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:info@newton.ac.uk END:VEVENT END:VCALENDAR