From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the initial billiard table is that the corresponding billiard ball map or an iterate of it has a Kronecker invariant torus with a Diophantine frequency and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we obtain C^1 families of Kronecker tori with Diophantine frequencies. If the family of the Laplace-Beltrami operators satisfies the weak isospectral condition, we prove that the average action on the tori and the Birkhoff Normal Form of the billiard ball maps remain the same along the perturbation. As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions two and three. Applications are obtained also for strictly convex billiard tables of dimension two as well as in the case when the initial billiard table admits an elliptic periodic billiard trajectory. Spectral rigidity of billard tables close elliptical billiard tables is obtained. The results are based on a construction of C^1 families of quasi-modes associated with the Kronecker tori and on suitable KAM theorems for C^1 families of Hamiltonians.Comment: 170 pages; new results about the spectral rigidity of elliptical billiard tables; new Modified Iterative Lemma in the proof of KAM theorem with parameter

Atomic coherence and interference phenomena in resonant nonlinear optical interactions

Interference effects in quantum transitions, giving rise to amplification without inversion, optical transparency and to enhancements in nonlinear optical frequency conversions are considered. Review of the relevant early theoretical and experimental results is given. The role of relaxation processes, spontaneous cascade of polarizations, local field effects, Doppler-broadening, as well as specific features of the interference in the spectral continuum are discussed.Comment: 13 pages, 13 eps figures, review paper, Proceedings of the 15th International Conference on Nonlinear Optics - ICONO'9

Asymmetric tunneling, Andreev reflection and dynamic conductance spectra in strongly correlated metals

Landau Fermi liquid theory predicts that the differential conductivity between metallic point and metal is a symmetric function of voltage bias V. This symmetry holds if the particle-hole symmetry is preserved. We show that the situation can be different when one of the two metals is a strongly correlated one whose electronic system can be represented by a heavy fermion liquid. When the heavy fermion liquid undergoes fermion condensation quantum phase transition, the particle-hole symmetry is violated making both the differential tunneling conductivity and dynamic conductance asymmetric as a function of applied voltage. This asymmetry can be observed when the strongly correlated metal is either normal or superconducting. We show that at small values of $V the asymmetric part of the dynamic conductance is a linear function of V and inversely proportional to the maximum value of the gap and does not depend on temperature provided that metal is superconducting, when it becomes normal the asymmetric part diminishes at elevated temperatures.Comment: 8 pages, 7 figure

Finite linear groups, lattices, and products of elliptic curves

Let $V$ be a finite dimensional complex linear space and let $G$ be an irreducible finite subgroup of \GL(V). For a $G$-invariant lattice $\Lambda$ in $V$ of maximal rank, we give a description of structure of the complex torus $V/\Lambda$. In particular, we prove that for a wide class of groups, $V/\Lambda$ is isogenous to a self-product of an elliptic curve, and that in many cases $V/\Lambda$ is isomorphic to a product of mutually isogenous elliptic curves with complex multiplication. We show that there are $G$ and $\Lambda$ such that the complex torus $V/\Lambda$ is not an abelian variety but one can always replace $\Lambda$ by another $G$-invariant lattice $\Delta$ such that $V/\Delta$ is a product if elliptic curves with complex multiplication. We amplify these results with a criterion, in terms of the character and the Schur $\mathbf Q$-index of $G$-module $V$, of the existence of a nonzero $G$-invariant lattice in $V$.Comment: 25 pages. Several examples are adde

Enhanced four-wave mixing via elimination of inhomogeneous broadening by coherent driving of quantum transition with control fields

We show that atoms from wide velocity interval can be concurrently involved in Doppler-free two-photon resonant far from frequency degenerate four-wave mixing with the aid of auxiliary electromagnetic field. This gives rise to substantial enhancement of the output radiation generated in optically thick medium. Numerical illustrations addressed to typical experimental conditions are given.Comment: LaTeX2e, hyperref, 7 pages, 5 figures, to appear in PRA 1 august 200

The experimental study of the surface current excitation under the influence of a relativistic electron electromagnetic field

The problem of the surface current excitation in a conductive targets by a relativistic electron electric field as the origin of such radiation mechanisms as diffraction and transition radiation of relativistic electron was considered in frame of both surface current and pseudo-photon methods. The contradiction between these viewpoints in respect to the surface current on the target downstream surface necessitated the experimental test of this phenomenon. The test performed on electron beam of the 6 MeV microtron showed, that not any surface current is induced on the target downstream surface under the influence of a relativistic electron electromagnetic field in contrast to the upstream surface. This is important implication for the understanding of the forward transition and diffraction radiation nature.Comment: Was Presented at the symposium Channeling 2008 (Charged and Neutral Particles Channeling Phenomena), October 25 - November 1, 2008 Erice (Trapani - Sicily), Italy, and was accepted for publication in proceedings of symposiu