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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Spectral theory of convolution operators on finite
intervals: small and large interval asymptotics -
Dmitry Ponomarev (Vienna University of Technology
\; Steklov Mathematical Institute\, Russian Academ
y of Sciences )
DTSTART;TZID=Europe/London:20190816T090000
DTEND;TZID=Europe/London:20190816T100000
UID:TALK128665AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/128665
DESCRIPTION:One-dimensional convolution integral operators pla
y a crucial role in a variety of different context
s such as approximation and probability
theory
\, signal processing\, physical problems in radiat
ion transfer\, neutron transport\, diffraction pro
blems\, geological prospecting issues and quantum
gases statistics\,.
Motivated by this\, we con
sider a generic eigenvalue problem for one-dimensi
onal convolution integral operator on an interval
where the kernel is
real-valued even $C^1$-smo
oth function which (in case of large interval) is
absolutely integrable on the real line.
We sho
w how this spectral problem can be solved by two d
ifferent asymptotic techniques that take advantage
of the
size of the interval.
In case of s
mall interval\, this is done by approximation with
an integral operator for which there exists a com
muting
differential operator thereby reducing
the problem to a boundary-value problem for second
-order ODE\, and often
giving the solution in
terms of explicitly available special functions su
ch as prolate spheroidal harmonics.
In case of
large interval\, the solution hinges on solvabili
ty\, by Riemann-Hilbert approach\, of an approxima
te auxiliary
integro-differential half-line eq
uation of Wiener-Hopf type\, and culminates in sim
ple characteristic equations for
eigenvalues\,
and\, with such an approximation to eigenvalues\,
approximate eigenfunctions are given in an explic
it form.
Besides the crude periodic approximat
ion of Grenander-Szego\, since 1960s\, large-inter
val spectral results were available
only for i
ntegral operators with kernels of a rapid (typical
ly exponential) decay at infinity or those whose s
ymbols
are rational functions. We assume the s
ymbol of the kernel\, on the real line\, to be con
tinuous and\, for the sake of
simplicity\, str
ictly monotonically decreasing with distance from
the origin. Contrary to other approaches\, the pro
posed
method thus relies solely on the behavio
r of the kernel'\;s symbol on the real line rat
her than the entire complex plane
which makes
it a powerful tool to constructively deal with a w
ide range of integral operators.
We note that\
, unlike finite-rank approximation of a compact op
erator\, the auxiliary problems arising in both sm
all-
and large-interval cases admit infinitely
many solutions (eigenfunctions) and hence structu
rally better represent
the original integral o
perator.
The present talk covers an extension
and significant simplification of the previous aut
hor'\;s result on
Love/Lieb-Liniger/Gaudin
equation.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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