1202.4126 (Nishu Lal et al.)

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

Nishu Lal, Michel L. Lapidus

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.

View original: http://arxiv.org/abs/1202.4126