The spectral problem (A + V(z))ψ = zψ is considered with A, a self-adjoint operator. The perturbation V(z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A' : V(z) = —B(A' — z)~(-1)B~*. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A' are separated. Conditions are formulated when the perturbation V(z) may be replaced with a "potential" W independent on z and such that the operator H = A + W has the same spectrum and the same eigenfunctions (more precisely, a part of spectrum and a respective part of eigenf unctions system) as the initial spectral problem. The operator H is constructed as a solution of the non-linear operator equation H = A + V(H) with a specially chosen operator-value function V(H). In the case if the initial spectral problem corresponds to a two-channel variant of the Friedrichs model, a basis property of the eigenfunction system of the operator H is proved. A scattering theory is developed for H in the case where the operator A has continuous spectrum.
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机译：自伴算子A考虑频谱问题（A + V（z））ψ=zψ。假设摄动V（z）依赖于频谱参数z作为另一个自伴算子A'的解析物：V（z）= -B（A'-z）〜（-1）B〜*。假定算符B具有有限的希尔伯特-施密特范数，并且算符A和A'的光谱是分开的。当扰动V（z）可以被独立于z的“势” W代替，并且使得算符H = A + W具有相同的光谱和相同的本征函数（更确切地说，一部分光谱和a本征函数系统的各个部分）作为初始频谱问题。算子H被构造为带有特殊选择的算子值函数V（H）的非线性算子方程H = A + V（H）的解。在初始频谱问题对应于弗里德里希斯模型的两通道变量的情况下，证明了算符H的本征函数系统的基本性质。在算子A具有连续光谱的情况下，针对H提出了散射理论。
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