Laplace Operators on Fractals and Related Functional Equations

Abstract

We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpiński gasket.

References in corpus (4)

• Uses of zeta regularization in QFT with boundary conditions: a cosmo-topological Casimir effect
• Thermodynamics of photons on fractals
• Heat Kernels and Zeta Functions on Fractals
• Spatial Log Periodic Oscillations of First-Passage Observables in Fractals

Cited by in corpus (16)

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• Flow of Fractal Fluid in Pipes: Non-Integer Dimensional Space Approach
• Dirac and magnetic Schrödinger operators on fractals
• Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space
• Energy and Laplacian on Hanoi-type fractal quantum graphs
• Singularly continuous spectrum of a self-similar Laplacian on the half-line
• Toward the mechanics of fractal materials: mechanics of continuum with fractal metric
• Regularized Laplacian determinants of self-similar fractals
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• Quantum Vacuum Energy of Self-Similar Configurations
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets
• Lower bound of the asymptotic complexity of self-similar fractal graphs
• Minimal gap in the spectrum of the Sierpinski gasket
• The fractal geometry and the mapping of Efimov states to Bloch states
• Hierarchical Schrödinger-type operators: the case of potentials with local singularities