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**Reflection operators and their applications to asymptotic investigations of semiclassical integral equations.**
*(English)*
Zbl 0743.45003

Estimates and asymptotics for discrete spectra of integral and differential equations, Pap. Semin. Math. Phys., Leningrad/Russia 1989-90, Adv. Sov. Math. 7, 107-157 (1991).

[For the entire collection see Zbl 0741.00011.]

The following integral equation is considered: (1) \(\varepsilon^{- 1}\int^ 1_{-1}A((x-y)/\varepsilon)f(y)dy=g(x)\), where \(\varepsilon>0\) is assumed to be a small parameter. Another form of the equation (1) being convenient in the investigations is the following one: (2) \(\int^{1/\varepsilon}_{-1/\varepsilon}A(x-y)f(y)dy=g(x)\).

In order to study the equation (1) it is worth to discuss the corresponding symbol given by the formula \(a(\varepsilon\xi)=\int^{+\infty}_{-\infty}\exp(- ix\xi)\varepsilon^{-1}A(x/\varepsilon)dx\). The considerations of this paper are inspired by the cases when symbols (or their derivatives) have jumps or roots at some points. Such cases were not satisfactory studied up to now.

The authors introduce the so-called reflection operators. They obtain several interesting results concerning the theory of this kind. Moreover, the results obtained are applied to the study of the asymptotic behavior of the solutions of the equations (1) or (2) as \(\varepsilon\to 0\). Several particular cases of the equation (1), arising in the theory of elasticity, in hydrodynamics, electrostatics, diffraction theory and quantum statistics, are discussed in details.

The following integral equation is considered: (1) \(\varepsilon^{- 1}\int^ 1_{-1}A((x-y)/\varepsilon)f(y)dy=g(x)\), where \(\varepsilon>0\) is assumed to be a small parameter. Another form of the equation (1) being convenient in the investigations is the following one: (2) \(\int^{1/\varepsilon}_{-1/\varepsilon}A(x-y)f(y)dy=g(x)\).

In order to study the equation (1) it is worth to discuss the corresponding symbol given by the formula \(a(\varepsilon\xi)=\int^{+\infty}_{-\infty}\exp(- ix\xi)\varepsilon^{-1}A(x/\varepsilon)dx\). The considerations of this paper are inspired by the cases when symbols (or their derivatives) have jumps or roots at some points. Such cases were not satisfactory studied up to now.

The authors introduce the so-called reflection operators. They obtain several interesting results concerning the theory of this kind. Moreover, the results obtained are applied to the study of the asymptotic behavior of the solutions of the equations (1) or (2) as \(\varepsilon\to 0\). Several particular cases of the equation (1), arising in the theory of elasticity, in hydrodynamics, electrostatics, diffraction theory and quantum statistics, are discussed in details.

Reviewer: J.Banaś (Rzeszów)

### MSC:

45A05 | Linear integral equations |

45M05 | Asymptotics of solutions to integral equations |

47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |