Ayman Kachmar

Professor

Education Background

Ph.D. (University Paris XI)

Research Field
Mathematical analysis, spectral theory, and partial differential equations, with a focus on questions related to quantum mechanics and phase transitions in condensed matter physics
Academic Area
Mathematics and Applied Mathematics
Personal Website
Email
akachmar@cuhk.edu.cn
Biography

Ayman Kachmar is a professor at the Chinese University of Hong Kong, Shenzhen, starting on September 1, 2023. He obtained a Ph.D. in mathematics from Paris XI University, France; afterwards, he held post-doctoral positions in Paris, France and in Aarhus, Denmark. Before joining CUHKSZ, Ayman Kachmar was professor at the Lebanese University, Lebanon, and has held visiting professor positions in Lebanon, France, and Sweden. He published more than 50 papers in peer reviewed journals on various topics such as the magnetic and Robin Laplacians, counting eigenvalues, quantum tunneling, phase transitions in Ginzburg--Landau and Landau--deGennes theories.

Academic Publications

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