Shifted lattices and asymptotically optimal ellipses
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.
KeywordsTranslated lattice Concave curve Convex curve p-ellipse Spectral optimization Dirichlet Laplacian Schrödinger eigenvalues Harmonic oscillator
Mathematics Subject ClassificationPrimary 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 .
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.
- 3.Antunes, P.R.S., and P. Freitas. 2013. Optimal spectral rectangles and lattice ellipses. In Proceedings of the Royal Society of London Series A Mathematical, Physical and Engineering Sciences, No. 2150, 20120492Google Scholar
- 6.Ariturk, S., and R. S. Laugesen. Optimal stretching for lattice points under convex curves. Portuguese Mathematical. arXiv:1701.03217 (to appear).
- 12.Gittins, K. and S. Larson. Asymptotic behaviour of cuboids optimising Laplacian eigenvalues. Integral Equations Operator Theory. arXiv:1703.10249 (to appear).
- 14.Huxley, M.N. 1996. Area, Lattice Points, and Exponential Sums. London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications.Google Scholar
- 15.Huxley, M.N. 2003. Exponential sums and lattice points. III. In: Proceedings of the London Mathematical Society (3) No. 87, 591–609.Google Scholar
- 16.Krätzel, E. 2000. Analytische Funktionen in der Zahlentheorie. Teubner–Texte zur Mathematik, vol. 139. Stuttgart: B. G. Teubner.Google Scholar
- 18.Larson, S. Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. arXiv:1611.05680.
- 19.Laugesen, R.S., and S. Liu. Optimal stretching for lattice points and eigenvalues. Ark. Mat., http://www.math.illinois.edu/~laugesen/ (to appear).
- 20.Liu, S. 2017. Asymptotically optimal shapes for counting lattice points and eigenvalues. Ph.D. dissertation, University of Illinois, Urbana–Champaign. http://hdl.handle.net/2142/98364.
- 21.Marshall, N.F. Stretching convex domains to capture many lattice points. arXiv:1707.00682.
- 22.Marshall, N.F., and S. Steinerberger. Triangles capturing many lattice points. arXiv:1706.04170.