arXiv subjects

Gustav Mårdby

Publications and source records attributed to Gustav Mårdby.

4 recordsLinked to original sources

arXiv2026

Beyond Three Terms: Exponential Bounds in the Neumann Heat Trace of Polygons

We study the short-time asymptotic behavior of the heat trace associated with the Neumann Laplacian on polygonal domains in the plane. By establishing locality principles for the heat kernel near corners, edges, and the interior, we approximate the heat kernel on the polygon by model heat kernels defined on infinite sectors, half-planes, and the full plane, respectively. Although it is known that the Neumann heat trace of polygons admits a three-term asymptotic expansion followed by an exponentially small remainder, an explicit estimate for the exponent in this remainder term is not known. In this article, we provide such an estimate. We also discuss whether the exponent is sharp, and how it relates to known results. Finally, we discuss issues that arise when trying to extend the results to Robin boundary conditions.

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arXiv2024

Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori

In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example.

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arXiv2024

Spectral invariants of integrable polygons

An integrable polygon is one whose interior angles are fractions of $π$; that is to say of the form $\frac πn$ for positive integers $n$. We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation.

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arXiv2024

112 years of listening to Riemannian manifolds

In 1910, Hendrik Antoon Lorentz delved into the enigmatic Laplace eigenvalue equation, also known as the Helmholtz equation, pondering to what extent the geometry in which one solves the equation can be recovered from knowledge of the eigenvalues. Lorentz, inspired by physical and musical analogies, conjectured a fundamental relationship between eigenvalues, domain volume, and dimensionality. While his conjecture initially seemed insurmountable, Hermann Weyl's groundbreaking proof in 1912 illuminated the deep connection between eigenvalues and geometric properties. Over the ensuing 112 years, mathematicians and physicists have continued to decipher the intricate interplay between eigenvalues and geometry. From Weyl's law to Milnor's example of isospectral non-isometric flat tori, and Kac's inspiring question about hearing the shape of a drum, the field has witnessed remarkable progress, uncovering spectral invariants and advancing our understanding of geometric properties discernible through eigenvalues. We present an overview of this field amenable to both physicists and mathematicians.

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