Three universal oscillatory asymptotic phenomena

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Three universal oscillatory asymptotic phenomena
Michael Berry Physics Department University of
Bristol United Kingdom
http//www.phy.bris.ac.uk/staff/berry_mv.html
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1. Dominance by subdominant exponentials (DSE)
z plane
exponentials comparable
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very common situation a and a- depend
differently on z (and hence on F) ignoring
logarithms,
no loss of generality with mgt0
dominance by subdominant exponential (DSE) if
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DDE
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example 1 saddle-point and end-point
end-point at tz, saddle at t0
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phase contours of g1(F)
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example 4 suppressed end-point
saddle at tz, end-point at z0, suppressed but
dominant for Imzgt0
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application of f4(z) to conical diffraction
light incident on a slab of crystal with three
different principal dielectric constants
wavelength l
M V Berry 2004 Conical diffraction asymptotics
fine structure of Poggendorff rings and axial
spike, J. Opt. A 6, 289-300
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conical-diffraction ring profile is
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again, main contribution from subdominant
exponential (saddle)
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2. Universal attractor of differentiation
as n increases, all smooth functions look like
cost
(cf. orthogonal polynomials)
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if z(t) analytic in a strip including the real
axis
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as t increases from -? to ?, zn(t) describes
Maclaurins sinusoidal spiral (1718) (universal
loops) in polar coordinates,
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universal attractor of hodograph map (velocity
dz/dt)
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as n increases, universal cosine emerges near
t0
instability of differentiation?
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universal loops in geometric phase physics state
vector Y driven by slowly-varying operator H(et)
(adiabatic quantum mechanics)
as t increases from -? to ?, H describes a loop
in operator space
Y starts in an eigenstate of H( ?)
for small e, Y returns to its original form,
apart from a phase factor
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geometric phase corrections are determined by
time-dependent transformation to eigenbasis of
H(et), giving H1(t), and then iterating to H2(t),
etc phase corrections are related to areas of Hn
loops
in the simplest nontrivial case, iterated loops
are given by the derivative (hodograph) map
phase corrections get smaller and then inevitably
diverge, universally
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if z(t) has no finite singularities, asymptotics
of zn(t) determined by saddle-points in integral
representation, e.g.
example (suggested by David Farmer) inverse
gamma derivatives
1/G(z) has zeros at z0, -1, -2
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3. Superoscillations
a band-limited function
nevertheless - counterintuitively - such
functions can oscillate arbitrarily faster than
kmax, over arbitrarily large intervals they can
superoscillate
several recipes, suggested by Aharonov, Popescu
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?-function, centred on uA
where k(u) is even with k(0)kmax and k(u)kmax
for real u (Aharonov suggested k(u)kmaxcosu)
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more careful small-? analysis saddle-point
method, with ?x?2
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deportment of saddle us(?,3)
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local wavenumber
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A4, ?0.2
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test of saddle-point approximation with
k(u)1-u2/2, u2
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demystification a simple function with
rudimentary superoscillations
cosx1
band-limited, with frequencies 0,1,-1
cosx1-?
pairs of close zeros, separated by ??
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a periodic superoscillatory function (with Sandu
Popescu)
superoscillations with a factor a faster than
normal oscillations
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alternative form
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superoscillations invisible in power spectrum
narrow spectrum, centred on k1/a
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generalization integrate over a to generate
arbitrary functions locally, as band-limited
function
example a locally narrow gaussian
narrower than cos(Nx) if Agt1
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A4, N40
locally, gexp-160x)2/2
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in signal processing, superoscillations in a
function f(t) emerging from a perfect low-pass
filter could generate the illusion that the
filter is leaky
Beethovens 9th symphony - a one-hour signal,
requiring up to 20kHz for accurate reproduction -
can be generated by a 1Hz signal (but after the
hour, f(t) rises by a factor exp(1019))
in quantum physics, the large-k superoscillations
in a function f(x) represent weak values of
momenta - values outside the spectrum of an
operator, obtained in a weak measurement - a
gamma ray emerging from a box containing only red
light
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summary
asymptotically, functions can be dominated by
their subdominant exponentials
M V Berry,2004 Asymptotic dominance by
subdominant exponentials, Proc. Roy. Soc. Lond,
A,460, 2629-2636
under repeated differentiation, functions
eventually oscillate universally
functions can oscillate arbitrarily faster than
their fastest fourier components