"Magnetic Edge and Semiclassical Eigenvalue Asymptotics" by Dr. Ayman Kachmar скачать в хорошем качестве

"Magnetic Edge and Semiclassical Eigenvalue Asymptotics" by Dr. Ayman Kachmar

What will be the energy levels of an electron moving in a magnetic field? In a typical setting, these are eigenvalues of a special magnetic Laplace operator involving the semiclassical parameter (a very small parameter compared to the sample’s scale), and the foregoing question becomes on searching asymptotic expansions of the eigenvalues in the singular limit of vanishing semiclassical parameter. This talk will discuss some answers to these questions including a joint work with W. Assaad and B. Helffer on magnetic fields varying on very short scales (such a magnetic field creates an edge separating two uniform but distinct values of the magnetic field's intensity). The main result is an accurate eigenvalue asymptotics displaying the splitting between the consecutive eigenvalues and featuring the geometry of the edge. One notable aspect is a purely variational proof not involving pseudo-differential calculus.