Variational Approach to Differential Invariants of Rank 2 Vector Distributions

In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his "reduction-prolongation" procedure. After Cartan's work the following questions remained open: first the geometric reason for existence of Cartan's tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan's tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4. For n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan's tensor.Comment: 37 pages, SISSA preprint, 12/2004/M, February 2004, minor corrections of misprint

Transmission Resonances Induced by Time-Periodic Driving of a Quantum Well with a V-Shaped Bottom

How the scattering dynamics of a quantum system is affected by an application of a time-periodic driving field, such as coherent electromagnetic radiation, has been a subject of increasing importance in the past three decades. Time-periodic fields can profoundly alter the dynamics of matter in ways that are relevant to the design of semiconductor structures, such as quantum dots and superlattices, that have possible applications to quantum computation and quantum information processing. This thesis is a theoretical treatment of a localized one-dimensional quantum system that is subject to an external driving field, such as an oscillating electric field of a laser or of some microwave radiation, that oscillates periodically in time. When not subject to the driving field, the system is characterized by a potential energy function that is a finite one-dimensional square well (i.e. a 1D quantum potential well with finite depth and a flat bottom. Also, the zero level of potential energy is set such that the potential in the well’s exterior is zero). To this well, we specifically introduce a driving field that causes the well’s bottom to bend from a flat shape to a V-shape, such that the potential energy as a function of position is held fixed at the endpoints of the well’s region of space and forms the spike of the letter “V” at the midpoint of the well. Our chosen external field makes the V-shaped bottom perpetually bend up and down, causing the potential energy in the region of the well to vary periodically in time. In this study, we analyze how a plane wave that propagates toward the region of our well and is with a fixed incident energy not only scatters into outgoing plane waves with the same energy — propagating in both directions of one-dimensional space — but is also induced by the driving field to have nonzero probabilities of transition to infinitely many other states with different energies. Due to the time-periodicity of our potential, an incoming wave-particle (i.e. a matter wave) of a given energy can only access energies that differ by integer multiples of ħω from its original energy, where ω is the angular frequency of the oscillation of the potential. In the case that the time-periodic driving is due to an electromagnetic field oscillating at an angular frequency ω, we can explain the transitions in which a particle can only gain or lose an amount of energy that is an integer multiple of ħω as follows: The electromagnetic field induces the particle to absorb or emit photons, each carrying a quantum of energy equal to an integer multiple of ħω. The analysis to be revealed comprises three main achievements, all of which would help one to accomplish suitable computational precision, accuracy, and efficiency when making a prediction about a scattering phenomenon in our chosen system. The first achievement is that my research supervisor and I managed to solve the Schrӧdinger equation for this system analytically (i.e. exactly), so for a choice of four intervals that together constitute one-dimensional space, we were able to find the actual space of (complex-valued) explicit solutions to the equation in every interval separately. Within that space of solutions, if we only consider functions that are continuously differentiable, square-integrable, and somewhere nonzero (that is, nonzero solutions for which their partial derivatives with respect to spatial variables exist and are continuous everywhere and the squares of their absolute values are integrable over all space), then we obtain the set of all possible states that a particle can access in our quantum system. By accurately finding the space of solutions to the Schrӧdinger equation and by imposing the requirement that a wavefunction must be continuously differentiable, we were able to deduce into what superposition of states a given incoming plane wave must scatter into in order to form a state that is possible to include in a physically allowed superposition of states. Accounting for the fact that a physically acceptable solution must be at least continuously differentiable is directly related to the second success of the study, which is to derive a formula enabling me to find the scattering matrix (or S-matrix), a mathematical construct that relates the incoming plane waves to the states of definite energy outside the well into which they scatter. Despite the complexity of the solutions, I managed to exploit the reflection symmetry of the system about the center of the well and other simplifying properties to come up with expressions for four matrix blocks that constitute a matrix that contains the S-matrix, such that all four expressions involve the same eight matrices.This in turn led to the third achievement, which was that I set up an efficient method for using computational software (in my case, I used Wolfram Mathematica 11.0) to find the elements of the S-matrix. Such a method was suitably fast at generating high-quality graphs of moduli squared of some matrix elements as functions of incident “energy” (actually, quasienergy, as we shall see later). Those graphs revealed the probability for an incoming wave (with a fixed energy and a unit probability current) to both transmit through the region of the well and to transition to a state, such as an outgoing wave or a negative-energy state, of some fixed energy. The energy of the new state did not necessarily have to be equal to the incident energy. Given these computational freedoms, I created a demo of my method by constructing these transmission graphs for a specific set of parameters expressed in Hartree atomic units: well width of 2, particle mass µ = 1, amplitude U0 = 0.5 and angular frequency ω = 4 of the oscillation of the V-shaped bottom, and unperturbed well depth V0 = 10. When I compared the different graphs of some of the combined transmission and transition probabilities provided by the elements of the S-matrix, I noticed two incident energies for which transmission resonances occur. Next, I exploited those resonances to determine some energies of quasibound states, which are states in which a particle’s probability is localized inside the region of the well for a finite amount of time. (In contrast, bound states have their probability localized for an infinite amount of time). For the version of our system without the external field (i.e. with U0 = 0), the bound state energies for the first and second excited states are -6.7791 and -3.0542 Hartrees, respectively, while for the driven system, I found that the corresponding energies of quasibound states were -6.7788 and -3.0534 Hartrees. The fact that these energies of quasibound states for the perturbed system are quite close to those of the unperturbed system is an indication that the chosen oscillation strength U0 = 0.5 Hartrees is weak enough to preserve many of the general properties of the unperturbed system.Physic

Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds

Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canonical frames for these structures can be given in the two prolongation steps: the first step, based on our previous works, gives the canonical bundle of moving frames for the fixed submanifolds of flags on each fiber and the second step consists of the prolongation of the bundle obtained in the first step. The bundle obtained in the first step is not as a rule a principal bundle so that the classical Tanaka prolongation procedure for filtered structures can not be applied to it. However, under natural assumptions on submanifolds of flags and on the ambient distribution, this bundle satisfies a nice weaker property. The main goal of the present paper is to formalize this property, introducing the so-called quasi-principle frame bundles, and to generalize the Tanaka prolongation procedure to these bundles. Applications to the equivalence problems for systems of differential equations of mixed order, bracket generating distributions, sub-Riemannian and more general structures on distributions are given.Comment: 49 pages. The Introduction was extended substantially: we demonstrate how flag structures appear in the geometry of double fibrations and, using this language, we discuss the motivating examples in more detai

A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions on a 5-dimensional manifold. We solve the analogous problems for rank 2 distributions on an n-dimensional manifold for arbitrary n greater than 5. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.Comment: 8 page

On local geometry of nonholonomic rank 2 distributions

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$ for n>5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold $W_D$ of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n-1)-dimensional fiber bundle over $W_D$ with the structure group of all M\"obius transformations, preserving 0.Comment: 21 pages, this is the long version of the short note math.DG/0504319 (the latter was published in C.R. Acad. Sci. Paris, Ser. I, Vol. 342, Issue 8 (15 April 2006), 589-59

Equivalence of variational problems of higher order

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the special class of rank 2 distributions associated with underdetermined ODEs z'=f(x,y,y',..., y^{(n)}); the equivalence problem for variational ODEs of order 2n. This leads to new results such as the fundamental system of invariants for all these problems and the explicit description of the maximally symmetric models. The central role in all three equivalence problems is played by the geometry of self-dual curves in the projective space of odd dimension up to projective transformations via the linearization procedure (along the solutions of ODE or abnormal extremals of distributions). More precisely, we show that an object from one of the three equivalence problem is maximally symmetric if and only if all curves in projective spaces obtained by the linearization procedure are rational normal curves.Comment: 20 page