My research focuses on the rigorous investigation of questions related to quantum mechanics, superconductivity and liquid crystals. I specialize in the areas of semi-classical analysis, magnetic Laplacian, Ginzburg-Landau functional and spectral iso-perimetric inequalities.
June 6, 2007: Ph.D. in Mathematics, Université Paris XI, France. Advisor: Bernard Helffer.
Employment:
September 1, 2007: ATER (Teaching and research associate), Université Paris XI, France.
September 1, 2008: Post-doc, University of Århus, Denmark. Advisor: Søren Fournais.
September 1, 2010: Lebanese University, Lebanon.
September 1, 2023: The Chinese University of Hong Kong, Shenzhen, China.
Service:
October 1, 2017 - September 30, 2019: Chairperson of the mathematics department, Faculty of Sciences Section 5, Lebanese University, Lebanon.
February 2019: Search committee, Faculty of Sciences, Lebanese University, Lebanon.
October 1, 2020 - August 31, 2023: Steering committee, Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Lebanon.
Events:
Organizer (with W. Assaad) of the Thematic Program in Mathematical Physics, American University of Beirut, Lebanon. Nov 2020 - March 2021.
Organizer (with W. Assaad and M. Correggi) Mathematics of Condensed Matter and Beyond (online conference), American University of Beirut. Feb 22-25, 2021.
Ph.D. students:
R. Fahs, University of Angers (defended July 2023). Co-advised with N. Raymond .
W. Assad, Lund University (defended September 2019). Co-advised with M. P.-Sundqvist.
K. Attar, Lebanese University and Université Paris XI (defended June 2015). Co-advised with B. Helffer.
M. Nasrallah. Lebanese University and University of Århus (defended November 2013). Co-advised with Søren Fournais.
Spectral Geometry:
A. Kachmar, V. Lotoreichik. A geometric bound on the lowest magnetic Neumann eigenvalue via the torsion function. SIAM J. Math. Anal. Vol. 56, No. 4, 5723--574 (2024).
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).
G. Habib, A. Kachmar. Eigenvalue bounds of the Robin Laplacian with magnetic field. Arch. Math. Vol. 110, 501--513 (2018).
Counting Eigenvalues:
S. Fournais, A. Kachmar. On the energy of bound states for magnetic Schrödinger operators. J. London Math. Soc. Vol. 80 (1) 233-255 (2009).
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).
S. Fournais, R.L. Frank, M. Goffeng, A. Kachmar, M. Sundqvist. Counting Negative Eigenvalues for the Magnetic Pauli Operator. Duke Math. J. Vol 174, No. 2, 313-353 (2025).
Quantum Tunneling:
B. Helffer, A. Kachmar. Quantum tunneling in deep potential wells and strong magnetic field revisited. Pure Appl. Anal. Vol. 6, No. 2, 319-352 (2024).
B. Helffer; A. Kachmar; M. Sundqvist. Flux and symmetry effects on quantum tunneling. Math. Ann. 390, 5185-5234 (2024).
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).
Spectral Asymptotics:
W. Assaad, A. Kachmar, B. Helffer. Semi-classical eigenvalue estimates under magnetic steps. Anal. PDE, Vol. 17, No. 2, 535-585 (2024).
B. Helffer, S. Fournais, A. Kachmar, N. Raymond. Effective operators on an attractive magnetic edge. J. Éc. Polytech., Math. Vol 10, 917-944 (2023).
B. Helffer, A. Kachmar. Eigenvalues for the Robin Laplacian in domains with variable curvature.
Tran. Amer. Math. Soc. Vol. 369 (5) 3253-3287 (2017).
V. Bonnaillie-Noël, S. Fournais, A. Kachmar, N. Raymond. Discrete spectrum for the magnetic Laplacian on perturbed half-spaces. Bull. Lond. Math. Soc. Vol. 56, No. 7, 2529--2551 (2024).
Phase transitions:
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).
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:
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).
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:
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).
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).
S. Fournais, A. Kachmar, X.B. Pan. Existence of surface smectic states in liquid crystals. J. Funct. Anal. 274, 900-958 (2018).
Graduate Course:
MEDP514 (28 hours, Lebanese University) Spectral Theory and Applications to PDE.
(Master 2) Lectures on the following topics: Unbounded operators, Kato-Rellich Theorem, Friedrichs' (Extension) Theorem, Spectrum, Compact Resolvent, Min-Max principle, Weyl's Theorem, Harmonic Oscillator, Harmonic Approximation.
Undergraduate courses:
American University of Beirut
Math201 (33 hours, fall 2025)
Sequences, Series, Taylor's Theorem, Functions of Several Varaibles, Optimization, Double and Triple Integrals
The Chinese University of Hong Kong, Shenzhen
MAT2050: Mathematical Analysis (numerical sequences and series, continuity and differentiability, point-set topology, sequences and series of functions, Riemann integrable functions, functions of several variables and vector-valued functions).
MAT2040: Linear Algebra (matrices, vector spaces, linear transformations, diagonalization and the spectral theorem).
MAT1001: Calculus I (functions, limits, continuity, derivatives, extrema of functions, indefinite and definite integrals, usual functions).
Lebanese University
M3301 (60 hours) & M3306 (30 hours)
Introduction to measure theory, Lebesgue integral, convergence theorems, product measures, Fubini-Tonelli Theorem, convolution product and Fourier transform in R.
M3311 (30 hours):
Introduction to Bounded Operators: Adjoints, Compact Operators and Ascoli's Theorem.
M3320 (30 hours):
Introduction to PDE: Separation of Variables, Integral Methods, Heat, Wave and Laplace equations, Uniqueness of solution, harmonic functions.
M2206 (60 hours):
Normed Vector spaces, Riesz Compactness Theorem, bounded operators, Hilbert spaces, projection Theorem, Riesz representation theorem, Hilbert basis, Fourier Series.
American University of Beirut
Math202 (33 hours):
Line integrals, vector fields, Green's theorem, surface integrals, divergence and Stoke's theorems, 1st order ODE, 2nd order ODE (reduction of order, characteristic equations, undetermined coefficients, etc), power series solutions of ODE, systems of ODE.
Visiting Professor, Lund University, Sweden (6 months: September 1, 2022 -- February 28, 2023).
Visiting Professor, University of Angers, France (1 month: January 16 -- February 17, 2022).
9th European Congress of Mathematics (Mini-Symposium: Spectral Theory of Differential Operators in Mathematical Physics, July 15-19, 2024)
Nantes Université, France. (45 days: November 1, 2021 -- December 15, 2021).
Politecnico di Milano, Italy. Three Months of Quantum Mathematics in Milano. (May 16-27, 2022).
Institute Mittag-Leffler, Sweden. Spectral Methods in Mathematical Physics. (Two weeks: Jan 21-Feb 02, 2019).
New York University at ECNU Shanghai, China. (Two weeks: November 5-15, 2018).
Banff International Research Station, Canada. Phase transition models. (One week: May 01-05, 2017).