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

Speaker(s) Dmitry Ponomarev Vienna University of Technology; Steklov Mathematical Institute, Russian Academy of Sciences
Date 16 August 2019 – 09:00 to 10:00
Venue INI Seminar Room 1
Session Title Spectral theory of convolution operators on finite intervals: small and large interval asymptotics
Event [WHTW01] Factorisation of matrix functions: New techniques and applications

One-dimensional convolution integral operators play a crucial role in a variety of different contexts such as approximation and probability<br> theory, signal processing, physical problems in radiation transfer, neutron transport, diffraction problems, geological prospecting issues and quantum gases statistics,.<br> Motivated by this, we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is<br> real-valued even $C^1$-smooth function which (in case of large interval) is absolutely integrable on the real line.<br> We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the<br> size of the interval.<br> In case of small interval, this is done by approximation with an integral operator for which there exists a commuting<br> differential operator thereby reducing the problem to a boundary-value problem for second-order ODE, and often<br> giving the solution in terms of explicitly available special functions such as prolate spheroidal harmonics.<br> In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary<br> integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for<br> eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.<br> Besides the crude periodic approximation of Grenander-Szego, since 1960s, large-interval spectral results were available<br> only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols<br> are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of<br> simplicity, 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's symbol on the real line rather than the entire complex plane<br> which makes it a powerful tool to constructively 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's result on<br> Love/Lieb-Liniger/Gaudin equation.

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