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Spectral theory of convolution operators on finite intervals: small and large interval asymptotics

Presented by: 
Dmitry Ponomarev
Friday 16th August 2019 - 09:00 to 10:00
INI Seminar Room 1
One-dimensional convolution integral operators play a crucial role in a variety of different contexts such as approximation and probability
theory, signal processing, physical problems in radiation transfer, neutron transport, diffraction problems, geological prospecting issues and quantum gases statistics,.
Motivated by this, we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is
real-valued even $C^1$-smooth function which (in case of large interval) is absolutely integrable on the real line.
We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the
size of the interval.
In case of small interval, this is done by approximation with an integral operator for which there exists a commuting
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 such as prolate spheroidal harmonics.
In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary
integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for
eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.
Besides the crude periodic approximation of Grenander-Szego, since 1960s, large-interval spectral results were available
only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols
are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of
simplicity, strictly monotonically decreasing with distance from the origin. Contrary to other approaches, the proposed
method thus relies solely on the behavior of the kernel's symbol on the real line rather than the entire complex plane
which makes it a powerful tool to constructively deal with a wide range of integral operators.
We note that, unlike finite-rank approximation of a compact operator, the auxiliary problems arising in both small-
and large-interval cases admit infinitely many solutions (eigenfunctions) and hence structurally better represent
the original integral operator.
The present talk covers an extension and significant simplification of the previous author's result on
Love/Lieb-Liniger/Gaudin equation.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons