The Journal of Analysis

, Volume 26, Issue 1, pp 71–102 | Cite as

Shifted lattices and asymptotically optimal ellipses

Original Research Paper


Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions, and consider a decreasing concave (or convex) curve in the first quadrant. Construct a family of curves by rescaling in the coordinate directions while preserving area, and identify the curve in the family that encloses the greatest number of the shifted lattice points. We find the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape depends explicitly on the lattice shift, except that when the shift is too negative, the maximizing curve fails to converge and instead degenerates. Our results handle the p-circle \(x^p+y^p=1\) when \(p>1\) (concave) and also when \(0<p<1\) (convex). The circular case \(p=2\) with shift \(-1/2\) corresponds to minimizing high eigenvalues in a symmetry class for the Laplacian on rectangles, while the straight line case (\(p=1\)) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.


Translated lattice Concave curve Convex curve p-ellipse Spectral optimization Dirichlet Laplacian Schrödinger eigenvalues Harmonic oscillator 

Mathematics Subject Classification

Primary 35P15 Secondary 11P21 52C05 



The material in this paper forms part of Shiya Liu’s Ph.D. dissertation at the University of Illinois, Urbana–Champaign [20].


This study was funded by the Simons Foundation (Grant #429422 to Richard Laugesen) and the University of Illinois Research Board (award RB17002).

Compliance with ethical standards

Human and animal rights

This research did not involve human participants or animals.

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Forum D'Analystes, Chennai 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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