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Edward R. Vrscay

Past Research Programme

Mathematical Physics/Quantum Mechanics

Perturbation Theory, Summability of Divergent Perturbation Expansions

My early interest in quantum mechanics (i.e., from undergraduate and graduate school days) was mostly practical and computational in nature. My Master's and Ph.D. thesis research focused on quantum mechanical perturbation theory. Very briefly, most perturbation expansions encountered in quantum mechanics are divergent, i.e. have zero radius of convergence. (Recall that for a series to be convergent at x, the sequence of partial sums S_n(x) at x must converge in the limit n -> infinity.) One would well wonder whether such series are useful at all. Indeed, they are. Firstly, they are often asymptotic to the function that they represent, for example, the energy. As such, the first few terms often provide useful approximations of the function for suitably small values of the perturbation parameter. (For example, the Stark effect in hydrogen is presented in all undergraduate quantum mechanics texts as an example of the utility of the perturbation method.)

Secondly, these divergent series are often "summable". What this means is that for a given x, one can use the series coefficients c_n to construct a new sequence T_n(x) that converges to a limit T(x) as n -> infinity. The divergent series is said to be summable to T(x) (according to the method used to construct the new sequence). An example of a summability method is that of Pade approximants (rational functions).

My early work focused on the continued fraction (CF) representations of quantum mechanical perturbation series. CF's can represent particular cases of Pade approximants.

  • E. R. Vrscay and J. Cizek, "Continued Fractions and Rayleigh-Schrodinger Perturbation Theory at Large Order", J. Math. Phys. 27, 185-201 (1986).
  • E. R. Vrscay, "Renormalized Rayleigh-Schrodinger Perturbation Theory", Theor. Chim. Acta 73, 365-382 (1988).
  • E. R. Vrscay, "Nonlinear Self-Interaction Hamiltonians of the form $H^0 + lambda r^q$ and their Rayleigh-Schrodinger Perturbation Expansions", J. Math. Phys. 29, 901-911 (1988).

Classical Limit of Quantum Mechanics

It is often written that classical mechanics (CM) follows from quantum mechanics (QM) in the limit h (Planck's constant) -> 0. By itself, of course, this is a meaningless statement since Planck's constant is a nonzero constant with a physical dimension - it simply cannot be set to zero. However, in particular physical regimes, e.g. systems with high energy, h may be considered to be small, hence negligible, compared to the action associated with periodic orbits. With such an understanding, one may then perform the ``mathematical trick'' of letting h -> 0. This is the basis of semi-classical methods, e.g., the so-called WKB method. When such limits are taken properly, the quantum mechanical probability densities associated with wave functions (of high quantum number) approach classical probability densities.

One can also apply such limiting processes to quantum mechanical perturbation expansions, again with care. For example, consider the well-known Rayleigh-Schrodinger perturbation expansion for the energy of a QM hamiltonian, for example, the energy E(n,a) of a simple one-dimensional quartic anharmonic oscillator p^2 + x^2 + ax^4, where n (0,1,2,...) is the energy level being considered and a is the perturbation parameter. The perturbation expansion will be a sum of terms that contain powers of a, n and h, Planck's constant. If one lets n -> infinity and h -> 0 so that the product nh = J, where J is constant, then the RS expansion becomes the classical Poincare-von Zeipel expansion for the energy of the periodic orbit of a classical quartic anharhmonic oscillator, where the action J of the orbit is kept constant.

Our early work focused on a method of using classical Hypervirial and Hellman-Feynman theorems to generate such classical mechanical expansions:

  • S. M. McRae and E. R. Vrscay, "Canonical perturbation expansions to large order from classical hypervirial and Hellman-Feynman theorems," J. Math. Phys. 33 (9), 3004-3024 (1992).

Later work examined the application of RS perturbation theory to study the time evolution of harmonic oscillator coherent states under an anharmonic oscillator Hamiltonian. From appropriate RS expansions, we were able to extract the Poincare-Lindstedt perturbation expansions that one obtains from classical mechanical differential equations:

  • S. M. McRae and E. R. Vrscay, ``Perturbation Theory and the Classical Limit of Quantum Mechanics,'' J. Math. Phys. 38, 2899-2921 (1997).

Dynamical Systems and Applications

Iteration Dynamics of Rational Functions

The Newton (-Raphson) function N(z) = z - g(z)/g'(z) associated with a function g(z) defines a discrete dynamical system z(n+1) = N(z(n)) for which the sequences {z(n)} locally converge quadratically to a root z* of g(z). For g(z) polynomial, N(z) is a rational function. The dynamics of Newton's method is then fully described by the Julia-Fatou theory of iteration of rational functions. The basins of attraction of each root of g(z) will share a common boundary, the so-called Julia set J of N(z). (One way to describe J is that it is the closure of the set of all repulsive cycles of N(z).)

In the very simple case g(z) = z^2 - 1, the Julia set boundary of the basins of attraction of the two roots, 1 and -1, is the imaginary axis. However, in the case of three roots, i.e., g(z) = z^3 - 1, the Julia set boundary is a very complicated set that lies within sectors centered along the three rays theta = pi/3, -pi, -pi/3.

In early works, I examined the Julia set basin boundaries of iteration schemes designed to accelerate the local convergence of sequences to roots:

  • E. R. Vrscay, "Julia sets and Mandelbrot-like sets associated with higher order Schroder rational iteration functions," Math. Comput. 46, 151-169 (1986).
  • E. R. Vrscay and W.J. Gilbert, "Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and Konig rational iteration functions," Numer. Math. 52, 1-16 (1988).

"Iterated Function Systems" or IFS

Suppose that w : X -> X is a contraction mapping of a complete metric space (X,d) to itself. Then, by the celebrated Banach Fixed Point Theorem, there is a unique point x' in X such that w(x') = x', the fixed point of w. Moreover, for any starting point x_0 in X, the sequence of points {x_n} defined by the iteration procedure x_{n+1}=w(x_n) converges to x' as n -> infinity. In other words, x' is an attractive fixed point.

Now suppose that we have more than one contraction mapping on X, i.e. w_i : X -> X, i=1,...N. Each of these maps w_i will have its own fixed point x'_i. And, of course, if we apply only one of these maps repeatedly, say w_P, then the iteration sequence {x_n} will converge to its fixed point x'_P. But what happens if you pick the maps at random? Clearly, each map will be fighting to bring the sequence to its own fixed point.

Assuming that you pick each map w_i with a nonzero probability p_i (sum of p_i = 1), the sequence will eventually approach a unique set, the "attractor" A of the "iterated function system" W = {w_1, ... , w_N}, performing a random walk over it (or at least getting nearer and nearer to it). The probabilities will play a role in determining how often various regions of the attractor are visited.

Examples on [0,1]:

  • w_1(x) = x/3, w_2(x) = x/3 + 2/3. Note that w_1(0)=0 and w_2(1)=1. Then the attractor of this IFS is the classical Cantor set on [0,1].
  • w_1(x) = x/2, w_2(x) = x/2 + 1/2. Note again that w_1(0)=0 and w_2(1)=1. Then the attractor of this IFS is the interval [0,1]. If p_1 = p_2 = 1/2, then the distribution of the random-walking iterates {x_n} over the interval [0,1] is uniform. If p_1 > p_2, then there is more probability of finding the iterates in the interval [0,1/2] than in [1/2,1]. But this means that there will be more probability of finding them in [0,1/4] vs. [1/4,1/2] and more in [1/2,3/4] than [3/4,1], and so on. The result is that the visitation frequencies over small sets demonstrate a complicated fractal-like pattern.

    • For a little more discussion on this subject, you may wish to look at my Hitchhiker's Guide to Fractal Image Compression. It was written for a general audience with some undergraduate mathematical knowledge.

    Other sample works:

    • P. Centore and E. R. Vrscay, "Continuity of Attractors and Invariant Measures of Iterated Function Systems", Canadian Math. Bull. 37, 315-329 (1993).
    • E. R. Vrscay and D. Weil (1991), "Missing Moment and Perturbative Methods for Polynomial Iterated Function Systems", Physica D 50, 478-492 (1991).

    Generalized Fractal Transforms, Inverse Problems for Fractal Transform, Fractal Image Coding

    The most popular "fractal-based" algorithms for both the representation representation as well as the compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) The "IFS component" of these methods is a set of N contraction maps {w_1, w_2,...,w_N}, w_i X -> X, over a complete metric space (X,d), the "base space" which represents the computer screen.

    The major developments of the research at Waterloo to date are:

    • A formulation of "Formal Inverse Problems" of approximation of functions, measures, etc. using IFS-type methods. By the "formal inverse problem", we imply that the approximation of a "target" function/measure/image may be produced to arbitrary accuracy.
    • Solutions to the various formal inverse problems as well as algorithms to achieve approximations to arbitrary accuracy.
    • The formulation of a consistent mathematical theory of "Generalized Fractal Transforms" over a suitable complete metric space (Y,d_Y) which could represent our space of images. This involves the determination of a suitable "fractal transform" operator T which will map images to images. A set of rules for constructing such an IFS-type operator is formulated including the condition that T be contractive with respect to the metric d_Y.
    • The above procedure has led to a unification of the various IFS-type methods over function spaces: IFS -> IFZS -> IFSM.
    • The IFS-type methods over function spaces have been linked with the method of IFSP (over measure spaces) by a formulation of Fractal Transforms over D'(X), the space of distributions on X.
    • The mathematical basis of the discrete "fractal-wavelet transform" where subtrees of wavelet coefficient trees are scaled and copied to lower subtrees. In general, fractal-wavelet transforms are equivalent to recurrent local IFSM with condensation functions.

    The above work is summarized in the following paper:

    For more mathematical presentations, see:

    • B. Forte and E. R. Vrscay, Theory of Generalized Fractal Transforms, in "Fractal Image Encoding and Analysis", edited by Y. Fisher (Springer Verlag, Heidelberg, 1998).
    • B. Forte and E. R. Vrscay, Inverse Problem Methods for Generalized Fractal Transforms, in "Fractal Image Encoding and Analysis", ibid.
    • C. A. Cabrelli, B. Forte, U. M. Molter and E. R. Vrscay, ``Iterated Fuzzy Set Systems: A New Approach to the Inverse Problem for Fractals and Other Sets'', J. Math. Anal. Appl. 171, 79-100 (1992).


Last Modified:  Wednesday 6 April 2011
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