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SUMMARY:Spectral theory of convolution operators on finite intervals: smal
 l and large interval asymptotics - Dmitry Ponomarev (Vienna University of 
 Technology\; Steklov Mathematical Institute\, Russian Academy of Sciences 
 )
DTSTART:20190816T080000Z
DTEND:20190816T090000Z
UID:TALK128668@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:One-dimensional convolution integral operators play a crucial 
 role in a variety of different contexts such as approximation and probabil
 ity<br> theory\, signal processing\, physical problems in radiation transf
 er\, neutron transport\, diffraction problems\, geological prospecting iss
 ues and quantum gases statistics\,.<br> Motivated by this\, we consider a 
 generic eigenvalue problem for one-dimensional convolution integral operat
 or on an interval where the kernel is<br> real-valued even $C^1$-smooth fu
 nction which (in case of large interval) is absolutely integrable on the r
 eal line.<br> We show how this spectral problem can be solved by two diffe
 rent asymptotic techniques that take advantage of the<br> size of the inte
 rval.<br> In case of small interval\, this is done by approximation with a
 n integral operator for which there exists a commuting<br> differential op
 erator thereby reducing the problem to a boundary-value problem for second
 -order ODE\, and often<br> giving the solution in terms of explicitly avai
 lable special functions such as prolate spheroidal harmonics.<br> In case 
 of large interval\, the solution hinges on solvability\, by Riemann-Hilber
 t approach\, of an approximate auxiliary<br> integro-differential half-lin
 e equation of Wiener-Hopf type\, and culminates in simple characteristic e
 quations for<br> eigenvalues\, and\, with such an approximation to eigenva
 lues\, approximate eigenfunctions are given in an explicit form.<br> Besid
 es the crude periodic approximation of Grenander-Szego\, since 1960s\, lar
 ge-interval spectral results were available<br> only for integral operator
 s with kernels of a rapid (typically exponential) decay at infinity or tho
 se whose symbols<br> are rational functions. We assume the symbol of the k
 ernel\, on the real line\, to be continuous and\, for the sake of<br> simp
 licity\, strictly monotonically decreasing with distance from the origin. 
 Contrary to other approaches\, the proposed<br> method thus relies solely 
 on the behavior of the kernel&#39\;s symbol on the real line rather than t
 he entire complex plane<br> which makes it a powerful tool to constructive
 ly deal with a wide range of integral operators.<br> We note that\, unlike
  finite-rank approximation of a compact operator\, the auxiliary problems 
 arising in both small-<br> and large-interval cases admit infinitely many 
 solutions (eigenfunctions) and hence structurally better represent<br> the
  original integral operator.<br> The present talk covers an extension and 
 significant simplification of the previous author&#39\;s result on<br> Lov
 e/Lieb-Liniger/Gaudin equation.
LOCATION:Seminar Room 1\, Newton Institute
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