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Mathematical Physics
Org: D. Brydges (UBC), S. Sontz (CIMAT) and Carlos Villegas (UNAM-Cuernavaca)
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- DAVID BRYDGES, University of British Columbia
A combinatorial generalisation of Cramer's rule
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We review a result of G. X. Viennot, Lecture Notes in Mathematics
1234(1986), and comment on its significance for statistical
mechanics: a ratio of generating functions for disjoint oriented loops
in a finite graph can be expressed in terms of the generating function
of a single path in the graph weighted according to loops in the path.
The result is a generalisation of Cramer's formula for the inverse of
a matrix.
- JAIME CRUZ SAMPEDRO, Universidad Autónoma del Estado de Hidalgo
Embedded Eigenvalues of Continuous and Discrete Schrödinger
Operators
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First we present general results about the instability of embedded
eigenvalues in the continuum, and then we treat in more detail the
instability of embedded eigenvalues of one dimensional Schroedinger
operators, with Wigner-von Neumann-like potentials, both in the
continuous and the discrete cases.
- RAFAEL DEL RIO, UNAM
Sturm-Liouville operators in the half axis with local
perturbations
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We give conditions which imply equivalence of the Lebesgue measure
with respect to a measure m generated as an average of spectral
measures corresponding to Sturm-Liouville operators in the half axis.
We apply this to prove that some spectral properties of these
operators hold for large sets of boundary conditions if and only if
they hold for large sets of positive local perturbations.
This is joint work with O. Tchebotareva.
- RICHARD FROESE, University of British Columbia
AC spectrum for Schrödinger operators on tree-like graphs
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I will discuss recent proofs, obtained with Hasler and Spitzer, of the
existence of absolutely continuous spectrum for Schrödinger
operators on graphs.
- ANTONIO HERNANDEZ, IIMAS-UNAM, Apdo. Postal 20-726, Mexico City 01000, Mexico
Symmetry breaking and adiabatic invariants
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We will discuss adiabatic momentum maps in the context of examples of
mechanical systems with approximate symmetry. A procedure for
averaging the "variational principle" will be described.
- DIMITRI JAKOBSEN, McGill, Dept. of Math, 805 Sherbrooke W, Montreal, QC,
H3A 2K6, Canada
Estimates from below for spectral function and error term in
Weyl law
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We obtain asymptotic lower bounds for the spectral function of the
Laplacian on compact manifolds. In the negatively curved case,
thermodynamic formalism is applied to improve the estimates. Our
results can be considered pointwise versions (on a general manifold)
of Hardy's lower bounds for the error term in the Gauss circle
problem. We next obtain a lower bound for the remainder in Weyl's law
on negatively curved surfaces. On higher-dimensional negatively
curved manifolds, we prove a similar bound for the oscillatory error
term. Our approach uses wave trace asymptotics, equidistribution of
closed geodesics and small-scale microlocalization.
- ROBERT MOODY, University of Victoria, Victoria, BC
Between order and disorder: the mathematics of quasicrystals
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Quasicrystals are materials that lie somewhere between crystals and
disordered materials. This talk is an introduction to the mathematics
that has been created to model and explain them. We will start with
various characterizations of mathematical crystals, where the
underlying structure is based on lattices, and show how nicely these
can be generalized to encompass some of this intermediate world of
aperiodic order.
A characteristic feature of quasicrystals is their crystal-like
diffraction, which is deeply related to their internal order. We will
indicate some of the known ways of producing aperiodic pure point
diffractive sets and the current state of trying to characterize them.
At the end of the day, aperiodic order seems to be a reconciliation of
precise local order and average global order. We will show how,
through the use of dynamical systems, this idea can be made more
precise.
- JEREMY QUASTEL, Dept. of Maths., University of Toronto, 40 St. George,
Toronto
Effect of Noise on Traveling Fronts in the Fisher-KPP
equation
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KPP-type reaction-diffusion equations perturbed by noise have random
traveling fronts. We compute the speed asymptotically for small
values of the noise. As conjectured by Brunet and Derrida, the
slowdown is as the inverse square of the logarithm of the noise, with
an explicit constant.
- LUIS SILVA, IIMAS-UNAM Apartado Postal 20-726 Mexico DF 01000
Applications of M. G. Krein's Theory of Entire Operators to
Sampling Theory
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The Whittaker-Shannon-Kotel'nikov Sampling Theorem gives a formula
for reconstructing Paley-Wiener functions from their values at a
discrete set of points (samples). This theorem has been extended and
generalized in various ways.
In this talk we consider a generalization of the
Whittaker-Shannon-Kotel'nikov Sampling Theorem on the basis of a
particular class of simple symmetric operators with deficiency indices
(1,1). The theory of this class of operators is due to M. G. Krein.
This is a joint work with Julio H. Toloza.
- GORDON SLADE, University of British Columbia, Vancouver
Invasion percolation on a tree
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Invasion percolation is a stochastic growth process which produces a
random infinite subgraph-the invaded region-of a given infinite
graph G. We consider the case where G is a regular tree and study
the large-scale properties of the invaded region. Viewed far from the
origin, the invaded region looks locally like a large critical
percolation cluster. But surprisingly, we prove that the global
structure of the invaded region is dramatically different than that of
the incipient infinite percolation cluster.
This is joint work with O. Angel, J. Goodman and F. den Hollander.
- STEPHEN SONTZ, CIMAT, Guanajuato, Gto., Mexico
Heat kernel analysis in a deformation of quantum mechanics
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We present a m-deformation of quantum mechanics based on Dunkl
operators, as studied by Rosenblum. This includes a m-deformed
Segal-Bargmann space and an asssociated m-deformed
Segal-Bargmann transform. We show the relation of these structures
to heat kernel analysis, following ideas introduced by Hall.
- JULIO TOLOZA, Universidad Nacional Autónoma de México, Ciudad
Universitaria, México D.F.
Absence of continuous spectrum on a class of unbounded Jacobi
operators
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We establish sufficient conditions for self-adjointness on a class of
unbounded Jacobi operators defined by matrices with main diagonal
sequence of very slow growth and rapidly growing off-diagonal entries.
With some additional assumptions, we also prove that these operators
have only discrete spectrum.
- ALEXANDER TURBINER, Nuclear Science Institute, UNAM
Anharmonic oscillator and double-well potential:
approximating eigenfunctions
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A simple uniform approximation of the logarithmic derivative of the
ground state eigenfunction for both the quantum-mechanical anharmonic
oscillator and the double-well potential given by V = m2 x2 + gx4
at arbitrary g ³ 0 for m2 > 0 and m2 < 0, respectively, is
presented. It is shown that if this approximation is taken as
unperturbed problem it leads to an extremely fast convergent
perturbation theory. A connection with WKB approximation is briefly
discussed.
Dedicated to the memory of Professor Felix A. Berezin.
- CARLOS VILLEGAS, UNAM, Matemáticas, Cuernavaca
Asymtotics of clusters of eigenvalues for perturbations of
the hydrogen atom Hamiltonian
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We present in this talk a limiting eigenvalue distribution theorem for
the Schrödinger operator of the hydrogen atom (with the Planck
parameter (h/2p) included) plus e times a bounded continuous
function Q. By considering suitable dilation operators, we prove
that taking e = O((h/2p)2) we obtain well defined clusters of
eigenvalues around the energy E = -1/2 whose limiting distribution
involves the Radon transform of the function Q along the classical
orbits of the Kepler problem with energy E=-1/2 with respect to an
integration over the space of geodesics of the 3-sphere S3. The
idea of the proof involves a well known unitary transformation from
the Hilbert space generated by the bound states of the hydrogen atom
onto L2(S3) and coherent states on the sphere S3. We will
comment on the generalization of the theorem above to the
n-dimensional case and when Q is a pseudodifferential operator of
order zero.
- RICARDO WEDER, UNAM
Inverse Scattering at a Fixed Energy
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We prove that the averaged scattering solutions to the Schrödinger
equation with short-range electromagnetic potentials (V,A) where
V(x) = O(|x|-r), A(x) = O(|x|-r), |x| ®¥, r > 1, are dense in the set of all solutions to the
Schrödinger equation that are in L2(K) where K is any connected
bounded open set in Rn, n ³ 2, with smooth boundary.
We use this result to prove that if two short-range electromagnetic
potentials (V1,A1) and (V2,A2) in Rn, n ³ 3, have
the same scattering matrix at a fixed positive energy and if the
electric potentials Vj and the magnetic fields Fj : = curlAj,
j = 1,2, coincide outside of some ball they necessarily coincide
everywhere.
In a previous paper of Weder and Yafaev the case of electric
potentials and magnetic fields in Rn, n ³ 3, that are
asymptotic sums of homogeneous terms at infinity was studied. It was
proven that all these terms can be uniquely reconstructed from the
singularities in the forward direction of the scattering amplitude at
a fixed positive energy.
The combination of the new uniqueness result of this paper and the
result of Weder and Yafaev implies that the scattering matrix at a
fixed positive energy uniquely determines electric potentials and
magnetic fields that are a finite sum of homogeneous terms at
infinity, or more generally, that are asymptotic sums of homogeneous
terms that actually converge, respectively, to the electric potential
and to the magnetic field.
- PETR ZHEVANDROV, University of Cartagena, Cartagena, Colombia
Water waves guided by underwater obstacles
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It is well-known that underwater obstacles such as ridges and
submerged horizontal cylinders can serve as waveguides for surface
water waves. It is also known that for large values of the wavenumber
k in the direction of the ridge or cylinder, there is only one
guided wave. We construct the corresponding eigenfunctions and
eigenfrequencies assuming that k®¥ by means of reducing the
initial problem to a pair of boundary integral equations and then
solving them by applying the method of Zhevandrov and Merzon (Amer. Math. Soc. Transl. (2) 208(2003), p. 235). The resulting
formulas are infinite convergent series of the Neumann type, which
reduce to quite simple asymptotics of the eigenfrequencies as
k®¥.
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