The spectral properties of a singularly perturbed self-adjoint Landau Hamiltonian in the plane with a delta-potential supported on a finite curve are studied. After a general discussion of the qualitative spectral properties of the perturbed Landau Hamiltonian and its resolvent, one of our main objectives is a local spectral analysis near the Landau levels.
This talk is based on joint works with P. Exner, M. Holzmann, V. Lotoreichik, and G. Raikov.[-]

The development of quantum information processing and quantum computation goes hand in hand with the ability of addressing and manipulating quantum systems. Quantum Control Theory has provided a successful framework, both theoretical and experimental, to design and develop the control of such systems. In particular, for finite dimensional quantum systems or finite dimensional approximations to them. The theory for infinite dimensional systems is much less developed.
In this talk I propose a scheme of infinite dimensional quantum control on quantum graphs based on interacting with the system by changing the self-adjoint boundary conditions. I will show the existence of solutions of the time-dependent Schrödinger equation, the stability of the solutions and the (approximate) controllability of the state of a quantum system by modifying the boundary conditions on generic quantum graphs.[-]

In this (hopefully) blackboard talk, we will discuss the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, I will explain how to get a uniform description of the spectrum located between the Landau levels. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, I will explain how to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and also how to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.
Joint work with R. Fahs, L. Le Treust and S. Vu Ngoc.[-]

We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. In this talk, we deepen the connection between the eigenfunctions and this landscape function $u$ by proving that its reciprocal $1/u$ acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.[-]

The Laplace–Beltrami operator in the curved Möbius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are approximated well by an unconventional flat model whose spectrum can be computed explicitly in terms of Mathieu functions. Contrary to the traditional flat Möbius strip, our effective model contains a geometric potential. A comparison of the three models is made and analytical results are accompanied by numerical computations.[-]