Asymptotically optimal shapes for counting lattice points and eigenvalues

Lattice points;Planar convex domain;P-ellipse;Spectral optimization;Laplacian;Dirichlet eigenvalues;Neumann eigenvalues;Translated lattice;Schrödinger eigenvalues;Harmonic oscillator

Liu, Shiya

关键词: Lattice points;    Planar convex domain;    P-ellipse;    Spectral optimization;    Laplacian;    Dirichlet eigenvalues;    Neumann eigenvalues;    Translated lattice;    Schrödinger eigenvalues;    Harmonic oscillator;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/98364/LIU-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result implies when1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity. 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: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity? In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number ofshifted lattice points in the same family of reciprocal stretching curves as in Part I.The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates.Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.