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Semiclassical integral equations on the semiaxis. (English) Zbl 0869.45003

Dobrushin, R. L. (ed.) et al., Topics in statistical and theoretical physics. F. A. Berezin memorial volume. Transl. ed. by A. B. Sossinsky. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 177(32), 45-49 (1996).
Summary: The asymptotic behavior of solutions on the semiaxis of integral equations of the form \[ \frac{1}{\varepsilon} \int^\infty_0 A\Biggl( \frac{x-y}{\varepsilon}; x,y\Biggr) f(y)dy= g(x), \qquad x>0, \] is described as \(\varepsilon\searrow 0\). It is assumed that the symbol \[ a(\xi;x,y)= \int^{+\infty}_{-\infty} e^{-i\xi z}{\mathcal A}(z;x,y)dz \] has a jump with respect to \(\xi\) at \(\xi=0\). The procedure proposed here can be regarded as an asymptotic generalization of the Wiener-Hopf method to the case in which the symbol \(a(\xi;x,y)\) depends nontrivially on \(x\) and \(y\).
For the entire collection see [Zbl 0853.00022].

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45M05 Asymptotics of solutions to integral equations
47G10 Integral operators
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