Correlations of chaotic eigenfunctions: a semiclassical analysis

Strong-field photoelectron holography denotes the interference of various electron paths in laser-induced ionization of atoms, leading to interference patterns in the final momentum distribution. For a quantitative description of holography beyond the electric dipole approximation and in the presence of the Coulomb potential, we develop a semiclassical model in which the initial conditions of outgoing electrons are set according to the beyond-dipole strong-field approximation for the ionization step. The phases associated with trajectories are evaluated following the prescription for semiclassical propagators. The validity of the method is confirmed by comparison to the numerical solution of the time-dependent Schr\"odinger equation in two spatial dimensions. The semiclassical model reproduces correctly the backward and forward shifts of the photoelectron momenta along the laser propagation axis that arise from beyond-dipole dynamics. The position of the central holographic interference fringe can be estimated already from a simplified Coulomb-free interference model that provides closed-form expressions for the beyond-dipole shifts. In three dimensions, Coulomb focusing causes a breakdown of the semiclassical model for final momenta with directions close to the polarization axis. We implement a beyond-dipole regularization procedure based on the concept of glory scattering, which was recently used to describe Coulomb focusing in the dipole approximation. While the position of the central maximum and higher-order fringes can already be obtained approximately by simpler semiclassical modeling, this glory model is able to predict the shape of the distribution in the close vicinity of the central maximum. The violation of the dipole approximation in holography should be observable with mid-infrared fields, for which the forward/backward shifts can be comparable with the fringe spacing.

There are a number of excellent texts available on semi-classical analysis. The focus of the present monograph, however, is an aspect of the subject somewhat less systematically developed in other texts: In semi-classical analysis, many of the basic results involve asymptotic expansions in which the terms can be computed by symbolic techniques, and the focus of this monograph is the symbol created thus. In particular, the techniques involved in this symbolic calculus have their origins in symplectic geometry, and the first seven chapters of the present work are a discussion of this underlying symplectic geometry. Another feature which differentiates this monograph from other texts is an emphasis on the global aspects of the subject: A considerable amount of time is spent here showing that the objects studied are coordinate-invariant and hence make sense on manifolds. Wherever possible, intrinsic coordinate-free descriptions of these objects are given. Topics discussed include wave and heat trace formulas for globally defined semi-classical differential operators on manifolds, and equivariant versions of these results involving Lie group actions.

We investigate the internal space of Bessel functions which is associated to thegroup Z of positive and negative integers defining their orders. As a result wepropose and prove a new unifying formula (to be added to the huge literatureon Bessel functions) generating Bessel functions of real orders out of integerorder ones. The unifying formula is expected to be of great use in appliedmathematics. Some applications of the formula are given for illustration

: The report contains tables of the first five roots of the following transcendental equations: (a) J0(alpha) Y0(alpha) = Y0(alpha) J0(alpha); (b) J1(alpha) Y1(alpha) = Y1(alpha) J1(alpha); (c) J0(alpha) Y1(alpha) = Y0(alpha) J1(alpha) where J0(alpha), Y0(alpha), J1(alpha), Y1(alpha) are Bessel functions of order 0 and 1 respectively. In these equations, alpha is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. Additional tables are included listing an auxiliary quantity gamma which is better suited to interpolation particularly when k is close to unity. (Author)

Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the ‘small values regime’ and the ‘asymptotic’ one, and covering, in this way, an area of great physical interest. The examples that illustrate our approach are accessible to the undergraduate student.

(1) What is semiclassical Analysis? (2) What is a Trace Formula? (3) Physical Motivations : “Quantum Chaos” (4) Gutzwiller Trace Formula. (Rigorous Approach)

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

TRAITEMENT AUTOMATIQUE DES SONDAGES ELECTRIQUES AUTOMATIC PROCESSING OF ELECTRICAL SOUNDINGS*

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,>4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.

From Mie's classical theory of the action of small spherical particles on plane waves of light, the expression giving the loss of light due to absorption and scattering is reduced to the formula involving only Bessel functions of orders given by half integral values. The result is used for calculating the absorption by small carbon particles whose diameter is comparable with the wave-length of the incident light, particles that can be measured only by interference methods. When the diameter is less than 0.2 μ the coefficient of absorption decreases toward the red end of the spectrum. The reverse is true for 0.3 and 0.4 μ particles.

We have found new exact solutions of the coupled nonlinear Schroedinger equation of Benney-Roskes which is written as i u t -β u x x +γ u y y +δ u * u u -2 w u =0, β w x x +γ w y y -βδ( u * u ) x x =0. Three types of new solutions have been obtained. They are described by the Hermite polynomials, the Bessel function of order ±1/4 and ±1/3 (Airy function). Physically these solutions represent exploding and decaying solitons.

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schr\"odinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.

Resonance-enhanced multiphoton ionization process of the argon atom by an 800-nm 30-fs linearly polarized laser field is investigated at intensities range from 1.1 to $4.55\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$. At $4.55\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$ intensity the experimental photoelectron energy spectrum is in a good agreement with the time-dependent Schr\"odinger equation (TDSE) calculation where the double structure originating from dressed $4p\ensuremath{-}4d$ coupled transition is clearly identified. At lower intensity of $1.1\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$, the resonant ionization process via the $4f$ state is observed, however, the expected peak (jet) at ${90}^{\ensuremath{\circ}}$ in the photoelectron angular distribution from the zeroth order above threshold ionization, has vanished completely. Such behavior is attributed to the destructive interference phenomenon in the coherent contributions of different partial waves of the photoelectron, namely, $\ensuremath{\epsilon}d$ and $\ensuremath{\epsilon}g$ states, and has been confirmed in our TDSE calculations.

Evaluation of the Modified Bessel Function of the First Kind and Zeroth Order