raymond. Tunneling for the Robin Laplacian in smooth planar domains. Commmun. Contemp. Math. Vol. 19 (1) 1650030, 38 pp. (2017).

Phase transitions:

(Proof of oscillations generated by Aharononv-Bohm potential, non-simple connectivity of domains and Robin condition; results consistent with the Little-Parks experiment).

 

A. Kachmar, X.B. Pan. Oscillatory patterns in the Ginzburg-Landau model driven by the  Aharonov-Bohm potential. J. Funct. Anal. Vol. 279, No. 10, Article ID 108718, 37 p. (2020).


B. Helffer, A. Kachmar. Thin domain limit and counterexamples to strong diamagnetism.  Rev. Math. Phys. Vol. 33, No. 2, Article ID 2150003, 35 p. (2021).


A. Kachmar, M. P.-Sundqvist. Counterexample to strong diamagnetism for the magnetic Robin Laplacian. Mathematical Physics, Analysis and Geometry. Vol. 23, art. no. 27 (2020).


(Transition from surface to bulk superconductivity; Abrikosov lattices)


S. Fournais, A. Kachmar. Nucleation of bulk superconductivity close to critical magnetic field.  Advances in Mathematics. 226 1213 - 1258 (2011).

Non-linear effective models, density of superconductivity:


(Distribution of density/order parameter)


A. Kachmar. The Ginzburg-Landau order parameter near the second critical field.  SIAM J. Math. Anal. 46 (1) 572-587 (2014).


B. Helffer, A. Kachmar. The density of superconductivity in the bulk regime. Indiana Univ. Math. J. 67 (6) pp. 2181-2198 (2018).


(New effective models and applications)


B. Helffer, A. Kachmar. The Ginzburg-Landau functional with vanishing magnetic field. Arch.  Rational Mech.  Anal. 218 (1) 55-122 (2015).

B. Helffer, A. Kachmar. From constant to non-degenerately vanishing magnetic fields in superconductivity. Annales de l'Institut Henri Poincaré - Analyse non-linèaire Vol. 34, 423-438 (2017). 
 

3D Ginzburg-Landau functional:

(Vortex energy in 3D)

A. Kachmar. The ground state energy of the three dimensional Ginzburg-Landau model in the mixed phase.  J. Funct. Anal. 261 (11) 3328 - 3344 (2011).


(Surface superconductivity in 3D; Abrikosov energy)


A. Kachmar, S. Fournais, M. Persson The ground state energy of the three dimensional Ginzburg-Landau functional. Part II: Surface regime. J. Math. Pures Appl. 99 343-374 (2013).

Landau-DeGennes model (liquid crystals):

S. Fournais, A. Kachmar, X.B. Pan. Existence of surface smectic states in liquid crystals. J. Funct. Anal. 274, 900-958 (2018).

Selected recent works:

Magnetic Laplacian in a sector:

(Proof of existence of discrete spectrum in a sector with Neumann boundary condition and under constant magnetic field).


V. Bonnaillie-Noël, S. Fournais, A. Kachmar, N. Raymond. Discrete spectrum for the magnetic Laplacian on perturbed half-spaces. arXiv:2208.13646

Electro-magnetic quantum tunneling:

(Accurate calculation of quantum tunneling under constant magnetic field and radial symmetric potentials)

 

B. Helffer, A. Kachmar. Quantum tunneling in deep potential wells and strong magnetic field revisited. arXiv:2208.13030