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We solve exactly the spectral problem for the non-Hermitian operator $H_Uf(x)\equiv f(U-1/x)/x^2$. Despite the absence of orthogonality, the eigenfunctions of this operator allow one to construct in a simple way the expansionof an arbitrary function in series. Explicit formulas for the expansioncoefficients are presented. This problem is shown to be connected with that ofcalculating the strange attractor's density for the map $x_{n+1}=1/(U-x_n)$.The explicit formula for the strange attractor's density for this map isderived. All results are confirmed by direct computer simulations.

Spectrum and eigen functions of the operator $H_U f(x)\equiv f(U-1/x)/x^2$ and strange attractor's density for the mapping $x_{n+1}=1/(U-x_n)$