We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in
bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We …

This is a comprehensive review of the nodal domains and lines of quantum billiards,
emphasizing a quantitative comparison of theoretical findings to experiments. The nodal …

Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study
of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Key …

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 …

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem
is to describe the relationship between the shape (geo metry) of the drum and its sound (its …

A quantum wave with probability density, confined by Dirichlet boundary conditions in a D-
dimensional box of arbitrary shape and finite surface area, evolves from the uniform state …

The present research monograph is a testimony to the fact that Fractal Analysis is deeply
connected to numerous areas of contemporary Mathematics. Here, we have in mind, in …

Motivated in part by the first author's work [23] on the Weyl‐Berry conjecture for the
vibrations of 'fractal drums'(that is,'drums with fractal boundary'), ML Lapidus and C …

The theory of fractal strings and their complex dimensions investigates the geometric,
spectral and physical properties of fractals and precisely describes the oscillations in the …

Let $ K $ be a self-similar set in $\mathbb R^ d $, of Hausdorff dimension $ D $, and denote
by $| K (\epsilon)| $ the $ d $-dimensional Lebesgue measure of its $\epsilon …