## Generalized method of eigenoscillations in diffraction theoryM. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, N. N. Voitovich Generalized Method of Eigenoscillations in Diffraction Theory The book presents a new method for solving various diffraction and scattering problems in acoustics, electrodynamics, and quantum mechanics. Each version of the method is based on the representation of the diffracted field in the form of a series in the eigenfunctions of an auxiliary homogeneous problem in which the spectral parameter is usually not the frequency. This allows one to treat problems not only in bounded but also in unbounded domains or in the entire space. For example, for the problem of diffraction on a metallic body, the homogeneous problem of the same form can be used with impedance as the spectral parameter. The transparency coefficient, the dielectric constant, etc. can also be used as the spectral parameter. The method is especially effective for the analysis of resonance systems, in particular, of open resonators and waveguides. The method permits one to represent the exact solution in unbounded domains in the form of a series (since the spectrum is discrete), without an additional integral with respect to the spectral parameter, and use the variational approach though the corresponding problems are usually nonselfadjoint. The formal exposition of the method is presented in Chapters 1 and 2. The variational approach is described and analysed in Chapter 3. Chapter 4 contains a number of examples with applications of the method to particular diffraction problems. Chapter 5 contains a rigorous mathematical treatment of the main versions of the method on the basis of modern tools of the theories of nonself-adjoint operators and elliptic pseudodifferential operators. This investigation provides more deep information about the properties of classical integral and integro-differential operators related to the Helmholtz equation and the Maxwell system than in well-known textbooks. |

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### Contents

Introduction | 9 |

Spectral Parameter in the Equation the eMethod | 21 |

Spectral Parameter in Boundary Conditions | 71 |

Copyright | |

6 other sections not shown

### Common terms and phrases

admissible functions arbitrary assertions asymptotic auxiliary boundary conditions Chapter closed resonator coefficients compact operator complete conditions at infinity consider contain convergence coordinates corresponding curve defined dielectric body diffraction problem Dirichlet problem domain V+ dvu+ e-method eigenfrequencies eigenfunctions eigenvalues elliptic pseudodifferential operator en(r expansion exterior problem fc-method finite follows formula Fourier given Green formula Green function Helmholtz equation Hence homogeneous problem infinitely smooth inner product integral equation interior kernel left-hand side Lipschitz method of eigenoscillations Neumann problem nonzero normal derivative obtain open resonator operator A(k operator of order orthogonality conditions permittivity principal symbol properties Proposition pseudodifferential operator radiation condition readily replaced respect right-hand side Ritz method root functions satisfies the equation satisfy the radiation Section selfadjoint selfadjoint operator Sobolev spaces solve space spectral parameter spectrum stationary Subsection subspace surface technique term Theorem tion waveguide zero