Abstract
We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues \(\lambda (n)\) of shapes with n edges that are of the form \(\lambda (n) \sim x\sum _{k=0}^{\infty } \frac{C_k(x)}{n^k}\) where x is the limiting eigenvalue for \(n\rightarrow \infty \). Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order \(C_k(x)\) and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).
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Acknowledgements
D.B. would like to thank Oliver Freyermuth for his help in carrying out the computations on a computing cluster, Fredrik Johansson for assistance with Arb, and Plamen Koev for discussions on possible improvements of the approach of Sect. 6. D.R. would like to thank Erik Panzer for his help with finding a good basis for single-valued MZVs in weight 16. D.R. would also like to thank Steven Charlton for his great help with calculations involving alternating MZVs. The authors would also like to thank the anonymous referees for their valuable comments. D.B. acknowledges financial support from the Bonn-Cologne Graduate School of Physics and Astronomy honors branch.
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Appendix 1. Brief introduction to multiple zeta values
Appendix 1. Brief introduction to multiple zeta values
The Riemann zeta function
is known to converge for all \(s\in \mathbb {C}\) that satisfy \(\text {Re}(s)>1\). The Riemann zeta function is arguably the most famous function in all of mathematics due to its tight connection to the distribution of prime numbers. For our treatment, we restrict ourselves to the special values of \(\zeta (s)\) at natural numbers \(s \in \mathbb {N}\), \(s>1\), which are sometimes simply called zeta values. For even zeta values, Euler has famously shown that
where \(B_n\) are Bernoulli numbers. For odd zeta values much less is known. It has been proven by Apéry in 1979 that \(\zeta (3)\) is irrational. His proof has not yet been extended to any other odd zeta values. It is known that at least one of \(\zeta (5),\zeta (7),\zeta (9),\zeta (11)\) has to be irrational and that there exists an infinite number of irrational odd zeta values. However, it is still unknown whether all odd zeta values are irrational and whether \(\pi , \zeta (3), \zeta (5), \dots , \zeta (2n+1),\dots \) have any algebraic relations among them (altough it is widely believed that there are none). As we have seen in Sect. 1, the arguments of products of zeta values that appear in the expansion coefficients of polygon eigenvalues add up to the index of the corresponding asymptotic coefficient. For example, the tenth coefficient for the Dirichlet case contains the products \(\zeta (7)\zeta (3)\) and \(\zeta (5)^2\), and we have \(7+3=2\cdot 5 = 10\). This number is called the weight of a zeta product. Note that the product of two zeta values can be written as
We can split the last sum into three terms
One of the terms is given as a zeta value
The remaining “crossing” terms are the so-called multiple zeta values \(\zeta (a,b)\) and \(\zeta (b,a)\). More generally, the multiple zeta values (MZVs) are defined as
The decomposition
is sometimes called the Nielsen reflection formula. Multiple zeta values satisfy a wide variety of interesting relations. For instance, the \(\mathbb {Q}\)-linear span of MZVs inside \(\mathbb {R}\) is closed under products, i.e., forms an algebra. An important subalgebra of MZVs is given by single-valued MZVs that were defined by Brown in [8]. These have been found to appear in amplitudes of Feynman diagrams of string theory. The relation between single-valued MZVs and odd zeta values is given by
We will not cover the theory of single-valued MZVs in this brief introduction but instead, only give an expression for the single-valued MZVs that appear in our expansion formulas
Some of the expressions, we give also involve alternating multiple zeta values which are special values of multiple polylogarithms
when all \(z_i\) are set to \(\pm 1\).
1.1 Remaining coefficients for regular polygons with Dirichlet boundary condition
1.2 Remaining coefficients for smooth star shapes with m = 2
1.3 Coefficients for smooth star shapes with m = 3
1.4 Coefficients for smooth star shapes with m = 4
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Berghaus, D., Jones, R.S., Monien, H. et al. Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry. Adv Comput Math 50, 38 (2024). https://doi.org/10.1007/s10444-024-10138-3
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DOI: https://doi.org/10.1007/s10444-024-10138-3