BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Anderson localization for one-dimensional ergodic
Schrdinger operators with piecewise monotonic samp
ling functions - Kachkovskiy\, I (University of Ca
lifornia\, Irvine)
DTSTART;TZID=Europe/London:20150324T113000
DTEND;TZID=Europe/London:20150324T123000
UID:TALK58544AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58544
DESCRIPTION:Co-author: Svetlana Jitomirskaya (University of Ca
lifornia\, Irvine) \n\nWe consider the one-dimensi
onal ergodic operator families\negin{equation}\nl
abel{h_def}\n(H_{lpha\,lambda}(x) Psi)_m=Psi_{m+1
}+Psi_{m-1}+lambda v(x+lpha m) Psi_m\,quad min ma
thbb Z\,\nnd{equation}\nin $l^2(mathbb Z)$. Such
operators are well studied for analytic $v$\, wher
e they \nundergo a metal-insulator transition from
absolutely continuous spectra (for small $lambda$
) to purely point spectra with exponentially decay
ing eigenfunctions (for large $lambda$)\; the latt
er is usually called Anderson localization. Very l
ittle is known for general continuous of smooth $v
$. However\, there are several well developed mode
ls with discontinuous $v$\, such as Maryland model
and the Fibonacci Hamiltonian.\n\n\n\nWe study th
e family $H_{lpha\,lambda}(x)$ with $v$ satisfyin
g a bi-Lipshitz type condition (for example\, $v(
x)={x}$). It turns out\nthat for every $lambda$\,
for almost every $lpha$ and all $x$ the spectrum
of the operator $H_{lpha\,lambda}(x)$ is pure po
int. This is the first example of pure point spect
rum at small coupling for bounded quasiperiodic-ty
pe operators\, or more generally for ergodic opera
tors with underlying systems of low disorder.\n\n\
n\nWe also show that the Lyapunov exponent of this
system is continuous in energy for all \n$lambda$
and is uniformly positive for $lambda$ sufficient
ly \n(but nonperturbatively) large. In the regime
of uniformly positive Lyapunov exponent\, our \nre
sult gives uniform localization\, thus providing t
he first\nnatural example of an operator with this
property.\nThis is a joint result with Svetlana J
itomirskaya\, University of California\, Irvine.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
END:VEVENT
END:VCALENDAR