# The One-Dimensional Modified Weyl-Berry Conjecture: An Elementary Approach

• Roland J. Etienne
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 276)

## Abstract

Let $$\varOmega$$ be a bounded domain in $$\mathbb {R}^n$$ with boundary $$\delta \varOmega$$ and consider the eigenvalue problem
$$-\varDelta u=\lambda u,$$
with Dirichlet boundary conditions, i.e. $$u\left| _{\delta \varOmega }\right. =0$$. Its set of eigenvalues, $$0<\lambda _1\le \lambda _2\le \cdots \le \lambda _k\le \cdots$$—each eigenvalue being repeated according to (algebraic) multiplicity—is countable and the eigenvalue counting function may be defined as
$$N(\lambda ):=\#\{(0<)\lambda _k<\lambda \},$$
for a given positive $$\lambda$$. The modified Weyl-Berry conjecture for the asymptotics of the eigenvalues of the Laplacian on bounded open subsets of the line (fractal strings) then states that
$$N(\lambda )=\pi ^{-1}\left| \varOmega \right| _1\lambda ^{\frac{1}{2}}+\mathscr {O}(\lambda ^{\frac{d}{2}}),$$
with $$\left| \varOmega \right| _1$$ being the one-dimensional Lebesgue measure of $$\varOmega$$ and $$d\in [0,1]$$ the Minkowski dimension of the boundary. Based upon a matrix representation of the Laplacian, it will be shown how to obtain some of the key results on the one-dimensional modified Weyl-Berry conjecture through elementary methods.

## Keywords

Modified Weyl–Berry conjecture Dirichlet boundary value problem Eigenvalue problems

## References

1. 1.
Betten, J.: Finite Elemente für Ingenieure 1: Grundlagen, Matrixmethoden, Elastisches Kontinuum. Springer, Heidelberg (1997)
2. 2.
Etienne, R.J.: Some numerical results for the spectrum of monoatomic cantor chains. In: 2nd Conference on Analysis and Probability on Fractals. Cornell University, Ithaca, NY (2005)Google Scholar
3. 3.
Etienne, R.J.: On the asymptotic distribution of the Dirichlet eigenvalues of Fractal Chains. Ph.D. thesis, University of Siegen, Siegen (2014)Google Scholar
4. 4.
Falconer, K.J.: Techniques in Fractal Geometry. Wiley, New York (1997)
5. 5.
Hellwege, K.H.: Einführung in die Festkörperphysik I, Heidelberger Taschenbücher, vol. 33. Springer, Heidelberg (1968)
6. 6.
Kittel, C.: Introduction to Solid State Physics. Wiley, New York (1996)Google Scholar
7. 7.
Lapidus, M.L., Maier, H.: The Riemann hypothesis and inverse spectral problems for fractal strings. J. Lond. Math. Soc. 52(2), 15–34 (1995)
8. 8.
Lapidus, M.L., Pomerance, C.: The Riemann Zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. Lond. Math. Soc. 66(3), 41–69 (1993)
9. 9.
Lapidus, M.L., van Frankenhuysen, M.: Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions. Birkhäuser, Berlin (2000)