Counterexamples to the modified Weyl–Berry conjecture on fractal drums

DOI10.1017/S0305004100074053zbMath0858.58052WikidataQ56455401 ScholiaQ56455401MaRDI QIDQ4877652

Michel L. Lapidus, Carl B. Pomerance

Publication date: 2 April 1997

Published in: Mathematical Proceedings of the Cambridge Philosophical Society (Search for Journal in Brave)

zbMATH Keywords

counterexamplemodified Weyl-Berry conjecture

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Asymptotic distributions of eigenvalues in context of PDEs (35P20) Fractals (28A80) Perturbations of PDEs on manifolds; asymptotics (58J37)

Related Items (29)

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets ⋮ On spectral asymptotics for domains with fractal boundaries ⋮ Fractal strings and multifractal zeta functions ⋮ Duality questions for operators, spectrum and measures ⋮ Beurling zeta functions, generalised primes, and fractal membranes ⋮ Complex dimensions of fractals and meromorphic extensions of fractal zeta functions ⋮ Tube formulas for self-similar and graph-directed fractals ⋮ Fractal zeta functions of orbits of parabolic diffeomorphisms ⋮ Computing eigenvalues of the Laplacian on rough domains ⋮ Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis ⋮ Generalisation of the modified Weyl-Berry conjecture for drums with jagged boundaries ⋮ Central limit theorems for the spectra of classes of random fractals ⋮ Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics ⋮ Truncated infinitesimal shifts, spectral operators and quantized universality of the Riemann zeta function ⋮ Measure-geometric Laplacians for partially atomic measures ⋮ Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces ⋮ Distance and tube zeta functions of fractals and arbitrary compact sets ⋮ On the existence of optimal shapes in architecture ⋮ Semiclassical bounds for spectra of biharmonic operators ⋮ Fractal zeta functions and complex dimensions of relative fractal drums ⋮ Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators ⋮ Tube formulas and complex dimensions of self-similar tilings ⋮ Lacunarity of self-similar and stochastically self-similar sets ⋮ The Sound of Fractal Strings and the Riemann Hypothesis ⋮ The Shape of Anisotropic Fractals: Scaling of Minkowski Functionals ⋮ Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory ⋮ Regularly varying functions, generalized contents, and the spectrum of fractal strings ⋮ An overview of complex fractal dimensions: from fractal strings to fractal drums, and back ⋮ Eigenvalues of the Laplacian on domains with fractal boundary

Cites Work

• Unnamed Item
• Can one hear the dimension of a fractal?
• Potential and scattering theory on wildly perturbed domains
• One-dimensional cohomologies of discrete subgroups
• The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums
• Fractal Drum, Inverse Spectral Problems for Elliptic Operators and a Partial Resolution of the Weyl-Berry Conjecture
• An Example of a Two-Term Asymptotics for the "Counting Function" of a Fractal Drum

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