Semiclassical integral equations with slowly decreasing kernels on finite domains. (English. Russian original) Zbl 0794.45001

Budylin, A. M.; Buslaev, V. S.

St. Petersbg. Math. J. 5, No. 1, 141-158 (1994); translation from Algebra Anal. 5, No. 1, 160-178 (1993).

This paper concerns the asymptotic behaviour of the solution of the integral equation \[ \varepsilon^{-1}\int^ 1_{-1}{\mathcal A}(\varepsilon^{-1}(x-y))f(y)dy=g(x) \] as \(\varepsilon\downarrow 0\). The cases when the corresponding symbol \[ a(\varepsilon\xi)=\int^ \infty_{-\infty}e^{-ix\xi}\varepsilon^{-1}{\mathcal A}(\varepsilon^{- 1}x)dx \] has jumps and/or roots is investigated. From the geometrical point of view, the approach proposed here may be regarded as a generalization of the alternating method of Schwarz.

Reviewer: J.F.Toland (Bath)

MSC:

45A05	Linear integral equations
45M05	Asymptotics of solutions to integral equations
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

semiclassical integral equations; slowly decreasing kernels; finite domains; Wiener-Hopf method; convolution equations; singular integral equations; Szegö formula; asymptotic behaviour; alternating method of Schwarz