Higher-Dimensional Complex Dynamics and Spectral Zeta Functions of Fractal Differential Sturm-Liouville Operators

Details Higher-Dimensional Complex Dynamics and Spectral Zeta Functions of Fractal Differential Sturm-Liouville Operators Higher-Dimensional Complex Dynamics and Spectral Zeta Functions of Fractal Differential Sturm-Liouville Operators

2012-02-19

arxiv.org/abs/1202.4126v1

Physics

Mathematical Physics

18 pages, no figures

Scientific paper

We investigate the spectral zeta function of a self-similar Sturm-Liouville operator associated with a fractal self-similar measure on the half-line and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane. The result generalizes to several complex variables and to the case of fractal Sturm-Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states.

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