  We solve exactly the spectral problem for the non-Hermitian operator $H_U
f(x)\equiv f(U-1/x)/x^2$. Despite the absence of orthogonality, the eigen
functions of this operator allow one to construct in a simple way the expansion
of an arbitrary function in series. Explicit formulas for the expansion
coefficients are presented. This problem is shown to be connected with that of
calculating 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 is
derived. All results are confirmed by direct computer simulations.
