Related papers: Eigenmodes of fractal drums: A numerical student e…
In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was…
In a family of drums used in the Indian subcontinent, the circular drum head is made of material of non-uniform density. Remarkably, and in contrast to a circular membrane of uniform density, the low eigenmodes of the non-uniform membrane…
The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several…
The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the $q$-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative…
The geometry of a physical system is intimately related to its spectral properties, a concept colloquially referred to as "hearing the shape of a drum". Three-dimensional topological insulator nanowires in a strong magnetic field $B$…
Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with…
Codes were written to simulate the propagation of monochromatic light through a bare optical resonator, using a computational Fourier method to solve the Huygens-Fresnel integral. This was used, in the Fox-Li method, to find the lowest-loss…
We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal…
We generalize Weyl's law to inhomogeneous bodies in $d$ dimensions. Using a perturbation scheme recently obtained by us in Ref. \cite{Amore09}, we have derived an explicit formula, which describes the asymptotic behavior of the eigenvalues…
While nuclear magnetic resonance diffusion experiments are widely used to resolve structures confining the diffusion process, it has been elusive whether they can exactly reveal these structures. This question is closely related to X-ray…
Bending-mode vibrations of carbon nanotube resonator devices were mechanically detected in air at atmospheric pressure by means of a novel scanning force microscopy method. The fundamental and higher order bending eigenmodes were imaged at…
Eigenstate multifractality is of significant interest with potential applications in various fields of quantum physics. Most of the previous studies concentrated on fine-tuned quantum models to realize multifractality which is generally…
The quest of the knowledge of the structure and dynamics of the solar interior has been possible thanks to the study of the resonant acoustic (p) modes that are trapped in the solar interior. Since the solar rotation lifts the azimuthal…
When immersed in a see of cold electrons, local impurities give rise to density modulations known as Friedel oscillations. In spite of the generality of this phenomenon, the exact shape of these modulations is usually computed only for…
We numerically investigate classical and quantum transport through a soft-wall cavity with mixed dynamics. Remarkable differences to hard-wall quantum dots are found which are, in part, related to the influence of the hierarchical structure…
Fractal dimensions have been used as a quantitative measure for structure of eigenstates of quantum many-body systems, useful for comparison to random matrix theory predictions or to distinguish many-body localized systems from chaotic…
The lattice Schwinger model (SM), the discrete version of QED in 1+1 dimensions, is a well-studied test bench for lattice gauge theories. Here we study the fractal properties of the SM. We reveal the self-similarity of the ground state,…
A highly unusual pattern of strong multiple resonances for bulk electromagnetic waves is predicted and analysed numerically in thick periodic holographic gratings in a slab with the mean permittivity that is larger than that of the…
The ubiquity of parametric resonance is continually evident in the repeated experimental observations of this phenomenon in multiple physical systems. The elementary case of 2 mode parametric resonance of order 1 involves the excitation of…
We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous model, where all characteristics and fields are defined everywhere in the volume but they follow some…