Spectral Analysis of Fractal Operators

Spectral analysis of fractal operators examines the eigenvalue spectra and associated functions of differential operators defined on sets with self-similar and often non-integer dimensional structure. Central to this field is the fractal Laplacian, a generalisation of the classical Laplace operator constructed via Dirichlet forms adapted to fractal measures. Analysis of its spectrum reveals non-trivial scaling laws, oscillatory behaviour in the eigenvalue counting function and the emergence of complex dimensions in spectral zeta functions. Heat kernel estimates on fractals further illuminate diffusion processes and their sub-Gaussian or anomalous regimes. This body of work has broad implications for models of porous media, wave propagation in irregular substrates, quantum graphs and network dynamics, where underlying geometries depart from Euclidean regularity.

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Spectral Analysis of Fractal Operators publication trend

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Technical terms

Fractal Laplacian: A generalised Laplace operator defined on fractal sets via self-similar Dirichlet forms, capturing diffusion and wave propagation on irregular geometries.

Spectral Dimension: An exponent governing the scaling of the eigenvalue counting function with eigenvalue magnitude, reflecting effective dimensionality of fractal domains.

Spectral Zeta Function: A meromorphic function obtained by summing inverse powers of eigenvalues, used to analyse complex dimensions and oscillatory spectral features.

Heat Kernel: The fundamental solution of the heat equation associated with a fractal Laplacian, encoding time-dependent diffusion behaviour and decay estimates.

References

1. Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. Discrete and Continuous Dynamical Systems (2018).
2. Harmonic calculus on fractals—A measure geometric approach II. Transactions of the American Mathematical Society (2005).
3. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure and Applied Analysis (2020).
4. Spectral zeta functions of fractals and the complex dynamics of polynomials. Transactions of the American Mathematical Society (2007).
5. Sierpiński Fractals and the Dimension of Their Laplacian Spectrum †. Mathematical and Computational Applications (2023).
6. Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians. Journal of Nonlinear Mathematical Physics (2023).