Vibrating quantum billiards on Riemannian manifolds

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical ($\hbar \longrightarrow 0$) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.Comment: 23 pages, 6 figures, a few typos corrected. To appear in International Journal of Bifurcation and Chaos (9/01

Ecuación de Helmholtz generalizada

In this paper we introduce the generalized Helmholtz equation and present explicit solutions to this generalized Helmholtz equation, these solutions depend on three holomorphic functions. As an application we present explicit solutions to the Helmholtz equation. We note that these solutions are not necessarily limited to certain domains of the complex plane C.En este artículo introducimos la ecuación de Helmholtz generalizada y presentamos soluciones explícitas para esta ecuación de Helmholtz generalizada, estas soluciones dependen de tres funciones holomorfas. Como aplicación presentamos soluciones explícitas para la ecuación de Helmholtz. Observamos que estas soluciones nonecesariamente estan limitadas a ciertos dominios del plano complejo C

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

We show that the non-linear semi-quantum Hamiltonians are the classical conjugated canonical variables) er commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.Facultad de Ciencias ExactasInstituto de Física La Plata (IFLP

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

We show that the non-linear semi-quantum Hamiltonians are the classical conjugated canonical variables) er commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.Facultad de Ciencias ExactasInstituto de Física La Plata (IFLP