Catálogo de publicaciones - libros

QMath9 gave a particular importance to summarize the state of the art of the field in a perspective to transmit knowledge to younger scientists. The main contributors to the field were gathered in order to communicate results, open questions and motivate new research by the confrontation of different view points. The edition of this book follows this spirit; the main effort of the authors and editors is to help finding an access to the variety of themes and the sometimes very sophisticated literature in Mathematical Physics.

We discuss the recent proof of Cantor spectrum for the almost Mathieu operator for all conjectured values of the parameters.

QMath9 gave a particular importance to summarize the state of the art of the field in a perspective to transmit knowledge to younger scientists. The main contributors to the field were gathered in order to communicate results, open questions and motivate new research by the confrontation of different view points. The edition of this book follows this spirit; the main effort of the authors and editors is to help finding an access to the variety of themes and the sometimes very sophisticated literature in Mathematical Physics.

We use recent results on precise coupling terms in the optimal superadiabatic basis in order to determine exponentially small transition probabilities in the adiabatic limit of time-dependent two-level systems. As examples, we discuss the Landau-Zener and the Rosen-Zener models.

We have developed a scattering matrix approach to coherent transport through an adiabatically driven conductor based on photon-assisted processes. To describe the energy exchange with the pumping filds we expand the Floquet scattering matrix up to linear order in driving frequency.

We give a short presentation of two recent results. The first one is a rigorous proof of the Landauer-Büttiker formula, and the second one concerns resonant quantum transport. The detailed results are in [2]. In the last section we present the results of some numerical computations on a model system.

We will discuss a new type of an isoperimetric problem concerning a Hamiltonian with point interactions in ℝ, = 2, 3, all with the same coupling constant, placed at vertices of an equilateral polygon . We show that the ground state energy is locally maximized by a regular polygon and conjecture that the maximum is global; on the way we encounter an interesting geometric inequality. We will also mention some extensions of this problem.

This lecture concerns limit cycles in renormalization group (RG) behavior of quantum Hamiltonians. Cyclic behavior is perhaps more common in quantum mechanics than the fixed-point behavior which is well-known from critical phenomena in classical statistical mechanics. We discuss a simple Hamiltonian model that exhibits limit cycle behavior.

We present some results concerning the Cantor structure of the spectrum of quasi-periodic Schrödinger operators. These are obtained studying the dynamics of the corresponding eigenvalue equations, specially the notion of reducibility and Floquet theory. We will deal with the Almost Mathieu case, and the solution of the “Ten Martini Problem” for Diophantine frequencies, as well as other models.

Isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing isothermal processes as processes (“”) is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined.