Optimal stretching for lattice points and eigenvalues

Richard S. Laugesen University of Illinois Shiya Liu University of Illinois

Spectral Theory and Operator Algebra mathscidoc:1912.43008

Arkiv for Matematik, 56, (1), 111-145, 2018
We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the “radius” approaches infinity. In particular, the result implies that among all p-ellipses (or Lamé curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity, for 1<p<∞. The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0<p<1 by building on our results here. The case p=1 remains open, and is closely related to minimizing energy levels of harmonic oscillators: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity?
lattice points, planar convex domain, p-ellipse, Lamé curve, spectral optimization, Laplacian, Dirichlet eigenvalues, Neumann eigenvalues
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  title={Optimal stretching for lattice points and eigenvalues},
  author={Richard S. Laugesen, and Shiya Liu},
  booktitle={Arkiv for Matematik},
Richard S. Laugesen, and Shiya Liu. Optimal stretching for lattice points and eigenvalues. 2018. Vol. 56. In Arkiv for Matematik. pp.111-145. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204094207996274564.
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