В четверг 17 декабря с 17-00 по 19-00 пройдет семинар кафедры высшей математики и математической физики.
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Meeting ID: 270 950 5573
Докладчик: М.А. Лялинов
Тема: Спектральные свойства функционально-разностных уравнений
и асимптотика собственных функций оператора Шредингера с
сингулярным $\delta’$-потенциалом с носителем на границе угловой области
This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter.
Consideration of this type of equations is motivated by diffraction in angular domains in R^2 (or conical in R^3) with `semitransparent’ boundary conditions. Another application deals with construction of eigenfunctions for the Schr\»odinger operator with singular potential having its support on the boundary of the angular (conical) domains.
For positive values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The results obtained are applied to study of the behavior of eigenfunctions
for the operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary.
Our analysis shows that an eigenfunction exponentially vanishes at infinity as was expected. However, the rate of decay depends on direction of observation. Indeed, in the angular domain there is a singular direction with an asymptotically small angular vicinity, where the leading term of the asymptotics is described by a Fresnel type integral. The latter plays the role of transition function that is responsible for switching regimes of the exponential decay across the singular direction. The generalized eigenfunctions of the essential (continuous) spectrum are similarly studied.