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 $x_{n+1}=1/(U-x_n)$

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.

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 $x_{n + 1} = 1 / (U - x_{n})$

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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 $x_{n + 1} = 1 / (U - x_{n})$

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Spectrum and eigen functions of the operator HUf(x)≡f(U−1/x)/x2HU​f(x)≡f(U−1/x)/x2 and strange attractor's density for the mapping xn+1=1/(U−xn)xn+1​=1/(U−xn​)

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Spectrum and eigen functions of the operator HUf(x)≡f(U−1/x)/x2HU​f(x)≡f(U−1/x)/x2 and strange attractor's density for the mapping xn+1=1/(U−xn)xn+1​=1/(U−xn​)

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|讲解博客

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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 $x_{n + 1} = 1 / (U - x_{n})$

讲解博客

AI智能校验

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 $x_{n + 1} = 1 / (U - x_{n})$

讲解博客

AI智能校验